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+[    {        "title": "1204.4632v1.Magnetic_order_and_spin_dynamics_in_the_proximity_of_a_ferromagnetic_quantum_critical_point__A_μSR_study_of_YbNi4P2.pdf",        "content": "arXiv:1204.4632v1  [cond-mat.str-el]  20 Apr 2012Magnetic Order and Spin Dynamics in the Proximity of a Ferrom agnetic Quantum\nCritical Point: a µSR study of YbNi 4P2\nJ. Spehling,1M. G¨ unther,1C. Krellner,2,3N. Y`eche,1H. Luetkens,4C. Baines,4C. Geibel,2and H.-H. Klauss1,∗\n1Institute for Solid State Physics, TU Dresden, D-01069 Dres den, Germany\n2Max Planck Institute for Chemical Physics of Solids, D-0118 7 Dresden, Germany\n3Cavendish Laboratory, University of Cambridge,\nJ J Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n4Laboratory for Muon-Spin Spectroscopy, Paul-Scherrer-In stitute, CH-5232 Villigen, Switzerland\n(Dated: November 26, 2018)\nThe local 4 f-electronic spin dynamics and magnetic order in YbNi 4P2were studied by means of\nmuon-spin relaxation measurements. Zero-ﬁeld muon-spin r elaxation proves static magnetic order\nwith a strongly reduced ordered Yb3+moment of (2 .5−4.6)×10−2µB, below TC= 140 mK.\nAboveTC, the muon spin polarization P(t,B) is dominated by quasihomogeneous spin ﬂuctuations\nand exhibits a time-ﬁeld scaling relation P(t,B) =P(t/Bγ), indicating cooperative critical spin\ndynamics in the system. At T= 190 mK, slightly above TC,γ= 0.81(5) suggesting time-scale\ninvariant power-law behavior for the dynamic electronic sp in-spin autocorrelation function.\nPACS numbers: 71.27.+a, 75.30.-m, 76.75.+i, 75.50.Cc\nLanthanide-based heavy-fermion (HF) systems are\nsuitable model systems to study emergent phenomena at\na quantum critical point (QCP), where collective quan-\ntum ﬂuctuations trigger the system continuously from a\nmagnetically ordered to a non-magnetic ground state1–6.\nHowever, despite intense research, to the best of our\nknowledge, no 4 f-based material is known with a contin-\nuous ferromagnetic (FM) to paramagnetic (PM) quan-\ntum phase transition (QPT). The existence of such a\nQPT is also controversially discussed from a theoretical\npoint of view7–11.\nRecently, Krellner et al.suggested that the HF metal\nYbNi4P2with a quasi-one-dimensional (1-D) electronic\nstructure exhibits FM quantum criticality above a low\nFM transition temperature TC= 170 mK12. YbNi 4P2\ncrystallizes in the tetragonal ZrFe 4Si2structure contain-\ning isolated chains of edge-connected Ni tetrahedra along\nthec−axis. The Yb atoms are located in the channels\nbetweenthese Nitetrahedralchains. Thereduceddimen-\nsionality in the Yb and Ni network and the geometrical\nfrustrationbetweenneighboringYbchainsgiverisetoen-\nhanced quantum spin ﬂuctuations of the magnetic Yb3+\nions. In the PM state above 50 K, the magnetic sus-\nceptibility shows Curie-Weiss behavior with an eﬀective\nmoment µeff= 4.52µBthat is characteristic for mag-\nnetic Yb3+ions. Analysis of the magnetic entropy re-\nveals a Kondo energy scale of TK≈8 K for the crys-\ntal electric ﬁeld ground state doublet. The FM transi-\ntion is evidenced by distinct anomalies in magnetic sus-\nceptibility, speciﬁc heat, and resistivity measurements.\nLow-Tmagnetization measurements suggest an ordered\nFM moment of mord≈0.05(4)µB. Pronounced non-\nFermi-liquid (NFL) behavior is reﬂected by a stronger-\nthan-logarithmic diverging Sommerfeld coeﬃcient and a\nlinear-in- Tresistivity state apparent in a Trange larger\nthan a decade above TC. In external magnetic ﬁelds,\nthe NFL behavior is suppressed and FL behavior grad-\nually recovers. Therefore, YbNi 4P2is considered as aclean system situated in the very close vicinity of a FM\nQCP, with FM quantum ﬂuctuations dominating ther-\nmodynamic and transport quantities at T > T C.\nThe present knowledge on YbNi 4P2is based on mea-\nsurementsofmacroscopicmagnetic,thermodynamic,and\ntransport properties. The next step in a deeper investi-\ngation of this prospective FM quantum critical system\nis to get insight on a microscopic level. Beside the na-\nture of the magnetic order, a central issue in the present\ncontext of critical behavior is the spin dynamics. Since\nin systems close to a QCP, the ordered moment is usu-\nally strongly reduced, muon spin relaxation ( µSR) has\nproven to be an extremely valuable technique to collect\nappropriate information13–15.\nHere, we present µSR experiments on polycrystalline\nYbNi4P2, providing microscopic evidence for static mag-\nnetism at T≤TC≈140 mK with an ordered mo-\nment of mord= (2.5−4.6)×10−2µB/Yb, depending\non the assumed muon site. Above TC, the muon-spin\npolarization P(t) obeys the time-ﬁeld scaling relation\nP(t) =P(t/B0.81(5)), indicating cooperative and critical\nspin dynamics.\nIn aµSR experiment positive spin-polarizedmuons are\nimplanted into the sample, and the subsequent time evo-\nlution of the muon spin polarization is monitored by de-\ntecting the asymmetric spatial distribution of positrons\nemitted from the muon decay16.µSR in longitudinal ap-\nplied magnetic ﬁelds is dominated by Yb-4 felectronic\nspin ﬂuctuations that couple to the implanted muons.\nTheµSR experiments on YbNi 4P2in zero ﬁeld (ZF)\nand longitudinal (LF) applied ﬁeld – with respect to the\ninitial muon spin polarization – were performed on the\nπM3 beam line at the Swiss Muon Source (S µS) at the\nPaul-Scherrer-Institut,Switzerland. Thesamplewaspre-\npared by crushing ∼270 mg of single crystalline mate-\nrial, grown in a self-ﬂux at 1400◦C in a closed Tantal\ncrucibleandcharacterizedbypowderx-raydiﬀractionex-\nperiments,provingtheabsenceofanyforeignphases. De-2\n/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s49\n/s48/s46/s49 /s49/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s46/s49/s49\n/s84\n/s67/s40/s100/s41\n/s32/s40/s98/s41/s40/s99/s41/s32/s40 /s115/s45/s49 \n/s41\n/s84 /s32/s40/s75/s41/s102 /s32/s40/s77/s72/s122/s41/s84/s105/s109/s101/s32 /s116/s32/s40/s181/s115/s41/s40/s97/s41\n/s32\n/s32/s50/s53/s32/s109/s75\n/s49/s50/s53/s32/s109/s75\n/s49/s54/s48/s32/s109/s75\n/s56/s48/s48/s32/s109/s75/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32 /s80 /s40/s116/s41\n/s84\n/s67\n/s32/s32/s32/s32/s49/s47 /s84\n/s49 /s84 /s32/s40 /s115/s45/s49 \n/s75/s45/s49 \n/s41/s32\n/s84 /s32/s40/s75/s41/s41 /s49 /s40 /s53 /s46 /s49 \n/s84\nFIG. 1: (Color online) (a) Corrected muon spin polariza-\ntionP(t) at ZF for representative Tabove and below TC≈\n140 mK. At T≤TC, solid lines are ﬁtting curves accord-\ning to Eq. (1). (b) Tdependence of the ZF µSR frequency\nfµ(T). Thesolid line is aﬁttothephenomenological function:\nfµ=fµ(0)·(1−T\nTC)n. (c)Tdependence of the ZF static in-\nternal ﬁeld distribution σin Eq. (1). The solid line is a guide\nto the eye. (d) T-dependence of 1 /T1Tin the PM regime.\nThe line describes power-law behavior as1\nT1T∝T−1.5.\ntailed low- Tmeasurements on polycrystalline YbNi 4P2\nwere reported elsewhere12.\nFigure 1(a) displays typical time dependencies of the\nZF muon-spin polarization P(t) in YbNi 4P2at repre-\nsentative temperatures above and below TC. A ﬁnite\nT-independent background signal due to muons that\nstopped in a Ag sample holder (signal fraction ≈50%)\nwas taken into account. At T≥160 mK, an exponen-\ntial muon-spin relaxation is associated with fast ﬂuctu-\nating paramagnetic electron spins with a relaxation rate\nλ(160 mK) ≈0.152(2)µs−1. Note, that dense static nu-\nclear dipole moments would give rise to a weak Gaussian\nrelaxation in the PM regime. While cooling through TC,\nan additional magnetic relaxation mechanism is appar-\nent, strongly increasing with lowering T. Below TC, a\nlow-frequency oscillation with a Gaussian relaxation of\nthe muon-spin polarization is observed indicating mag-\nnetic ordering of weak electronic Yb3+moments. The\nmuon-spin asymmetry data in the FM regime can be de-\nscribed best using the functional form17,18:\nP(t) =1\n3+2\n3[cos(2πfµt)−σ2t\n2πfµ·sin(2πfµt)]·e−1\n2σ2t2,(1)\nwherefµandσare the muon spin precession fre-\nquency and the Gaussian ﬁeld width, respectively. The/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s49\n/s49/s51/s32/s71/s50/s51/s32/s71\n/s90/s70 /s49/s52/s51/s32/s71\n/s53/s51/s32/s71/s32\n/s32/s32/s32/s32/s32/s90/s70\n/s32/s32/s32/s32/s50/s32/s71\n/s32/s32/s49/s51/s32/s71\n/s32/s32/s50/s51/s32/s71\n/s32/s32/s53/s51/s32/s71\n/s49/s52/s51/s32/s71\n/s50/s57/s51/s32/s71\n/s32/s80 /s32/s40/s116/s41\n/s116/s32/s40 /s115/s41\nFIG. 2: (Color online) Corrected muon spin polarization at\nT= 20 mK and various longitudinal magnetic ﬁelds BLF.\nThe lines represent theoretical depolarization curves for the\nstatic GKT function in corresponding longitudinal ﬁelds.\n2/3 oscillating and the 1/3 non-oscillating terms origi-\nnate from the spatial averaging in polycrystalline sam-\nples, where 2/3 (1/3) of the internal magnetic ﬁeld com-\nponents are directed perpendicular (parallel) to the ini-\ntial muon spin, causing a precession (no precession) of\nthe muon spin. The observation of a 2/3 and 1/3 sig-\nnal fraction below TC, implies dense magnetic moments\nand proof that 100% of the sample volume shows static\nmagnetic order. The latter is supported by LF- µSR mea-\nsurements as discussed in detail below. In the limit\n2πfµ≫σ, Eq. (1) becomes a Gaussian damped cosine\nfunction. For 2 πfµ→0, close to the magnetic transi-\ntion, Eq. (1) is equivalent to the Gaussian Kubo-Toyabe\n(GKT) function19, which describes a muon-spin relax-\nation due to a static Gaussian ﬁeld distribution centered\naroundBlocal= 0. ZF- µSR on the antiferromagnetically\nordered system YbRh 2Si214reveals a similar crossover\nfrom a Lorentzian to a Gaussian damped µSR signal\nin the vicinity of the PM to magnetic phase transition,\nattributed to a transition from dynamic to static mag-\nnetism of magnetic Yb3+moments.\nFor YbNi 4P2a ﬁniteµSR frequency is clearly observed\nbelow 150 mK. From the measured frequency value fµ=\n0.188(1) MHz at 20 mK one can determine the internal\nlocal ﬁeld at the muon site to Blocal= 13.87 G using\nBlocal= 2πfµ/γµwithγµ= 2π×13.55 kHz/G as the\nmuon gyromagnetic ratio. The local ﬁeld Blocalas well\nas the local static ﬁeld width ∆ Blocal=σ/γµ≈6 G are\nverysmall forconventionalrare-earthmagnets with large\nordered moments. The fractional width ∆ Blocal/Blocal\nof the spontaneous ﬁeld distribution is ∼0.4 at low T\nand remains constant as T→TC, which is a reason-\nable value for a magnetically ordered HF system, as e.g.\nin CeRhIn 5∆Blocal/Blocal= 0.5 is observed20. Thus,\nthe local ﬁeld distribution is nearly uniform and homo-\ngeneous in the FM regime. The spontaneous muon-spin\nprecession and the Gaussian shape of the internal ﬁeld\ndistribution below TCarise from a dense system of weak\nmagnetic moments with small, static magnetic inhomo-\ngeneities. The presence of a ﬁnite Blocal∝negationslash= 0 proves co-\nherent magnetic order.3\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s69/s45/s52/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49\n/s48 /s49 /s50 /s51 /s52 /s53/s49/s49\n/s84 /s32/s61/s32/s49/s57/s48/s32/s109/s75/s76/s32/s61/s32/s49/s47/s84\n/s49 /s32/s40/s181/s115/s45/s49 \n/s41\n/s66\n/s76/s70 /s32/s40/s71/s41/s32\n/s116/s32/s40/s181/s115/s41/s32/s49/s52/s51/s32/s71/s32\n/s32/s32/s55/s51/s32/s71/s32 /s32\n/s32/s32/s51/s51/s32/s71\n/s32/s32/s49/s51/s32/s71\n/s32/s32/s32/s32/s32/s90/s70\n/s32/s80/s32 /s40/s116/s41\nFIG. 3: (Color online) Main panel: Field dependence of the\ndynamic muon spin relaxation rate λL. The solid curve rep-\nresents a Redﬁeld ﬁt. For display reasons, the ZF value is set\natBLF= 0.01 G. Inset: Field dependence of the corrected\nmuon spin polarization P(t) atT= 190 mK.\nZF-µSR allows a precise determination of the Tde-\npendence of the magnetic order parameter, which is pro-\nportional to the measured µSR frequency fµ. TheT\ndependence of fµandσis shown in Figs. 1(b) and 1(c).\nForT≤140 mK, both observables exhibit a continu-\nous increase. The Tdependence of fµcan be ﬁt to the\nphenomenological function fµ=fµ(0)·(1−T\nTC)nfor\nT < T Cwithn= 0.208±0.02,fµ(T) = 0.199(3) MHz,\nandTC= 140(2) mK. The value of the eﬀective critical\nexponent n, describing the critical behavior close to TC,\nis between n= 0.125 and 0.325, which are theoretically\nexpected for two-dimensional (2D) and isotropic three-\ndimensional (3D) Ising magnets, respectively. This is not\nin contradiction with the claim of a quasi-1D system. In\nsuch a system, the weak inter-chain coupling results in\nan evolution from a 1D behavior at high Tto a 2D Ising\nor 3D behavior at low T, which is intimately linked with\n(andisaprerequisitefor)thelong-rangeorderingatﬁnite\nT. The low data point density between 0 .6≤T\nTC≤1,\nhowever, precludes the determination of the precise criti-\ncal exponent. The obtained value for TCagrees well with\nthe value found in speciﬁc heat measurements on these\nsingle crystals21.\nFor all examined T≤TC, the sample signal is an-\nalyzed with a well-deﬁned single fµandσ, signaling\nthat the magnetic order is a bulk eﬀect and that only\none dominant muon stopping site is present. In gen-\neral, for the determination of the muon stopping site(s)\nit is important to deduce the hyperﬁne coupling con-\nstant. One way to ﬁnd potential muon sites is to com-\nparecalculatedandmeasuredquantitiesforthelocalﬁeld\nBlocalat the muon site. The muon preferentially settles\nat tetrahedra or octahedra interstitial crystallographic\nsites. From simple symmetry arguments the most prob-\nable muon stopping sites, using Wyckoﬀ’s notation,\nare 4f(1/4,1/4,0), 8 j(1/4,1/4,1/4), 4 f(1/4,1/4,1/2),\n8i(1/4,1/2,1/2), 4 c(1/2,0,0), 4 c(1/2,0,1/2), 2 b(0,0,1/2),\nand 2a(1/2,1/2,1/2). For a particular FM structure with\nthe magnetic Yb3+moments aligned within the a bplane/s49/s69/s45/s51 /s48/s46/s48/s49 /s48/s46/s49 /s49/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32\n/s32/s32/s32/s49/s51/s32/s71/s32/s32/s32 /s32/s32/s32/s57/s51/s32/s71/s32\n/s32/s32/s32/s50/s51/s32/s71/s32/s32/s32 /s32/s49/s52/s51/s32/s71\n/s32/s32/s32/s51/s51/s32/s71/s32/s32/s32 /s32/s50/s57/s51/s32/s71\n/s32/s32/s32/s55/s51/s32/s71/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84 /s32/s61/s32/s49/s57/s48/s32/s109/s75/s80/s32 /s40/s116/s41\n/s116/s47/s66\n/s76/s70 /s32/s40 /s115/s32/s71/s45/s49\n/s41\nFIG. 4: (Color online) Corrected muon decay asymmetry at\nT= 190 mK for various magnetic ﬁelds as function of the\nscaling variable t/B0.81\nLF. The dashed-dotted line is a ﬁt of the\n13 G data with P(t)/P(0) = exp[ −λLt]−0.9.\nand a dominant 4 f-µdipolar interaction, one can de-\ntermine the expected internal ﬁeld values for the pro-\nposed sites. Our lattice sum calculations reveal that only\nat the 4c(1/2,0,1/2) site and 8 i(1/4,1/2,1/2) site a lo-\ncal ﬁeld Blocalof the measured absolute magnitude is\nfound. For the 4 cand 8isites the measured local ﬁeld of\nBlocal= 13.87 G corresponds to a static ordered moment\nof the Yb ions of mord= 0.046µBand 0.025 µB, respec-\ntively. Both values are in good agreement with the value\ndeduced from recent magnetization measurements12.\nThe temperature dependence of the exponential relax-\nation rate λL=1\nT1, observed above TC, is plotted in\nFig. 1(d) on a log-log scale as1\nT1T. Cooling down from\n800 mK,1\nT1Texhibits power-law behavior according to\n1\nT1T∝T−1.40(6). AtT≤190 mK, the power-law behav-\nior in1\nT1Tpersists in the PM regime down to TC, how-\never, with a slight change of the critical exponent, i.e.,\n1\nT1T∝T−1.5(1)(dashed-dotted line). The observed1\nT1T\nbehavior is close to the T−4/3temperature dependence\npredicted by the self-consistent renormalization (SCR)\ntheory for a system close to a 3D ferromagnetic QCP22.\nThere is no prediction for an itinerant quasi1D system in\ntheTrangebetweentheexchangeenergyscaleandorder-\ningtemperature. Foran isolatingferromagneticquasi1D\nspin chain the Tdependence of the relaxation rate above\nTCdepends strongly on the details of the interactions –\nsee, e.g.,23.\nLF-µSR experiments allow to separate the dynamic\ncontribution to the relaxation of the muon-spin polar-\nization. Investigations of the low- Tmuon-spin dynamics\nyield additional information about the origin of the NFL\nbehavior in YbNi 4P2. Figure 2 displays the muon-spin\nasymmetry function P(t) atT= 20 mK for diﬀerent\napplied LF’s. The muon-spin relaxation is completely\nsuppressed in an applied ﬁeld BLF≈300 G, demonstrat-\ning that the internal ﬁeld distribution is static in nature.\nHowever, the observed decoupling can not be described\naccurately by a standard muon asymmetry function that\nconsiders an internal ﬁeld distribution which is symmet-4\nric around Blocal= 0. For comparison, Fig 2. shows the-\noreticaldepolarizationcurvesforthestaticGKTfunction\nin the corresponding longitudinal magnetic ﬁelds. This\nsupportstheZFdata, i.e., theobservationofabroadﬁeld\ndistribution centered around a ﬁnite but small internal\nﬁeldBlocal(20 mK) ≈13.87 G in the FM phase. Finally,\nwhenBLF≫Blocal, the muon spin relaxation is decou-\npled from the static Blocalas observed for BLF≥23 G.\nAtT > T C, the ﬁeld dependence of the muon-spin\nrelaxation probes the Fourier transform of the dynamic\nspin-spin autocorrelation function q(t) =∝angbracketleftSi(t)·Si(0)∝angbracketright,\nwhich exhibits exponential behavior for homogeneous\nsystems and power-law(or cutoﬀ power-law)or stretched\nexponential behavior for inhomogeneous systems. The\ninset of Fig. 3 displays the muon-spin polarization P(t)\natT= 190 mK, both in magnetic LF between 13 and\n143 G and ZF. The relaxation rate λLis reduced with\nincreasing ﬁeld. The ﬁeld dependence of λLis given in\nthe main panel of Fig. 3. It shows nearly no ﬁeld de-\npendence for magnetic ﬁelds of less than ∼13 G, but\nvaries more strongly, as H−κwithκ≈0.79(7), for higher\nﬁelds. Fromthe ﬁeld dependence of λL, the spin autocor-\nrelation time τccan be estimated using the Redﬁeld for-\nmalism for λL(BLF) = (2γ2\nµ∝angbracketleftB2\nfluc∝angbracketrightτc)/[1 + (γ2\nµB2\nLFτ2\nc)]\nconsidering τcas independent of the applied ﬁeld BLF.\nHere,Bfluc(t) describes the time-varying local magnetic\nﬁeld at the muon site due to ﬂuctuations of neighboring\nYb3+moments, with a local time averaged second mo-\nment ∆2=γ2\nµ∝angbracketleftB2\nfluc∝angbracketrightand a single ﬂuctuation time τc.\nFor ¯hω≪kBT(ωgiving the spin ﬂuctuation rate), the\nﬂuctuation-dissipation theorem24relatesτcto the imagi-\nnarycomponentofthe local q-independent f-electrondy-\nnamic susceptibility, i.e. τc(B) = (kBT)[χ′′(ω)/ω]. The\nﬁt to the data (solid curve in the main panel of Fig. 3)\nyields ∆2≈0.1 (MHz) and τc≈6×10−7s, the latter\nvalue nearly three orders of magnitude larger than the\none obtained for YbRh 2Si2atT= 20 mK14, suggesting\nvery slow critical ﬂuctuations.\nTheµSR time spectra in Fig. 3 are well described with\na stretched exponential relaxation function of the form\nP(t) =P(0)exp[−(λt)β]. An exponent of β≈0.9 shows\nthat the relaxation rate is nearly uniform throughout the\nsample, indicating that YbNi 4P2exhibits quasihomoge-\nneous spin ﬂuctuations for T≪TK. The spin dynam-\nics is characterized by a narrow distribution of correla-\ntion times ( β= 1 corresponds to one single correlation\ntime). Thus, disorder-driven theories, including Kondo\ndisorder25,26and the Griﬃth phase scenario27as primary\nmechanisms for the observed NFL behavior, can be ruled\nout. It further implies that the crystalline disorder in\nYbNi4P2is quite small, which is consistent with a small\nresidual resistivity ( ρ0∼2.4µΩcm) and the stoichio-\nmetric occupation of the crystallographic lattice sites re-\nvealed by the x-ray structure reﬁnement12.\nA sensitive test to identify power-law or stretched ex-ponential behavior of q(t) is a time-ﬁeld scaling analysis\nof the muon-spin relaxation function. In both cases a\nspeciﬁc time-ﬁeld scaling can be found, i.e., the muon-\nspin relaxation function P(t,BLF) obeys the scaling re-\nlationP(t,BLF) =P(t/Bγ\nLF). This relation applies\nonly in the asymptotic strong ﬁeld limit, i.e., as long as\n2πfµ=γµBLF≫λL28. If time-ﬁeld scaling is obeyed,\na plot of P(t,BLF) versus t/Bγ\nLFatT > T Cwill be\nuniversal for the correct choice of γ, and distinguishes\nbetween power-law ( γ <1) and stretched exponential\n(γ≥1) correlations. For small BLF, the ﬁeld depen-\ndence is expected to be due to the change of fµrather\nthan an eﬀect of ﬁeld on q(t). A breakdown of time-ﬁeld\nscaling would occur for high ﬁelds where q(t) is directly\neﬀected by the applied ﬁelds. Figure 4 shows the same\nasymmetry data, as displayed in Fig. 3, as functions of\nthe scaling variable t/Bγ\nLF. Forγ= 0.81(5) the data\nscale well over ∼2.5 orders of magnitude in t/Bγ\nLFand\nfor all ﬁelds between 13 and 143 G, except for 293 G.\nHere, at large t, the data fall above the low-ﬁeld scaling\ncurve. Fields µBBLF≥kBT(withkB=Boltzmann‘s\nconstant) would be expected to aﬀect the spin dynam-\nics. The scaling exponent γ= 0.81(5)<1 implies that\nwithin the µSR frequency range, the spin-spin correla-\ntion function q(t) is approximated by a power law (or a\ncutoﬀ power law) rather than a stretched exponential or\nexponential28, consistent with the Redﬁeld analysis. The\npower-law is time-scale invariant and dynamical modu-\nlations should therefore be observable in any time win-\ndow. The obtained time-ﬁeld scaling of the relaxation\ndata is a signature of slow homogeneous spin dynamics.\nIt strongly indicates that the critical slowing down of\nspin ﬂuctuations at the magnetic phase transition occurs\ncooperatively throughout the sample. In stoichiometric,\nhomogeneousNFL systems such behaviormay arise from\nthe eﬀect of disorder on quantum critical ﬂuctuations in-\nherent to a QCP. This is suggested for the NFL com-\npound YbRh 2Si214,15.\nIn conclusion, ZF- µSR in the stoichiometric NFL com-\npound YbNi 4P2clearly proves static magnetic ordering\nof strongly reduced Yb3+moments below TC= 140 mK.\nAboveTC, the muon spin polarization P(t) obeys the\ntime-ﬁeld scaling relation P(t) =P(t/B0.81(5)) for ap-\nplied magnetic ﬁelds Bbetween 13 and 143 G, indicat-\ning cooperative and critical spin dynamics. Power-law\nbehavior of the dynamic spin-spin autocorrelation func-\ntion is implied by the observation of γ <128. The LF-\nµSR results suggest that the NFL behavior observed at\nT > T Cis induced by quasi homogeneous critical spin\nﬂuctuations.\nWe acknowledgewith thanksthe help ofA. Amato and\nthe PSI accelerator crew as well as ﬁnancial support by\nthe German Science Foundation (DFG) in the framework\nof the priority program 1458, Grant No. KL1086/10-1.5\n∗Electronic address: h.klauss@physik.tu-dresden.de\n1Focus issue on quantum phase transitions, Nat. Phys. 4,\n167 - 204 (2008).\n2N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker,\nD. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich,\nNature (London) 394, 39 (1998).\n3G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001); 78, 743\n(2006).\n4T. Park, F. Ronning, H. Q. Yuan, M. B. Salamon,\nR. Movshovich, J. L. Sarrao, and J. D. Thompson, Na-\nture (London) 440, 65-68 (2006).\n5S. Friedemann, T. Westerkamp, M. Brando, N. 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Grollier2\n1King Abdullah University of Science and Technology (KAUST) ,\nPhysical Science and Engineering Division, Thuwal 23955-6 900, Saudi Arabia;\n2Unit´ e Mixte de Physique CNRS/Thales and Universit´ e Paris Sud 11,\nRoute D´ epartementale 128, 91767 Palaiseau, France; and\n3National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan\n(Dated: June 27, 2018)\nSpin transport in magnetic tunnel junctions in the presence of spin diﬀusion is considered theo-\nretically. Combining ballistic tunneling across the barri er and diﬀusive transport in the electrodes,\nwe solve the spin dynamics equation in the metallic layers. W e show that spin diﬀusion mixes the\ntransverse spin current components and dramatically modiﬁ es the bias dependence of the eﬀective\nspin transfer torque. This leads to a signiﬁcant linear bias dependence of the out-of-plane torque,\nas well as a non-conventional thickness dependence of both s pin torque components.\nPACS numbers: 75.60.Jk,85.75.Dd,72.25.-b\nCurrent-driven control of the magnetization direction\nin magnetic nanodevices has been made possible by\nthe prediction of Spin Transfer Torque (STT), by Slon-\nczewski and Berger1. The transfer of spin angular mo-\nmentumbetweenthespin-polarizedelectricalcurrentand\nthe local magnetization has been observed in various de-\nvices such as metallic spin-valves, magnetic tunnel junc-\ntions (MTJs) and magnetic domain walls2,3. A number\nof technological applications have been proposed, based\non these devices, such as STT-MRAM, Race track mem-\nory etc.4.\nThe most promising candidate for memory applica-\ntions to date is the MTJ5due to its high perfor-\nmances and good compatibility with the existing C-MOS\ntechnology6. Therefore, understanding the nature of the\nspin transfer torque in such devices is of seminal im-\nportance. The ﬁrst theoretical studies showed that in\nthe ballistic limit of symmetric MTJs comprising semi-\ninﬁnite ferromagnets, the spin torque is on the form7–13\nT=T/bardblM×(P×M)+T⊥M×P, (1)\nT/bardbl=a1V+a2V2, T⊥=b0+b2V2.(2)\nThe ﬁrst term T/bardblin Eq. (1), referred to as the in-plane\ntorque, competes with the magnetic damping allowing\nfor self-sustained magnetic precessions and switching,\nwhereas the second term T⊥, referred to as the out-of-\nplane torque, acts like an eﬀective ﬁeld applied along\nP. Here, MandPare the magnetization direction of\nthe free and pinned layers, respectively and Vis the\nbias voltage applied across the junction. Interestingly,\nas shown in Eq. (2), the bias dependence of the spin\ntorque components is well deﬁned, the in-plane torque\npossessing both linear ( a1) and (small) quadratic com-\nponents ( a2≪a1), and the out-of-plane torque being\nonly quadratic ( b2), besides the zero-bias exchange cou-\npling term ( b0). This bias dependence has been con-\nﬁrmed experimentally in FeCoB/MgO/FeCoB by spin-\ndiode measurements14. The quadratic bias dependence\nof the out-of-plane torque has serious implications oncurrent-driven magnetization dynamics. Whereas the\nsign of the in-plane torque depends on the polarity of\ncurrent injection, the out-of-plane torque is always in the\nsame direction. Consequently, at positive polarity both\ntorques favor the antiparallel conﬁguration, whereas at\nnegative polarity, the in-plane torque favors the parallel\nconﬁgurationwhile the out-of-planetorque favorsthe an-\ntiparallel one. This competition leads to back-hopping of\nthe magnetization state which is detrimental for applica-\ntions such as MRAM15,16.\nHowever, recent experiments have reported important\ndiscrepancies between the actual bias dependence of the\nspin torque and the one proposed in Eqs. (1)-(2)15–18,20.\nIn particular, Oh et al.16showed that in an asymmetri-\ncally designed MTJ, the bias dependence of the out-of-\nplane torque acquires a linear contribution. This con-\ntribution reduces the competition between in-plane and\nout-of-plane torques for negative polarity and results in a\nnet reduction ofthe back-hoppingprocess. Severalmech-\nanisms have been proposed to alter the bias dependence\nof the spin torque, such as ﬁnite layer thickness10,11,13,\nbarrier and electrode asymmetry10,11,13,19, magnons19–21\nand spin-ﬂip scattering in the electrodes19,20. Neverthe-\nless, these mechanismsfail to explain the largelinear bias\ndependence usually observed in the out-of-plane torque.\nIn the present study, we propose a model of spin trans-\nfer in a diﬀusive spin transport approach taking place\nin the metallic electrodes of MTJs. We will show that\nthe diﬀusion processes (i) mix the two spin components\ntransversetothelocalmagnetizationwhichhastheresult\nto (ii) dramatically modify the bias dependence of the ef-\nfective spin torqueas well asits thickness dependence. In\nparticular, we show that the out-of-plane torque acquires\na signiﬁcant linear bias dependent term. This result is\nobtained by solving the spin diﬀusion equation for the\nspin accumulation transverse vector in the metallic lay-\ners adjacent to the insulator and imposing the interfa-\ncial spin current as a boundary condition. In particu-\nlar, our model adopts the point of view of Zhang et al.2\nwherebymagnon-assistedtunneling process, at high bias,\nquenches any transport of hot electrons over a short dis-\ntance inside the ferromagnetic electrode resulting in a\npure diﬀusive transport within the electrode22.\nLet us consider a MTJ composed of N/F/I/F/N stack,\nwhere I is the insulating spacer, F are the ferromagnetic\nlayers and N the non magnetic metallic leads connected\ntotheFlayer. Thetunnelingprocessthroughtheinsulat-\ning barrier imposes a ballistic injection of carriers at I/F\ninterface which can be described by several calculation\ntechniques like the free electron model, a tight-binding\ntreatment or a density functional theory. These diﬀerent\napproaches are well documented in a recent review paper\nRef. 23. From the above arguments, it results that the\ntwo components of the transversespin current J0are im-\nposed at the I/F interface. It constitutes a boundary con-\nditionto the coupled diﬀusive spin transport-relaxation\nequations for the transverse component of the spin ac-\ncumulation vector m. Along these guidelines, the spin\ndynamics of the transverse spin accumulation in the fer-\nromagnetic electrode from the I/F interface is governed\nby the following time-dependent coupled equations\n∂m\n∂t=−∇·J −1\nτJm×M−1\nτφM×(m×M)−m\nτsf,(3)\nJ=−D∇⊗m, (4)\nwheremis the spin accumulation, Mis the direction of\nthe localized 3dmagnetization, tis the time, and Jis\nthe spin current tensor. In Eq. (4), the expression of\nthe spin current is limited to Ohm’s law viathe diﬀusion\nconstant D. Theseequationsaccountforspatialvariation\nof spin current, spin precession, spin dephasing and spin\nrelaxation through the respective spin precession time\nτJ, spin decoherence time τφand spin relaxation time\nτsf. Whereas the spin relaxation aﬀects the three spin\ncomponents, the spin precession and spin decoherence\nterms only aﬀect the two transverse components of the\nspin accumulation vector. In the transient regime of spin\ninjection dynamics, these equationscan be solvedanalyt-\nically using Green’s functions techniques in presence of\na source term J0δ(z)δ(t) (δis the Dirac distribution and\nzis the direction of the tunneling current where z= 0\ncorrespond to the exact I/F interface position). In the\nsteady-state regime of spin injection ( ×/integraltext∞\n0dt), it simply\ngives\n∇·J=−1\nτJm×M−1\nτφM×(m×M)−1\nτsfm,(5)\nJ=−D∇⊗m, (6)\nThe solutions for the spin accumulation in the ferromag-\nnet, when the magnetization is along zreads\nmt=m/bardbl+im⊥=Aex/L+Be−x/L(7)\n1\nL2=−i\nλ2\nJ+1\nλ2\nφ+1\nλ2\nsf(8)\nwhereλ2\ni=Dτi. We are now interested in the spin ac-\ncumulation in the ferromagnetic layer, displayed in Fig.1. On the left interface I/F, the boundary conditions are\ngiven by the interfacial spin current injected through the\nbarrierJ0=J0\n/bardbl+iJ0\n⊥. The two components are the\nin-plane and out-of-plane tunneling spin currents deﬁned\nbyJ0=J0\n/bardblM×(P×M) +J0\n⊥M×P. On the right\ninterface F/N, the connection is given by the continuity\nof the spin accumulation and spin current. In principle,\ninterfacial resistance and spin-ﬂip give additional contri-\nbutions to the interfacial boundary conditions. In the\npresent model, we disregard these eﬀects since they can\nbe addedbysimplyinsertinganeﬀectiveinterfaciallayer.\nAfter somealgebra,the twocomponentsofthetransverse\nspin density read\nm/bardbl+im⊥=J0L\nDFsinhd−x\nL+ηcoshd−x\nL\ncoshd\nL+ηsinhd\nL, x < d (9)\nm/bardbl+im⊥=J0L\nDFηe−x−d\nλsf\ncoshd\nL+ηsinhd\nL, x > d (10)\nwhereη=DF\nDNλN\nsf\nL,Diis the conductivity of the i-th layer\nanddis the thickness of the ferromagnetic layer. Figure\n1 displays the spatial proﬁle of the spin accumulation m/bardbl\nandm⊥transverse to the local magnetization for diﬀer-\nent spin dephasing length λφ=0.5 nm, 1 nm, and 1.5 nm.\nInthisﬁgure,weassumed J0\n⊥=0(noballisticout-of-plane\nspin current). For the chosen parameters, the spin accu-\nmulation is not fully absorbed in the ferromagnet and\ndecays away from the I/F interface. Furthermore, due to\nthe spin precession, the spin accumulation displays both\nin-plane and out-of-plane components.\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32 /s61/s49/s46/s53/s32/s110/s109\n/s32/s49/s32/s110/s109\n/s32/s48/s46/s53/s32/s110/s109\n/s109\n/s124/s78/s111/s114/s109/s97/s108/s32/s77/s101/s116/s97/s108\n/s32/s32/s84/s114/s97/s110/s115/s118/s101/s114/s115/s101/s32/s83/s112/s105/s110/s32/s68/s101/s110/s115/s105/s116/s121/s32/s40/s48 /s124/s124\n/s47\n/s70/s41\n/s80/s111/s115/s105/s116/s105/s111/s110/s32/s40/s110/s109/s41/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116\n/s109\n/s124/s124\nFIG. 1. (Color online) Transverse spin accumulation as\na function of the distance in the electrode in F(2 nm)/N.\nThe parameters are λJ=1 nm,λF\nsf=15 nm, λN\nsf=2 nm and\nDF/DN= 1.\nNote that the out-of-plane component m⊥increases\nwhen enhancing the spin dephasing length. Therefore,\neven in the absence of out-of-plane interfacial spin cur-\nrent (J0\n⊥=0), the interplay between spin dynamics and\nspin diﬀusion is expected to produce a torque with both3\nin-plane and out-of-plane components. The spin torque\nis deﬁned as the spatial change of spin current, compen-\nsated by the spin relaxation term\nT=1\nΩ/integraldisplay\nΩdΩ/parenleftbigg\n−∇·J −1\nτsfm/parenrightbigg\n.(11)\nHere, Ω is the volume of the magnetic layer. Therefore,\nthe total spin torque exerted on the ferromagnet is\nT/bardbl+iT⊥=J0\ndL2\nL2\n0coshd\nL+ηsinhd\nL−1\ncoshd\nL+ηsinhd\nL,(12)\nwhere1\nL2\n0=−i\nλ2\nJ+1\nλ2\nφ.\nIn the limit of inﬁnite ferromagnetic layer thickness,\nthe torque in Eq. (12) reduces to\nT/bardbl=1\ndξ/parenleftBig\n((1+χ)β2+χ2)J0\n/bardbl+χ2βJ0\n⊥/parenrightBig\n,(13)\nT⊥=1\ndξ/parenleftBig\n((1+χ)β2+χ2)J0\n⊥−χ2βJ0\n/bardbl/parenrightBig\n,(14)\nwhereχ=τφ/τsf,β=τJ/τsfandξ= (1+χ)2β2+χ2.\nIn the case of very short spin dephasing χ≪1, as in\nFe/MgO/Fe12, the spin torque reduces to the ballistic\nlimit,T/bardbl=J0\n/bardbl/dandT⊥=J0\n⊥/d. In the limit of inﬁnite\nspin dephasing χ≫1, such as the one considered in\nRef. 19 and 20, the spin torque arises from a mixture\nofJ0\n/bardblandJ0\n⊥,T/bardbl= (J0\n/bardbl+βJ0\n⊥)/d(1 +β2) andT⊥=\n(J0\n⊥−βJ0\n/bardbl)/d(1 +β2). Therefore, in the case of semi-\ninﬁnite ferromagnets,alinearbiasdependence ofthe out-\nof-plane torquearisesonly if both the spin dephasing and\nthe spin precession are comparable to the spin diﬀusion\nlength.\nThe case ofa ﬁnite ferromagneticlayerpresentssigniﬁ-\ncant diﬀerences due to the adjacent normal metal. When\nthespinprecessionanddephasingoccuronalengthmuch\nsmaller than the thickness of the ferromagnet, the trans-\nverse spin current is mostly absorbed at the interface be-\ntween the insulator and the ferromagnet and the spin\ntorque bias dependence is described by Eqs. (1)-(2).\nConversely, when the spin dephasing and precession ex-\ntend on a length comparable to the layer thickness, the\ntransverse spin current is no more conﬁned to the inter-\nface and extends on the layer thickness to the second in-\nterface with the normal metal, as shown in Fig. 1. This\nnew dynamics mixes the two transverse components of\nthe spin current and is therefore responsible for the de-\nviation from the ballistic bias dependence shown in Eq.\n(2). Figure 2 displays the two components of the spin\ntorque,T/bardblandT⊥, obtained from Eq. (12) in the case\nJ0\n⊥=0 as a function of the thickness of the ferromag-\nnetic layer. In this ﬁgure, the two contributions have\nbeen multiplied by the distance dto remove the 1 /dde-\npendence coming from the interfacial nature of the spin\ntorque. Interestingly, it appears that both in-plane and\nout-of-plane torques possess a thickness-dependence that\nis not simply ∝1/d, as it is expected in the caseof purely\ninterfacial spin torque./s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32\n/s74/s61/s48/s46/s53/s32/s110/s109\n/s32/s49/s32/s110/s109\n/s32/s49/s46/s53/s110/s109\n/s32\n/s74/s61/s48/s46/s53/s32/s110/s109\n/s32/s49/s32/s110/s109\n/s32/s49/s46/s53/s32/s110/s109/s49 /s50 /s51 /s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48\n/s32/s32/s83/s112/s105/s110/s32/s84/s111/s114/s113/s117/s101/s32/s82/s97/s116/s105/s111\n/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s32/s84/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s79/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s116/s111/s114/s113/s117/s101\n/s32/s32/s83/s112/s105/s110/s32/s84/s114/s97/s110/s115/s102/s101/s114/s32/s84/s111/s114/s113/s117/s101/s32/s40/s100/s32/s120/s32/s48 /s124/s124/s41\n/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s32/s84/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s73/s110/s45/s112/s108/s97/s110/s101/s32/s116/s111/s114/s113/s117/s101\nFIG. 2. (Color online) Thickness dependence of the normal-\nized spin torques dT/bardblanddT⊥for diﬀerent spin precession\nlengths, λJ= 0.5 nm, 1 nm and 1.5 nm. The parameters are\nthe same as in Fig. 1, except for the spin dephasing length:\nλφ= 0.5 nm (solid lines) and λφ=1 nm (dotted lines). In-\nset: Thickness dependence of the ratio between out-of-plan e\ntorque and in-plane torque for the same parameters.\nA deviation from the 1/ d-thickness dependence of the\nspin torque has been identiﬁed previously in calculations\nassuming a ballistic regime11,12. For example, Ref. 12\nobserves a deviation that is attributed to resonant quan-\ntum states in the ﬁnite ferromagnetic layer. Note that\nthe model proposed in the present work assumes a diﬀu-\nsive transport in the metallic layers, i.e. defects and dis-\norder are strong enough to quench quantum coherence.\nTherefore, the deviation we observe here is not related\nto quantum states, but rather to the incomplete absorp-\ntion of the spin current: when the thickness of the free\nlayer is on the order of or smaller than the spin dephas-\ning length λφ, the transverse spin current responsible for\nthe spin torque is not fully absorbed in the free layer and\ndiﬀuses towards the capping layer. This induces a devia-\ntionfromtheusual1/ d-thicknessdependence. Increasing\nthe thickness of the free layer improves the absorption of\nthe spin current and for thicknesses much largerthan the\nspin dephasing length, the thickness dependence of the\ntorque recovers the 1/ dlimit (see Fig. 2). Note that\nin the case of half-metallic behavior, as in Fe/MgO/Fe\ntunnel junctions, the minority band with ∆ 1symmetry\ndoes not propagate in the ferromagnet which results in\na quenching of the spin dephasing length ( d≫λφ) and\nreduces the spin torque to a 1/ dbehavior.\nSince the transverse spin accumulation is not fully ab-\nsorbed in the ferromagnet, the remaining unabsorbed\ntransverse spin accumulation diﬀuses into the normal\nmetal and modiﬁes the actual spin torque exerted on\nthe ferromagnet. More speciﬁcally, the mixing that gives\nriseto both componentsincreasesforsmallerthicknesses.\nTherefore, the ratio between the out-of-plane and the in-\nplane components, T⊥/T/bardbl, increases when decreasing the4\nthickness of the ferromagnet, as displayed in the inset\nin Fig. 2. The nature of the spin torque asymptoti-\ncallytendstowardsthesemi-inﬁniteferromagnetlimitfor\nthicknesses much larger than the spin dephasing length.\nLet us now illustrate the inﬂuence of the normal metal\non the spin diﬀusion in the ferromagnetic layer. The role\nof the normal metal can be qualitatively understood if\nonemodels the F/Nbilayerbyan equivalentsemi-inﬁnite\nF layer with an eﬀective spin diﬀusion length λ∗\nsf. In the\nsemi-inﬁnite limit, the spin torque reduces to\nT/bardbl+iT⊥d→∞−−−→J0\ndL∗2\nL2\n0, (15)\nwhere1\nL∗2=−i\nλ2\nJ+1\nλ2\nφ+1\nλ∗2\nsf,λ∗\nsfbeing the eﬀective\nspin diﬀusion length arising from the presence of the ad-\njacent normal metal. By equating Eq. (15) with Eq.\n(12), one can deﬁne the eﬀective spin diﬀusion length\nλ∗\nsfas a function of the parameters of the normal layer.\nTo obtain a tractable analytical result, we consider an\nultrathin ferromagnet, so that\nT/bardbl+iT⊥d→0− −− →J0\ndL2\nL2\n01\n1+L\nηd. (16)\nThis provides the analytical expression\n1\nL∗2=1\nL2+DN\nDF1\nλN\nsfd, (17)\nor equivalently\n1\nτF∗\nsf=1\nτF\nsf+1\nτN\nsfλN\nsf\nd=1\nτF\nsf+pN\n1−pN1\nτN\nsf,(18)\nwherepNis the probability to ﬁnd the particle in N.\nThe bulk spin relaxation time in the ﬁnite ferromagnet\nτF\nsfis renormalized by the presence of the normal metal\nthrough the probability pN. Increasing the spin diﬀu-\nsion length of the normal metal λN\nsfor decreasing the\nthickness of the ferromagnet dreduces the spin diﬀusion\nlength in the ferromagnet. The second term of Eq. (18)\ncan be viewed as the eﬀective spin lifetime in the ﬁnite\nferromagnetic layer.\nThis thickness dependence has a very important im-\nplication on the bias dependence of the spin torque. If\none assumes a bias dependence of the spin current on\nthe form J0\n/bardbl=a1VandJ0\n⊥=b2V2, as expected and\nobserved for systems such as Fe/MgO/Fe12,14, then both\nin-plane and out-of plane spin torque components will be\na mixture of linear and quadratic bias dependences. For\nexample, Sankey et al.and Kubota et al.experimentally\nfoundthat T⊥/T/bardbl|200mV≈10%to15%14. Assumingthatthe bias dependence of the interfacial spin current J0is\ngiven by the spin torque bias dependence measured by\nSankeyet al., the in-plane and out-of-plane torques are\nreproduced in Fig. 3 for diﬀerent thicknesses ofthe ferro-\nmagnetic layer. As expected from the discussion above,\nthe linear character increases when decreasing the fer-\nromagnetic thickness and at large thicknesses it reduces\n/s45/s53/s48/s48 /s48 /s53/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s100/s61/s49/s32/s110/s109\n/s32/s49/s46/s53/s32/s110/s109\n/s32/s50/s32/s110/s109\n/s32/s49/s48/s32/s110/s109/s40/s98/s41/s32/s32/s73/s110/s45/s80/s108/s97/s110/s101/s32/s84/s111/s114/s113/s117/s101/s32/s40/s104/s47/s50/s101/s32/s120/s49/s48/s32/s107/s45/s49\n/s41\n/s66/s105/s97/s115/s32/s118/s111/s108/s116/s97/s103/s101/s32/s40/s109/s86/s41/s40/s97/s41\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s32\n/s45/s53/s48/s48 /s48 /s53/s48/s48/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s32\n/s32/s32/s79/s117/s116/s45/s111/s102/s45/s80/s108/s97/s110/s101/s32/s84/s111/s114/s113/s117/s101/s32/s40/s104/s47/s50/s101/s32/s120/s49/s48/s48/s32/s107/s45/s49\n/s41\n/s66/s105/s97/s115/s32/s118/s111/s108/s116/s97/s103/s101/s32/s40/s109/s86/s41\nFIG. 3. (Color online) Bias dependence of the in-plane (a)\nand out-of-plane torques (b) for diﬀerent thicknesses of th e\nferromagnetic layer. The parameters are λJ=λφ=1 nm,\nλF\nsf=10 nm, λN\nsf=1 nm and DF/DN=1. The bias dependence\nof the interfacial spin current J0is taken from the spin torque\nbias dependence measured by Sankey et al.14. The inset dis-\nplay the zoom in the range ±200 mV.\nto the bulk value given by Eq. (14). Since the ballistic\nout-of-plane spin current J0\n⊥is small compared to the\nin-plane spin current J0\n/bardbl, only slight modiﬁcation of the\nin-plane torque is expected.\nIn conclusion, the inﬂuence of spin diﬀusion in the\nmetallic layers of MTJs on the spin transfer torque has\nbeen addressed theoretically. Assuming an interfacial\nbias-driven spin current at the interface between the in-\nsulator and the ferromagnet, the spin diﬀusion equation\nis solved and describes a complex spin dynamics in the\nmetallic layers. It is found that this dynamics mixes\nthe components of the spin current tranverse to the lo-\ncal magnetization which results in a superposition be-\ntween linear and quadratic bias dependence for both the\nin-plane and out-of-plane torques. The thickness depen-\ndence of the spin transfer torque is also altered for small\nthicknesses.\nThe authors acknowledge fruitful discussions with A.\nFert.\n∗aurelien.manchon@kaust.edu.sa\n1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);L. Berger, Phys. Rev. B 549353, (1996).\n2D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,5\n11901216 (2008); J. Z. Sun and D. C. Ralph, J. Magn.\nMagn. Mater. 320, 1227 (2008).\n3A. Manchon, and S. Zhang, ’Spin Torque in Magnetic Sys-\ntems: Theory’, Handbook of Spin Transport and Mag-\nnetism, Eds. E.-Y. Tsymbal and I. Zutic, Chap. 8, CRC\nPress, August 2011.\n4C. Chappert, A. Fert and F. Nguyen Van Dau, Nature\nMaterials 6, 813 (2007); S. S. P. Parkin et al., Science\n320, 190 (2008).\n5J.Z. Sun, J. Magn. Magn. Mater. 202, 157 (1999); Y. Huai\net al., Appl. Phys. Lett. 84, 3118 (2004); G. D. Fuchs et\nal., Appl. Phys. Lett. 85, 1205 (2004); D. Chiba et al.,\nPhys. Rev. Lett. 93, 216602 (2004).\n6S. Ikeda, J. Hayakawa, Y. M. Lee, F. Matsukura, Y. Ohno,\nT. Hanyu, and H. Ohno, IEEE Trans. Elec. Dev. 54, 991\n(2007).\n7I. Theodonis et al., Phys.Rev. Lett. 97, 237205 (2006).\n8J. C. Slonczewski, Phys. Rev. B 71, 024411 (2005); J.C.\nSlonczewski and J.Z. Sun, J. Magn. Magn. Mater. 310,\n169-175 (2007); See also, J. C. Slonczewski, Phys. Rev. B\n39, 6995 (1989).\n9A. Manchon et al., J. Phys.: Condens. Matter 20, 145208\n(2008);ibid19, 165212 (2007).\n10J. Xiao, G. E. W. Bauer, and A. Brataas, Phys. Rev. B\n77, 224419 (2008).\n11M. Wilczynski, J. Barnas, and R. Swirkowicz, Phys. Rev.B77, 054434 (2008).\n12C. Heiliger and M. D. Stiles, Phys. Rev. Lett. 100, 186805\n(2008).\n13Y.-H. Tang et al., Phys. Rev. Lett. 103, 057206 (2009);\nPhys. Rev. B 81, 054437 (2010).\n14J. C. Sankey et al., Nature Physics 4, 67 (2008); H. Kub-\notaet al., Nature Physics 4, 37 (2008).\n15J. Z. Sun et al., J. Appl. Phys. 105, 07D109 (2009); T.\nMinet al., J. Appl. Phys. 105, 07D126 (2009).\n16S.-C. Oh et al.,, Nature Physics 5, 898 (2009).\n17A. M. Deac et al., Nature Physics 4, 803 (2008).\n18S. Petit et al., Phys. Rev. Lett. 98, 077203 (2007).\n19A. Manchon, S. Zhang and K.-J. Lee, Phys. Rev. B 82,\n174420 (2010).\n20Z. Liet al., Phys. Rev. Lett. 100, 246602 (2008).\n21P. M. Levy and A. Fert, Phys. Rev. Lett. 97, 097205\n(2006); Phys.Rev. B 74, 224446 (2006); A. Manchon and\nS. Zhang, Phys. Rev. B 79, 174401 (2009).\n22S. Zhang, P. M. Levy, A. C. Marley, and S. S. P. Parkin,\nPhys. Rev. Lett. 79, 3744 (1997).\n23K. D. Belashchenko, and E.-Y. Tsymbal, ’Tunneling Mag-\nnetoresistance: Theory’, Handbook of Spin Transport and\nMagnetism, Eds. E.-Y. Tsymbal and I. Zutic, Chap. 12,\nCRC Press, August 2011."    },    {        "title": "0812.4956v3.Theory_of_spin_polarized_scanning_tunneling_microscopy_applied_to_local_spins.pdf",        "content": "Theory of spin-polarized scanning tunneling microscopy applied to local spins\nJ. Fransson,1,\u0003O. Eriksson,1and A. V. Balatsky2, 3,y\n1Department of Physics and Materials Science, Uppsala University, Box 530, SE-751 21 Uppsala\n2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n3Center for Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: August 30, 2021)\nWe provide a theory for scanning tunneling microscopy and spectroscopy using a spin-polarized\ntip. It it shown that the tunneling conductance can be partitioned into three separate contributions,\na background conductance which is independent of the local spin, a dynamical conductance which\nis proportional to the local spin moment, and a conductance which is proportional to the noise spec-\ntrum of the local spin interactions. The presented theory is applicable to setups with magnetic tip\nand substrate in non-collinear arrangement, as well as for non-magnetic situations. The partitioning\nof the tunneling current suggests a possibility to extract the total spin moment of the local spin\nfrom the dynamical conductance. The dynamical conductance suggests a possibility to generate\nvery high frequency spin-dependent ac currents and/or voltages. We also propose a measurement\nof the dynamical conductance that can be used to determine the character of the e\u000bective exchange\ninteraction between individual spins in clusters. The third contribution to the tunneling current is\nassociated with the spin-spin correlations induced by the exchange interaction between the local spin\nmoment and the tunneling electrons. We demonstrate how this term can be used in the analysis of\nspin excitations recorded in conductance measurements. Finally, we propose to use spin-polarized\nscanning tunneling microscopy for detailed studies of the spin excitation spectrum.\nPACS numbers: 72.25.-b,73.63.-b\nI. INTRODUCTION\nPushing the limits for detection of electronic, mag-\nnetic, and vibrational properties towards the quantum\nlimit requires appropriate experimental tools and tech-\nniques. In particular, single atomic spins1{3and mag-\nnetic nanostructures consisting of few magnetic atoms4{8\non nonmagnetic substrates are frequently studied as\nmodel systems for miniature data storage devices, spin-\ntronics applications, and qubits which are crucial for\nquantum information technology. Being well-de\fned and\ncontrollable on the atomic scale, they ideally serve their\npurpose for studies of fundamentals of their local prop-\nerties and interactions.\nThe scanning tunneling microscope (STM) was in-\nvented in the 1980's9for the purpose of imaging metallic\nsurfaces with atomic resolution, and it was theoretically\ndescribed by Terso\u000b and Hamann,10relating the tunnel-\ning conductance to the density of states (DOS) of the\nlocal environment. More than being a scanning tool, one\ncan park the tip over an object and perform di\u000berential\nconductance measurements in order to reveal informa-\ntion of the local electronic structure. Using the STM\nequipment as a means for spectroscopical measurements\nwas previously discussed, see e.g. Refs. 11{16, the ap-\nproach that is commonly known as scanning tunneling\nspectroscopy (STS). This technique has since been suc-\ncessfully applied in several directions, e.g. detection of\nnoise,17indirect measurements of the Kondo e\u000bect,18and\nthe observation of exchange splitting.19\nExtensions of the STM/STS techniques have been pro-\nvided by using a spin-polarized tip (SP-STM/SP-STS),\ne.g. magnetic CrO 2tips,20or e.g nonmagnetic W tipswhich are coated with a ferromagnetic metal, e.g. Fe,21\nor an anti-ferromagnetic metal, e.g. Cr.22To this end,\nthe theory by Terso\u000b and Hamann was extended to\nalso account for the spin-polarization in the tip and\nsubstrate.23{25The latter formulation was for instance\nused in Ref. 26 in analyzing experimental SP-STM re-\nsults of spin-moments for single Co atoms on a Pt sub-\nstrate. The lack of explicit reference to the spin moments\nof the adatoms located on the substrate in this theory,\nhowever, calls for an advancement in the theoretical for-\nmulation. In this paper we present a theory which sug-\ngest to use the SP-STM technique for directly extracting\nquantitative information about the magnetic moments of\nnano-scale objects.\nThe theory presented here is discussed in the context\nof recent experimental observations, using both STM and\nSTS approaches and using both a nonmagnetic and spin-\npolarized tip. Our discussion will be cast in the light\nof the theoretical description of the SP-STM, particu-\nlarly for measurements performed in presence of local\nspin moments, Sn, located at rnon the substrate. We\n\fnd that indeed STM tunneling is in principle capable\nof detecting single spin and moreover detect the spin ori-\nentation. This conclusion can be drawn, for instance,\nwhen we consider the tunneling electrons to be interact-\ning with the local spins through exchange. For this type\nof interaction mechanism, the tunneling matrix element\ncan be separated into one spin-independent and one spin-\ndependent component. Under such conditions, we \fnd\nthat the tunneling conductance can be separated into\nthree components, of which the \frst provides a conduc-\ntance depending on the electron and magnetic densities\nof the tip and substrate, the second yields a conductancearXiv:0812.4956v3  [cond-mat.mes-hall]  22 Feb 20102\nwhich is directly proportional to the local spin moment,\nand the third being proportional to the noise produced\nby the local spin \ructuations. The last contribution to\nthe current, or conductance, was recently discussed in\nRefs. 27{29.\nWe point out that we are interested in the qualitative\ne\u000bects caused by the presence of local spin, or magnetic,\nmoments located on the substrate surface. For this rea-\nson, we will build upon previous results concerning the\nmatrix elements for the tunneling electrons between the\ntip and substrate. In particular, the spin-dependence in\nthe tunneling current generated by the local spin mo-\nment via exchange interaction is of interest, while other\nfeatures that are pertinent to the tunneling, e.g. geom-\netry of the tip, surface, and adsorbate, bias voltage de-\npendence of the matrix element, etc, will be treated with\nless accuracy.\nWe begin the paper by a derivation of the tunneling\ncurrent and the corresponding (di\u000berential) conductance\nin Sec. II. We continue by discussing the main properties\nof the conductance contributions in Sec. III, and there-\nafter we discuss possible experimental measurements sug-\ngested by the theoretical results in Sec. IV, and we con-\nclude the paper in Sec. V.\nII. PROBING THE LOCAL SPIN MOMENT\nThe wave function of the tunneling electrons in the\ntip, separated from the substrate by distance d, has an\nexponentially small overlap with the substrate electron\nwave functions. The spin-dependent tunneling matrix\nelement can be calculated using Bardeen's result,30and\nis given by31{33\n\u0000n=\u00000exp0\n@\u0000s\n\b\u0000JSn(t)\u0001\u001b\n\b01\nA; (1)\nwhich is understood as a matrix in spin space. Here,\n\u001b= (\u001bx;\u001by;\u001bz) is the Pauli matrix vector, whereas \u0000 0\ndescribes the spin-independent tunneling in absence of J.\n\b is the tunneling barrier height, while \b 0=~2=8md2\nis the energy related to the distance dbetween the tip\nand surface. In principle, the spin-dependent tunnel-\ning matrix element depends on energy, position, ap-\nplied bias voltage, and quantum numbers of the elec-\ntron states, which have been discussed extensively in the\nliterature.11{13,34{37In the present study we shall omit\nthose dependencies, however, for the sake of focusing on\nthe dependence of the tunneling conductance on the local\nspin moments.\nThe exchange energy JjSjis small compared to the\nbarrier height, so that we can expand the exponent and\n\fnd the e\u000bective tunneling matrix element\nTn=T0+T1\u001b\u0001Sn; (2)where\nT0=\u00000e\u0000p\n\b=\b0coshJjSj\n2\br\n\b\n\b0; (3a)\nT1=\u00000e\u0000p\n\b=\b0sinhJjSj\n2\br\n\b\n\b0; (3b)\nsuch thatT1=T0\u0018J=\b. For metals and semiconductors\nit is reasonable to use J\u00180:1 eV,38while the tunneling\nbarrier \b\u00181 eV, giving typical values of T1=T0\u00181=10.\nIn the following discussion, the tunneling rates T0andT1\nare treated as constants, except that we allow T1to carry\na spatial dependence which is related to the positions of\nthe local spin moments. More details of this description\nof the tunneling matrix elements can be found in e.g.\nRefs. 31{33.\nWe next assume that the substrate surface is metallic\nfor whichHsub=P\nk\u001b\"k\u001bcy\nk\u001bck\u001bis su\u000ecient, where cy\nk\u001b\ncreates a surface electron with energy \"k\u001b, momentum k,\nand spin\u001b. The energy-momentum dispersion relation\nneed not be of free-electron character but may assume\nany general form, for which the speci\fc details are unim-\nportant for the present derivation. The energies \"k\u001bare\ngiven relative to the Fermi level \"F, which is common\nfor the system as a whole. We associate the electronic\nand magnetic densities N(r;\") and M(r;\"), respectively,\nwith the electrons in the substrate. For simplicity, we will\nassume in the following that those densities are slowly\nvarying with energy. Analogously, we model the elec-\ntrons in the tip by Htip=P\np\u001b\"p\u001bcy\np\u001bcp\u001b, and de\fne\nits corresponding electronic and magnetic densities n(\")\nandm(\"), respectively.\nIn general, the magnetic moments of the substrate and\ntip may be in a non-collinear arrangement. Thus, de\fn-\ning thez-direction of the global reference frame in e.g.\nthe spin-quantization axis of the tip, the operators of the\nsubstrate are transformed according to\n\u0012\nck\"\nck#\u0013\n=\u0012\ncos(\u0012=2)e\u0000i\u001e=2\u0000sin(\u0012=2)e\u0000i\u001e=2\nsin(\u0012=2)ei\u001e=2cos(\u0012=2)ei\u001e=2\u0013\u0012\nck+\nck\u0000\u0013\n(4)\nwhere\u001eand\u0012are the azimuthal and polar angles between\nthe local substrate reference frame (spins s=\u0006) and the\nglobal one (spins \u001b=\";#).\nTunneling of electrons between the tip and substrate\nin the presence of the local spin moments is captured by\nthe model\nHT=X\npkn\n\u001b\u001b0cy\np\u001b[T0\u000e\u001b\u001b0+T1(r\u0000rn)\u001b\u001b\u001b0\u0001Sn]\n\u0002ck\u001b0eik\u0001r+ieVt+H:c:; (5)\nwhere the bias voltages applied across the junction is\ndenoted by eV. Due to the local nature of the spins,\nwe have included a spatial dependence in the inter-\nacting tunneling rate, and we use e.g. T1(r\u0000rn) =\nT0exp (\u0000jr\u0000rnj=\u0015), where\u0015is the decay length.3\nThe charge current running between the tip and the\nsubstrate is derived using non-equilibrium Green func-\ntions on the Keldysh contour, starting from the funda-\nmental relation I(t) =\u0000e@tP\np\u001bhnp\u001bi, wherenp\u001b=\ncy\np\u001bcp\u001b, giving\nI(r;t;V) =\u00002e\n~ImX\npkn\n\u001b\u001b0hcy\np\u001b(t)^T\u001b\u001b0(rn;t)ck\u001b0(t)i\n\u0002eik\u0001r+ieVt; (6)\nwhere we have de\fned ^T\u001b\u001b0(rn;t) =T0\u000e\u001b\u001b0+T1(r\u0000\nrn)\u001b\u001b\u001b0\u0001Sn(t). The current expresses the rate of\nchange of the expectation value of np\u001b, correspond-\ning to the number of charges, e, in the tip. Expand-\ning the average in the above expression according to\nhA(t)i\u0019(\u0000i)R\nCh[A(t);HT(t0)]idt0, and by converting to\nreal times, the current can be written as\nI(r;t;V) =2e\n~ReX\npp0\nkk0X\n\u001b\u001b0\n\u001b00\u001b000X\nnmZt\n\u00001ei(k\u0000k0)\u0001r+ieV(t\u0000t0)\n\u0002h[cy\np\u001b(t)^T\u001b\u001b0(rn;t)ck\u001b0(t);\ncy\nk0\u001b000(t0)^T\u001b000\u001b00(rm;t0)cp0\u001b00(t0)]idt0: (7)\nHere, notice the appearance of the commutator taken\nbetween the operators inside the average. Separating\nthe tunneling operator ^Tinto its components T0and\nT1, we \fnd that the current can be naturally written\nas a sum of three terms, i.e. I(t) =P2\ni=0Ii(t). We\nconsider the di\u000berential conductance @I(r;t;V)=@V =P2\ni=0@Ii(r;t;V)=@V of stationary source drain voltages\nsince such conditions are predominant in experimental\nsituations.\nA. Background conductance\nThe \frst contribution from the average in Eq. (7) is\ngiven by\nT2\n0\u000e\u001b\u001b0\u000e\u001b000\u001b00h[cy\np\u001b(t)ck\u001b(t);cy\nk0\u001b(t0)cp0\u001b(t0)]i\n=T2\n0\u000e\u001b\u001b0\u000e\u001b000\u001b00(hcy\np\u001b(t)cp0\u001b(t0)ihck\u001b(t)cy\nk0\u001b(t0)i\n\u0000hcp0\u001b(t0)cy\np\u001b(t)ihcy\nk0\u001b(t0)ck\u001b(t)i); (8)\nwhere we have decoupled the averages in the integral\nin terms of correlation functions for the substrate and\ntip electrons. Under the assumption that scattering\nbetween di\u000berent momentum and spin states can be\nneglected, and introducing the notations g<\n\u0014\u001b(t;t0) =\nihcy\n\u0014\u001b(t0)c\u0014\u001b(t)iandg>\n\u0014\u001b(t;t0) = (\u0000i)hc\u0014\u001b(t)cy\n\u0014\u001b(t0)i, the\n\frst contribution to the tunneling current is de\fned by\nI0(r;t;V) =2eT2\n0\n~ReX\npkX\n\u001bZt\n\u00001[g<\np\u001b(t0;t)g>\nk\u001b(t;t0)\n\u0000g>\np\u001b(t0;t)g<\nk\u001b(t;t0)]eieV(t\u0000t0)dt0: (9)Here, the lesser and greater correlation functions of the\ntip electrons are given by\ng<=>\np\u001b(t;t0) =(\u0006i)f(\u0006\"p\u001b)e\u0000i\"p\u001b(t\u0000t0); (10)\nrespectively, and analogously for the correlation func-\ntions of the substrate electrons. Here, f(x) is the Fermi\nfunction. Thus, performing the time-integration and re-\nplacing the momentum summations by integrations over\nthe corresponding spin-resolved density of electron states\n(DOS)n\u001b(\") andN\u001b(r;\") in the tip and substrate, re-\nspectively, we \fnally arrive at the tunneling current\nI0(r;t;V) =2\u0019eT2\n0\n~X\n\u001bZ\nn\u001b(\"\u0000eV)N\u001b(r;\")\n\u0002[f(\"\u0000eV)\u0000f(\")]d\"\n=2\u00192eT2\n0\nhZ\n[f(\")\u0000f(\"\u0000eV)][n(\"\u0000eV)\n\u0002N(r;\") +m(\"\u0000eV)\u0001M(r;\"))d\":(11)\nThe last expression is obtained by noting that n\u001b= (n+\n\u001bmz)=2 andN\u001b= (N+\u001bMzcos\u0012)=2 (the factor \u001b=\u00061)\nwhich thus provides a description that is independent of\nthe coordinate system.\nWe are, ultimately, interested in the tunneling (dif-\nferential) conductance, @I(r;V)=@V, to which I0con-\ntributes\n@I0(r;t;V)\n@V=\u00192\u001b0T2\n0\n4kBTZ\ncosh\u00002\"\u0000eV\n2kBT[n(\"\u0000eV)\n\u0002N(r;\") +m(\"\u0000eV)\u0001M(r;\")]d\"\n!\u00192\u001b0T2\n0[n(\"F\u0000eV)N(r;\"F)\n+m(\"F\u0000eV)\u0001M(r;\"F)); T!0;(12)\nwhere\u001b0= 2e2=his the fundamental conductance unit,\nwhereaskBandTis the Boltzmann constant and temper-\nature, respectively. The expression is obtained under the\nassumption that the electronic and magnetic densities in\nthe tip vary slowly with energy. While this conductance\ncontribution relates the DOS and magnetic density in the\ntip (n(\"F\u0000eV),m(\"F\u0000eV)) and substrate ( N(r;\"F),\nM(r;\"F)), it is independent of the local spin moments.\nB. Dynamical conductance\nWe next consider the expression for the \frst order con-\nductance in the total local spin moment. This is provided\nby a similar consideration of the second contribution to4\nthe tunneling current, i.e. beginning from\nI1(r;t;V) =2eT0\n~ReX\npk\u001b\u001bz\n\u001b\u001bX\nnT1(r\u0000rn)Zt\n\u00001[f(\"p\u001b)\n\u0002f(\u0000\"k\u001b)hSz\nn(t0)i\u0000f(\u0000\"p\u001b)f(\"k\u001b)\n\u0002hSz\nn(t)i]ei(\"p\u001b\u0000\"k\u001b+eV)(t\u0000t0)dt0: (13)\nHere, we again have assumed that scattering between\ndi\u000berent momentum and spin channels within the tip\nand substrate is negligible. We replace the time-\ndependent spin by its Fourier transform, i.e. hSz\nn(t)i=R\nhSz\nn(!)iei!td!=(2\u0019). Along with the replacement of the\nmomentum summations by energy integration over the\ncorresponding spin resolved DOS of the tip and substrate,\nwe \fnd that this current can be written\nI1(r;t;V) =2\u0019eT0\nhX\nnT1(r\u0000rn)Z\nhSz\nn(!)icos!t\n\u0002[n(\")Mz(r;\"0) cos\u0012+mz(\")N(r;\"0)]\n\u0002[f(\")f(\u0000\"0)\u000e(\"\u0000\"0\u0000!+eV)\n\u0000f(\u0000\")f(\"0)\u000e(\"\u0000\"0+eV)]d\"d\"0d!\n2\u0019:(14)\nWe, thus, obtain the conductance\n@I1(r;t;V)\n@V=\u00192\u001b0T0\n4kBTX\nnT1(r\u0000rn)Z\nhSz\nn(!)icos(!t)\n\u0002[mz(\")N(r;\"0) +n(\")Mz(r;\"0) cos(\u0012)]\n\u0002[\u000e(\"\u0000\"0\u0000!+eV) +\u000e(\"\u0000\"0+eV)]\n\u0002cosh\u00002\"\n2kBTd\"d\"0d!\n2\u0019:\n!\u00192\u001b0T0T1X\nnZ\nhSz\nn(!)icos(!t)h\nmz(\")\n\u0002N(r;\"F) +n(\")Mz(r;\"F) cos(\u0012)i\n\u0002[\u000e(\"+eV\u0000!) +\u000e(\"+eV)]d\"d!\n2\u0019;\n(15)asT!0.\nC. Conductance with spin-spin correlations\nThe last term contained in the original expression given\nin Eq. (7) can be written as\nI2(r;t;V) =2e\n~ReX\n\u001b\u001b0X\nnmT1(r\u0000rn)T1(r\u0000rm)Z\nn\u001b(\")\n\u0002N\u001b0(\"0)Zt\n\u00001ei(\"\u0000\"0+eV)(t\u0000t0)[f(\")f(\u0000\"0)\n\u0002\u001b\u001b\u001b0\u0001hSn(t)Sm(t0)i\u0001\u001b\u001b0\u001b\u0000f(\u0000\")f(\"0)\n\u0002\u001b\u001b0\u001b\u0001hSm(t0)Sn(t)i\u0001\u001b\u001b\u001b0]dt0d\"d\"0;\n(16)\nexplicitly expressed in terms of the spin-spin correlation\nfunctions of the local spins. Although the spin-spin cor-\nrelation function e.g. hSn(t)Sm(t0)iprovides all infor-\nmation from the local spin correlations to the tunneling\ncurrent, it is convenient to rewrite this general function in\nterms of the propagators \u001f\u0006\u0007\nnm(t;t0) = (\u0000i)hS\u0006\nn(t)S\u0007\nm(t0)i\nand\u001fz\nnm(t;t0) = (\u0000i)hSz\nn(t)Sz\nm(t0)i. We notice in the sta-\ntionary regime, that these correlation functions depends\non the time-di\u000berence t\u0000t0, which allow us to express\nthem through the Fourier transforms, e.g. \u001fz\nnm(t;t0) =R\n\u001fz\nnm(!)e\u0000i!(t\u0000t0)d!=(2\u0019). These remarks lead to that\nwe can write the third contribution to the tunneling cur-\nrent according to5\nI2(r;t;V) =i\u0019e\n8~X\nnmT1(r\u0000rn)T1(r\u0000rm)Z\u0010\nf(\")f(\u0000\"0)\u000e(\"\u0000\"0\u0000!+eV)\u0000f(\u0000\")f(\"0)\u000e(\"\u0000\"0+!+eV)\u0011\n\u0002\u0010\n[\u001f+\u0000\nnm(!) +\u001f\u0000+\nnm(!)][n(\")N(\"0)\u0000m(\")\u0001M(\"0)]\n+ [\u001f+\u0000\nnm(!)\u0000\u001f\u0000+\nnm(!)][n(\")Mz(\"0) cos\u0012\u0000mz(\")N(\"0)]\n+ 4\u001fz\nnm(!)[n(\")N(\"0) +m(\")\u0001M(\"0)]\u0011\nd\"d\"0d!\n2\u0019: (17)\nDespite the unappealing length of this expression, it provides a convenient starting point for analyses of spin inelastic\ntunneling spectroscopy. The conductance corresponding to this tunneling current becomes27\n@I2(r;t;V)\n@V=i\u0012\u0019\n4\u00132\u001b0\n2kBTX\nnmT1(r\u0000rn)T1(r\u0000rm)Z\u0010\nf(\")\u000e(\"\u0000\"0\u0000!+eV) +f(\u0000\")\u000e(\"\u0000\"0+!+eV)\u0011\n\u0002\u0010\n[\u001f+\u0000\nnm(!) +\u001f\u0000+\nnm(!)][n(\")N(\"0)\u0000m(\")\u0001M(\"0)]\n+ [\u001f+\u0000\nnm(!)\u0000\u001f\u0000+\nnm(!)][n(\")Mz(\"0) cos\u0012\u0000mz(\")N(\"0)]\n+ 4\u001fz\nnm(!)[n(\"F)N(\") +m(\")\u0001M(\"0)]\u0011\ncosh\u00002\"0\n2kBTd\"d\"0d!\n2\u0019!\n!i\u0012\u0019\n2\u00132\u001b0\n2X\nnmT1(r\u0000rn)T1(r\u0000rm)Z\u0010\nf(\")\u000e(\"\u0000!+eV) +f(\u0000\")\u000e(\"+!+eV)\u0011\n\u0002\u0010\n[\u001f+\u0000\nnm(!) +\u001f\u0000+\nnm(!)][n(\")N(\"F)\u0000m(\")\u0001M(\"F)]\n+ [\u001f+\u0000\nnm(!)\u0000\u001f\u0000+\nnm(!)][n(\")Mz(\"F) cos\u0012\u0000mz(\")N(\"F)]\n+ 4\u001fz\nnm(!)[n(\")N(\"F) +m(\")\u0001M(\"F)]\u0011\nd\"d!\n2\u0019; T!0: (18)\nWe summarize this section by pointing out that Eqs.\n(12), (15), and (18) serve as the main results of this pa-\nper. Below, we shall clarify the use of the partitioning of\nthe conductance and how we can interpret experimental\nresults in terms of the di\u000berent contributions.\nWe notice, however, that the electronic structures in\nthe tip and substrate are described in a simple fashion, in\nthat we assume dependence on a single momentum vector\nin terms of the homogeneous GFs, e.g. gk\u001b(t;t0) for the\nsubstrate electrons. Beginning from the current given in\nEq. (7) we can, however, repeat the above derivation al-\nlowing for an inhomogeneous description of the electronic\nstructures, using e.g. g\u001b\u001b0(k;k0;t;t0) for the substrate\nelectrons. This will modify the formulas given for the\nconductance, something which goes beyond the scope of\nthe present study.\nIn connection to this, we also notice that we, here, have\ndisregarded that the electronic and magnetic densities in\nthe substrate are in\ruenced by the local spin moments of\nthe adatoms. In reality, the local spins and the substrate\nmutually in\ruence one another and self-consistently cre-\nate a total e\u000bective magnetic moment locally around theadatoms. The basic formulas, Eqs. (12), (15), and (18),\nremain unchanged, however, by a self-consistent treat-\nment of the substrate and adatoms. From the point of\nview we take in this paper, we can assume that the self-\nconsistently created magnetic moment in the substrate is\nalready included in the density M.\nIII. DISCUSSION OF THE TUNNELING\nCONDUCTANCE\nThe \frst contribution to the conductance, see Eq. (12),\nessentially captures the Terso\u000b-Hamann theory10,23{25\nfor magnetic tip and substrate. One should notice, how-\never, that we here have neglected the speci\fc relation\nbetween the local tunneling matrix element T0and the\nlocal DOS in the substrate, since we here are interested\nin the possibility to resolve its dependence on local spin\nmoments located on the substrate. In studies of mag-\nnetic surfaces, this conductance provides a su\u000ecient tool\nfor analysis of the magnetic structure of the surface, by\nusing the spin-polarization of the electronic structure at6\nthe Fermi level, since the density M(\"F) contains all such\ninformation. An analysis of local spin moments adsorbed\nonto the substrate can, in this model, only be performed\non the level of the DOS and spin splitting of electron\nstates at an energy de\fned by the Fermi level and the ap-\nplied bias voltage. The total spin moment of the adatom\ncan, nevertheless, not be accessed through this conduc-\ntance. This fact is based on that although one can include\nthe adsorbate into the electronic and magnetic densities\nof the surface, one only obtains energy dependent infor-\nmation of the adsorbate, while the spin moment is de\fned\nby the spin-polarization integrated over all energies. Any\ndeeper information about the adsorbate can, thus, not be\nachieved. In this sense, we can consider the contribution\n@I0=@V as generating a back-ground conductance which\nincludes features originating from the substrate.\nThe second conductance contribution, see Eq. (15),\ncontains terms which depend on the spin-polarization in\nthe tip,mz(\"), and of the substrate, Mz(r;\"F), respec-\ntively, and vanishes in case both the tip and substrate\nare non-magnetic. An important result here is that this\ncontribution explicitly identi\fes a linear relationship be-\ntween local spin moment, hSz\nn(!)i=R\nhSz\nn(t)ie\u0000i!tdt,\nand the tunneling conductance. Eq. (15) implies that\none, in principle, can obtain a direct estimation of the size\nof the local spin moment simply by measuring the di\u000ber-\nential conductance at its location. In order to achieve\nthis functionality, it is necessary, as seen in Eq. (15),\nthat the spin-polarizations of the tip and substrate are\nknown. We further discuss this application below. It\nis worth pointing out, however, that although we have\nassumed stationary, i.e. time-independent, conditions,\nthere is nevertheless a time-dependent component in the\ntunneling conductance. This time-dependence is gener-\nated by the dynamics of the local spin and it is this fea-\nture that would enable a read-out of the local spin mo-\nment. The occurrence of the time-dependent component\nin the conductance, and tunneling current, also suggests\na mechanism which may be employed for generation of\nhigh frequency electrical currents. The e\u000bective magnetic\n\felds acting on the local spins, e.g. anisotropy \felds etc.,\nmay be of the order of meV, see below, which would en-\nable generation of high GHz to THz ac currents.\nThe last conductance contribution, see Eq. (18), pro-\nvides signatures that are generated by the spin-spin cor-\nrelations, or spin \ructuations, occurring in the adsorbate.\nIt is important to note that this conductance is \fnite for\nany polarization of the tip and the substrate, even when\nboth electrodes are non-magnetic. Hence, regardless of\nthe electronic and magnetic conditions of the system, this\nconductance directly depends on spin \ructuations in the\nadsorbate. It is also noticeable that the formulation of\n@I2=@V suggests one, in principle, will be able to study\nand distinguish between particular spin excitations in the\nadsorbate. This functionality is expressed from the fact\nthat the sum and di\u000berence of the correlation functions\n\u001f+\u0000\nnmand\u001f\u0000+\nnmare multiplied by di\u000berent combinations\nof the electronic and magnetic structures of the tip andsubstrate. One is therefore capable to con\fgure the STM\nset-up in order to probe particular spin excitations in the\nlocal spin. This will be discussed further below.\nIV. PHYSICAL INFORMATION CONTAINED\nIN THE TUNNELING CONDUCTANCE\nWe now discuss a few di\u000berent physical examples which\nare introduced in order to shine some light on di\u000berent\naspects of the tunneling conductance. Before we go into\nthe examples, however, we introduce the model of the\nlocal spins adsorbed on the surface.\nWe consider a cluster of spins on the substrate and\nwrite the Hamiltonian for the spin Snin the cluster\naccording toHn=g\u0016BB\u0001Sn, where Bis an exter-\nnal magnetic \feld. The e\u000bective exchange interaction\nbetween the spin moments in the cluster is given by a\nHeisenberg model HJ=\u0000JP\nn6=mSn\u0001Sm. This ef-\nfective exchange comprise a combination of e.g. direct\nHeisenberg exchange and RKKY-like (Ruderman-Kittel-\nKasuya-Yosida) exchange. The sign of the e\u000bective J\nmay, thus, vary with distance between the spins in the\ncluster.26In this way we describe clusters of spins with a\ntotal spin moment S. We de\fne the resulting eigensystem\nf\"\u001b;jS;\u001big,\u001b=\u0000S;\u0000S+ 1;:::;S , for the eigenenergies\nand eigenstates, respectively, of the model\nHS=X\nnHn+HJ+X\nn[D(Sz\nn)2+Ef(Sx\nn)2\u0000(Sy\nn)2g];\n(19)\nwhere we have added the spin anisotropy \felds DandE.\nThis model is pertinent to the recent studies of transi-\ntion metal elements adsorbed onto surfaces e.g. Fe and\nMn on CuN surfacem,5,6and Fe and Co on Pt surface.8\nIn those elements, the magnetism is constituted by the\nelectrons in the d-shell whereas the electrons in the s-\nandp-shell strongly hybridize with the surface states in\nthe substrate. We, therefore, assume that the s- and\np-electrons in the adsorbate are included in the Hamilto-\nnian for the surface electrons and, below, we shall refer\nto them as conduction electrons. The electrons in the\nd-shell can be regarded as only weakly hybridizing with\nthe conduction electrons which, therefore, enables us to\ndisregard their contribution to the conductance.\nUnder the assumption of very weak coupling between\nthed-electrons and the conduction electrons, which is\npertinent to the recent experiments on e.g. Fe and Mn\non CuN,5,6we write the total dynamical local spin mo-\nment in terms of the eigensystem de\fned for the cluster\naccording to\nhSz(!)i= 2\u0019SX\n\u001b=\u0000S\u001bf(\"\u001b)\u000e(!\u0000\"\u001b); (20)\nwheref(\"\u001b) provides the occupation of the state\njS;\u001bi. Hence, the conductance is proportional toR\nhSz(!)id!=(2\u0019) =P\n\u001b\u001bf(\"\u001b), c.f. Eq. (21) below.7\nIn order to also describe the experiments conducted\non e.g. a Pt surface,8we phenomenologically introduce\na width of the local spin excitations by replacing Dirac\ndelta functions by Lorentzian functions. Such a treat-\nment can be physically motivated by that this would be\nthe essential e\u000bect of a mean \feld approximated dressing\nof the bare spin averages and spin-spin correlation func-\ntions. This phenomenological model is su\u000ecient for our\npresent purposes since we do not attempt to explain all\nthe details of the experiments.\nA. Estimating the spin moment\nThe conductance in Eq. (15) can be directly linked to\nthe total local spin moment hSz\nni. This is simplest seen\nby assuming that the electronic and magnetic densities in\nthe tip and substrate vary slowly with the energy. Calcu-\nlating the Fourier transform of @I1(t)=@V and integrating\nover all frequencies gives\n@I1(r;V)\n@V=2\u00192\u001b0T0X\nnT1(r\u0000rn)hSz\nnih\nmz(\"F\u0000eV)\n\u0002N(r;\"F) +n(\"F\u0000eV)Mz(r;\"F) cos(\u0012)i\n;\n(21)\nwhere we have identi\fed the total spin moment of the\nnth spin byhSz\nni=R\nhSz\nn(!)id!=(2\u0019). In this fashion,\nwe obtain a linear relationship between the di\u000berential\nconductance @I=@V and the average spin moment hSz\nni.\nOur theory suggests that the STM conductance gener-\nates a time-dependent signal, in agreement with earlier\nstudies which suggest time-dependent noise spectroscopy\nusing STM tunneling current,31,32regardless of the time-\ndependence, or time-independence, of the bias voltage.\nThe Fourier transform of this signal provides an energy\nresolved signal from which additional understanding, i.e.\ndynamics, about the spin systems can be extracted.\nConsider the simple example, given for e.g. a single\n(n= 1) spin moment of S= 1, for which \u001b= 0;\u00061.\nThe anisotropy \felds DandEbreak up the spin sym-\nmetry, such that the spin levels are given by \"0= 0 and\n\"\u00061=D\u0006p\nE2+ (g\u0016BB)2, forB=B^z. The dynamical\nspin moment, thus, becomes hSz(!)i=\u0019[f(\"1)\u000e(!\u0000\"1)\u0000\nf(\"\u00001)\u000e(!\u0000\"\u00001)], which generates a non-vanishing cur-\nrent when the populations f(\"\u00061) of the statesj1;\u00061iare\ndi\u000berent. This observation holds for an adatom with any\nvalue of the spin and emphasizes the fact that the local\nspin has to have a de\fnite moment in order to be mea-\nsurable through the SP-STM. Using this dynamical spin\nmoment in Eq. (15) and assuming slowly varying elec-\ntronic and magnetic densities in the tip and substrate,\nresults for low temperatures in\n@I1(t)\n@V\u0018f(\"1) cos\"1t\u0000f(\"\u00001) cos\"\u00001t: (22)\n−2−1012\n25 50 75\ntime (ps)∂ I1/∂ V3456 (a) (b)\n(c)\nBmtipBmtip\nω (meV)B (T)\n−2 −1 0 1 2−4−2024\n123456B (T)\n−4−2024\n2468\nE=0\nE=2 meVT=0.3 K, E=0\nT=4.3 K, E=0FIG. 1: (Color online) Various aspects of @I1=@V for a sin-\ngleS= 1 adatom. (a) @I1(t;V)=@V, Eq. (15), for B=B^z\nandmtipanti-parallel, B=\u00001 T (lower plots), and par-\nallel,B= 1 T (upper plots). The plots correspond to\nT= 0:3 K,E= 0 (dashed), T= 4:3 K,E= 0 (solid).\n(b) and (c)j@I1(!;V;B )=@Vjas function of !(horizontal\naxis) andB(vertical axis) at T= 0:3 K, and (b) E= 0\nand (c)E= 2 meV. Bright (dark) colors correspond to large\n(small) amplitude. The conductances have been normalized\nby 2\u00192\u001b0T0T1[mzN+nMz]. Here,D=\u000010 meV,g= 2,\nmz= 3n(\"F)=4,Mz= 0, andV= 0:3 meV\nMore generally, the time-dependence of the conduc-\ntance can be written\n@I1(t)\n@V\u0018X\n\u001b\u001bf(\"\u001b) cos\"\u001bt: (23)\nThe period of the conductance oscillations are, thus,\ndirectly linked to the energy levels of the spin states, see\nFig. 1 (a) for an example of the time-dependent conduc-\ntance for two di\u000berent set-ups of the SP-STM. The period\nof the conductance oscillations can, thus, be changed by\napplying an external magnetic \feld Bzin order to vary\nthe energy levels \"\u001b, as is illustrated in Fig. 1 (b), show-\ning the magnetic \feld dependence of the Fourier trans-\nformed dynamical conductance. The time scale associ-\nated with the conductance oscillations is given by the set\nof eigenenergies \"\u001b, which means that only very low ly-\ning excitation energies (0.066 meV corresponding to 100\nGHz) are reachable by means of the state-of-the-art ex-\nperimental technology. Although detection of spin mo-\nments using this method de\fnitely challenges todays ex-\nperimental resources and capabilities, it should be an ac-\ncessible regimes within the nearest future.\nIt is important to notice, however, that the presence of\nthe time-dependent component in the conductance opens\nthe possibility to generate high frequency ac currents.\nDue to the high anisotropy \felds acting on the local\nspin moment, of the order to meV (see Sec. IV B), one\nwould be able to generate electrical and spin-dependent\nac currents and/or ac voltages with frequencies in the8\nTHz regime.\nB. Anisotropy parameters\nOur next example of the usefulness of Eq. (21) is to\nconsider an S= 1 adatom, and we plot for this sys-\ntem the conductance @I1=@V, see Fig. 1 (a), where the\nanisotropy \felds D=\u000010 meV and E= 0 meV for dif-\nferent temperatures, i.e. T=0.3 K (dashed) and T=4.3\nK (solid). These parameters are pertinent to the recent\nstudies of Co/Pt(111).1,8,26The plots in (a) clearly show\nthe phase shift of \u0019when the magnetic \feld Bis reversed\nfrom anti-parallel (lower) to parallel (upper) orientation\nwithmtip. The magnetization curves are S-shaped (not\nshown), which correspond to a paramagnetic behavior of\nthe local spin and which was observed in experiments.26\nFrom the dynamical conductance, @I(t)=@V or\n@I(!)=@V, one can obtain information about the mag-\nnetic anisotropy \felds DandEacting on the local spin.\nIn the case of D< 0 andE= 0, the spin moment points\nperpendicular to the surface, which generates a maxi-\nmal conductance. The response remains strong for val-\nues ofBsu\u000eciently large to maintain a spin-splitting of\nthe adatom which is larger than the thermal excitation\nenergykBT. This is illustrated in Fig. 1 (b), where\nj@1I(!;B)=@Vjis plotted for an S= 1 adatom, as func-\ntion of!andB. Bright (dark) colors correspond to large\n(small) amplitude of j@1I(!;B)=@Vj. The lines that rep-\nresent the ground state of the adatom cross at zero mag-\nnetic \feld, which is expected since the ground state is\nspin-degenerate at B= 0.\nA \fnite value of E, on the other hand, provides a tilting\nof the spin moment away from the perpendicular orien-\ntation, which leads to a weakened conductance. This is\nillustrated in Fig. 1 (c), where E= 2. The \fniteness\nofEbreaks the spin-degeneracy of the ground state at\nvanishing magnetic \feld. The ground state is, however,\npointing parallel to the substrate, which leads to that\nhSz(!)i= 0 atB= 0. The resulting magnetization line\nremains S-shaped (not shown). Despite the low temper-\nature, however, the magnetization curve is reminiscent of\nthe magnetization curve for higher temperature. This is\nexpected since the anisotropy \feld Eintroduces an en-\nergy barrier for the adatom to overcome in order to point\nits spin perpendicular to the substrate.\nIn the production of the plots in Fig. 1 (b) and (c) we\nreplaced the Dirac delta function by a Lorentzian func-\ntion with a broadening of 0.01 meV. This broadening is\nroughly 2 order of magnitudes smaller than what is ex-\npected from the experimental set-up with Co/Pt(111),\nsee Sec. IV D for estimates, and is chosen in order to\nemphasize the di\u000berent behavior of the conductance of\ndi\u000berent anisotropy parameters. Using more realistic\nanisotropy parameters blurs the resulting image, how-\never, the main di\u000berence between the character of the\nconductance for vanishing and \fnite \feld Ecan be re-\nsolved.C. Character of exchange interaction parameter\nStudies of the dynamical conductance can also be used\nto reveal the sign of the e\u000bective exchange interaction\nbetween magnetic adatoms located on the surface. Con-\nsider for example a spin dimer. In case of ferromag-\nnetic exchange, J > 0, the ground state of the spin\ndimer is a spin triplet. Varying the magnetic \feld, the\nground state of the dimer acquires a magnetic moment\ne.g.hSzi=Sor\u0000S, depending on whether the mag-\nnetic \feld is anti-parallel or parallel with the magnetic\nmoment in the tip, in analogy with the single spin case\ndiscussed above. Hence, there is a measurable dynamical\nconductance. In case of an anti-ferromagnetic exchange,\nhowever, the ground state of the spin dimer is a spin sin-\nglet, with zero magnetic moment. Then, the dynamical\nconductance vanishes, c.f. Eq. (15), and the total con-\nductance is strictly time-independent, except for possible\nnoise \ructuations.\nD. Spin \ructuations\nTo illustrate the e\u000bect of spin \ructuactions, we con-\nsider a local spin moment Scomprising two coupled spins\nSn,n= 1;2, a spin dimer, and consider them to be anti-\nferromagnetically coupled. The ground state is a spin\nsingletjS= 0;\u001b= 0i, while the \frst excited states con-\nstitute a spin triplet jS=S1+S2;\u001b= 0;\u0006(S1+S2)i.\nAssuming an exchange energy jJj=jES\u0000ETj> kBT,\nwhereET(S)denotes the triplet (singlet) energy, in order\nto prevent thermal excitations at zero bias, the equilib-\nrium conductance is given by the elastic tunneling be-\ntween the tip and the substrate only, i.e. dI=dV =\ndI0=dV. E\u000bects from tunneling electrons scattering o\u000b\nthe local spin moment averages to zero.\nThe coupling to the tunneling electrons via the spin-\nspin interaction e.g. cy\np\u001b\u001b\u001b\u001b0\u0001Snck\u001benables, on the\nother hand, each individual spin constituting Sto un-\ndergo spin-\rip transitions which are assisted by spin-\rips\nof the tunneling electrons. Due to this coupling, the cor-\nrelation function e.g. \u001b\u001b\u001b0\u0001hSn(t)Sm(t0)i\u0001\u001b\u001b00\u001bis non-\nvanishing, in general. The spin-spin interaction, thus,\nprovides a coupling between the singlet and triplet states\nwhich supports transitions between them. As a result of\nthese transitions, a new channel for conductance opens\nat bias voltages V\u0015jJj=e.\nThe spin-spin correlation functions \u001f\u0000+\nnm(!),\u001f+\u0000\nmn(!),\nand\u001fz\nnm(!) are calculated in terms of the eigensystem\nfor the total spin, giving\n\u001f\u0007\u0006\nnm(!) =(\u0000i)2\u0019X\nivhijS\u0007\nnjvihvjS\u0006\nmjiiP(Ei)\n\u0002[1\u0000P(Ev)]\u000e(!+Ei\u0000Ev); (24a)\n\u001fz\nnm(!) =(\u0000i)2\u0019X\nivhijSz\nnjvihvjSz\nmjiiP(Ei)\n\u0002[1\u0000P(Ev)]\u000e(!+Ei\u0000Ev); (24b)9\ndI/dV (arb. units)\n0.970.980.9911.011.02(a) (b)\nB=Bz B=Bx0T5T\n3T1T7T\n0T5T\n3T1T7T\n0.6\n−8−4 04 8\nbias voltage (mV)Fe/CuN\nS=2\nD=-1.55, \nE=0.31 (meV)Γ=100 μeV\n50 μeV\n30 μeV\n5 μeV10 μeV\n11.41.8\n−8−4 04 8\nbias voltage (mV)−8−4 04 8\nbias voltage (mV)(c)\nFIG. 2: (Color online) dI=dV for Fe/CuN under di\u000berent external magnetic \felds at T= 0:5 K; (a) B=Bz, (b)B=Bx, and\n(c) for di\u000berent \u0000. Plots are o\u000b-set for clarity.\nwherei(v) denotes the initial (intermediate) state,\nwhereasP(x) accounts for the population in the corre-\nsponding state. Eq. (24) provides the general qualitative\nfeatures of the spin-spin correlation function, and shows\nthat there will appear steps in dI=dV whenever the bias\nvoltage matches the transition energy \u0006(Ei\u0000Ev). In the\ncase of a spin dimer, for instance, there appear steps in\ndI=dV when the bias voltage supports the inelastic tran-\nsitions between the single and triplet states. In case of\ne.g. Fe/CuN, the calculated results are plotted in Fig. 2,\nfor a non-magnetic tip and substrate, showing that the\npresented theory reproduces the results discussed in Ref.\n28 and shows an excellent agreement with experiments.5,6\nSignatures of the excitations in the experimental mea-\nsurements do have a \fnite width, which corresponds to\nthat the intermediate states have \fnite lifetimes. Replac-\ning the delta functions in Eq. (24) by Lorentzian funtions\n1=(x2+\u00002), we phenomenologically include the (uniform)\nlifetime ~=\u0000 for all intermediate states. In case of a single\nFe on CuN with spin S= 2,6and the anisotropy param-\neters given in Tab. I, we \fnd that the Fe spin is weakly\ncoupled (\u0000\u001810\u000030\u0016eV) to the Cu(100) through the\nCuN layer, see 2 (c). This value is extracted by compar-\ning the ratio between the maximal and minimal conduc-\ntance with the experimental result ( \u00181=2). Similarly,\nwe also extract the widths of the single Fe ( S= 3=2)\nand Co (S= 1) adsorbed onto Pt(111) surface.8In Fig.\n3 (a) and (b), we have plotted d2I=dV2for Fe (upper)\nand Co (lower) and we \fnd the best correspondence with\nTABLE I: Anisotropy parameters, DandE, used for the con-\nductance plots given in Fig. 2 (c) and 3, the best widths, \u0000,\nfor the intermediate states in the spin-spin correlation func-\ntions.\nS D (meV) E (meV) \u0000 (meV)\nFe/CuN 2 -1.55 0.31 0 :01\u00000:03\nFe/Pt(111) 3/2 -3.25 0 5\nCo/Pt(111) 1 -10 2 10experiments using the parameters given in I. In Fig. 3\n(c), we \fnally provide a computation of the Fe dimer on\nPt(111) reported in Ref. 8. Because of the presence of\ntwo Fe atoms, we make use of space dependence of the\ntunneling rate and the decay length \u0015, and plot the com-\nputedd2I=dV2for di\u000berent values of the decay length.\nIn comparison with the experimental results, the decay\nlength of 0.5 \u0017A gives the best agreement, suggesting a\nvery rapid spatial decay of the spin-dependent tunneling\nrate. In the \fgure, we for reference have also included\nthe resulting d2I=dV2in absence of the broadening.\nIn the calculations we have estimated the popula-\ntion numbers P(Ei(v)) of the statesji(v)iinvolved in\nthe spin-spin correlation functions, c.f Eq. (24), by\nmaking use of the following observation. By expand-\ning the correlation function e.g. \u001f+\u0000\nnm(t;t0) in terms\nof its eigenstates, we can write it as \u001f+\u0000\nnm(t;t0) =P\nivhijS\u0000\nnjvihvjS+\nmjii(\u0000i)h(dy\nidv)(t)(dy\nvdi)(t0)iwheredy\ni\n(di) creates (annihilates) a particle in the state jii.\nIn the atomic limit, we can employ the decou-\nplingh(dy\nidv)(t)(dy\nvdi)(t0)i=hdy\ni(t)di(t0)ihdv(t)dy\nv(t0)i=\nP(Ei)[1\u0000P(Ev)]ei(Ei\u0000Ev)(t\u0000t0), where the population\nnumberP(Ei) =hdy\nidiican be estimated by using e.g.\nthe Gibbs distribution P(E) =e\u0000\fE=P\nie\u0000\fEi.\nE. Probing speci\fc spin excitations\nWe observe in Eqs. (18) and (24) that the transi-\ntion sequenceshijS+\nnjvihvjS\u0000\nmjii,hijS\u0000\nnjvihvjS+\nmjii, and\nhijSz\nnjvihvjSz\nmjiiare associated with di\u000berent projections\nof the spin-resolved LDOS in the tip and substrate.\nThe sum and di\u000berence of the \frst two sequences cou-\nple ton(\")N(\"F)\u0000m(\")\u0001M(\"F) andmz(\")N(\"F)\u0000\nn(\")Mz(\"F) cos\u0012, respectively, while the last sequence\ncouples ton(\")N(\"F)+m(\")\u0001M(\"F). The di\u000berent cou-\nplings re\rect an ability to enhance or attenuate the re-\nsponse of certain inelastic transitions, at will, by using\ndi\u000berent combinations of electronic and magnetic densi-\nties in the tip and substrate.10\nIn STM without spin-polarization ( m;M= 0), for in-\nstance, the excitation spectrum can be analyzed to cer-\ntain detail by means of applying an external magnetic\n\feld e.g. along the z-direction, of the spin quantization\naxis of the sample. Such application of the magnetic\n\feld introduces a Zeeman splitting of the levels, which\nthus leads to a separation of the peaks in the d2I=dV2.\nTo be speci\fc, consider a single adatom with S= 1,\nand anisotropy \felds D < 0 andE= 0, which is de-\nscribed by the states jS= 1;\u001b= 0;\u00061i, whereE\u001b<E 0\natB= 0. In this system, the leading contribution to\nthe term in dI2=dV containing\u001f\u0000+\nnm(!)+\u001f+\u0000\nnm(!), c.f Eq.\n(18), is proportional toP\n\u001b=\u00061[P(E\u001b)\u0000P(E0)][f(eV\u0000\nE\u001b+E0)\u0000f(eV+E\u001b\u0000E0)], while the term contain-\ning\u001f\u0000+\nnm(!)\u0000\u001f+\u0000\nnm(!) vanishes. The former contribu-\ntion generates steps in the conductance at bias volt-\nageseV=\u0006(E\u001b\u0000E0). Those steps are separated by\njE+1\u0000E\u00001jwhenever the states j1;0iandj1;\u001biare un-\nequally occupied. Hence, the spin splitting imposed by\nthe external magnetic \feld is detectable through sepa-\nrate but equally high steps in the conductance. This\nsplitting can be seen in Fig. 4, where we plot d2I=dV2of\nthis scenario, where the lowermost curve correspond to\nnon-magnetic conditions, whereas the second curve from\nbelow is obtained with B=BzandB= 1.\nUsing SP-STM opens further possibilities in the studies\nof spin systems, since then the term in dI2=dVcontaining\n\u001f\u0000+\nnm(!)\u0000\u001f+\u0000\nnm(!) is \fnite. In case of the S= 1 adatom,\nthe leading contribution to this term is proportional\ntoP\n\u001b=\u00061\u001b[P(E\u001b) +P(E0)\u00002P(E\u001b)P(E0)][f(eV\u0000\nE\u001b+E0)\u0000f(eV+E\u001b\u0000E0)], where it is important\nto notice that the contributions to this term have op-\nposite signs. Hence, by combining the electronic and\nmagnetic densities in the tip and substrate such that\nmzN\u0000nMzcos\u0012 < 0, this term leads to an attenu-\nated (intensi\fed) signal from the transition j1;0ih1;+1j\n−0.8−0.400.40.8\n−20 −10 0 10 20−0.8−0.400.40.8\nbias voltage (mV)Γ=10.0 meV\n    7.5 meV\n       5.0 meV\n       2.5 meVFe/Pt(111)\nS=3/2\nD=-3.25, E=0 (meV)(a)d2/dV2 (arb. units) d2/dV2 (arb. units)Co/Pt(111)\nS=1\nD=-10, E=0 (meV)\nΓ=10.0 meV\n    7.5 meV\n       5.0 meV\n       2.5 meV(b)\n−75 −50 −25 025 50 75−1−0.500.51(c)\nbias voltage (mV)d2/dV2 (arb. units)\nλ=0.5 Å \nλ=2.5 Å λ=1.5 Å 2Fe/Pt(111)\nS=2×3/2\nD=-3.25, E=0 (meV)\nJ=18 meV\nΓ=5 meV\nFIG. 3: (Color online) d2I=dV2for (a) Fe/Pt(111) and (b)\nCo/Pt(111) for di\u000berent widths \u0000, and (c) 2Fe/Pt(111) for\ndi\u000berent decay lengths \u0015, atT= 4:3 K. In panel (c), d2I=dV2\nwith zero broadening and \u0015= 0:5\u0017A is plotted for reference.\n−1 0 1−0.400.4\nbias voltage (mV)d2I/dV2 (arb. units)0.81.2\n2B=1, mz/n=0.5, Mz/N=0.75, θ=4π/5\nB=1, mz/n=0.5, Mz/N=0.75, θ=π/5\nB=1, mz/n=0.5, Mz/N=0,      θ=0\nB=1, mz/n=0,    Mz/N=0,      θ=0\nB=0, mz/n=0,    Mz/N=0,      θ=0FIG. 4: (Color online) d2I=dV2for a spinS= 1 impurity, for\nvarious conditions of the external magnetic \feld B, magne-\ntizations of the tip ( m) and substrate ( M), and the relative\nangle (\u0012) between the magnetization directions. Plots o\u000b-set\nfor clarity.\n(j1;0ih1;\u00001j), c.f. third and fourth curves from below\nin Fig. 4. By the same token, the signal is intensi\fed\n(attenuated) when mzN\u0000nMzcos\u0012>0, see uppermost\ncurve in Fig. 4. While the details of the terms containing\nthe sum and di\u000berence of the spin-spin correlation func-\ntions\u001f\u0000+\nnmand\u001f+\u0000\nnmdi\u000ber from system to system, the\ngeneral conclusion we can draw out of this observation\nis that the intensity of the signal from any speci\fc tran-\nsition depends on the magnetic densities of the tip and\nsubstrate, and on their relative orientation.\nF. Maximizing the signal from \ructuations\nFinally, we notice that the conductance dI0=dV van-\nishes fornN+m\u0001M= 0, which corresponds to the\ncase with a half-metallic tip and substrate such that their\nmagnetic moments are in anti-parallel alignment. In this\nset-up also the conductance dI1=dV = 0, which implies\nthat the measured signal is generated solely by dI2=dV.\nAs can be seen in Eq. (18), this conductance only de-\npends on the transverse components, i.e. the sum and\ndi\u000berence of \u001f\u0000+\nnmand\u001f+\u0000\nnm, since the term containing\n\u001fz\nnmis proportional to nN+m\u0001M(= 0). Therefore,\ndespite the presence of possible thermal noise, such a\nset-up would bene\ft from a very low current noise since\nmost of the noise would be related to the spin \ructua-\ntions, that is, the noise we want to measure. This can be\nseen by identifying the spin-dependent current operator\nwith\u000e^I(t) =T1S(t)\u0001s, where s=P\npk\u001b\u001b0cy\np\u001b\u001bck\u001b0. The\ncurrent-current correlation function is then given by31,32\nh\u000e^I(t)\u000e^I(t0)i=T2\n1s\u0001hS(t)S(t0)i\u0001s, where we average over\nthe dynamics of the localized spins and over the ensem-\nble of the tunneling electrons. Under the condition that11\nnN+m\u0001M= 0, the total dc current Iis proportional\ntoT2\n1Rt\n\u00001s\u0001hS(t)S(t0)i\u0001sd(t\u0000t0), and since the shot\nnoise is approximately hI2\nshot(!)i\u0018I, the signal-to-noise\nratio is about unity.\nV. CONCLUSIONS\nThe theory we propose here for STM/STS mea-\nsurements (essentially) contains the well-known Terso\u000b-\nHamann term,10,24,25Eq. (12), which maps the energy\ndependence electronic and magnetic densities of states,\nfor energies de\fned by the Fermi level, of the substrate,\nand the applied bias voltage. For the situation where\nmagnetic impurities are located on the substrate, our the-\nory also contains a contribution which is proportional to\nthe magnetic moment of the local spins, and allows for\na quantitative analysis of spin moments using SP-STM.\nThis is, hence, di\u000berent from the currently most common\nanalysis, which is solely focused on the energy depen-\ndent spin-polarization of the DOS. We show here that\nstudies and analyses of both the dynamical conductance,\n@I(V;t)=@V or@I(V;!)=@V, and the total conductance\n@I(V)=@V can be linked to the dynamical, hSz(t)ior\nhSz(!)i, and total,R\nhSz(!)id!=2\u0019, magnetic moments,\nrespectively, of the local spins. This contribution gen-\nerates di\u000berent response at the local spins depending on\ntheir relative orientation compared to the spin moment\non the SP-STM tip. To our knowledge, this contribution\nhas not been discussed before, and the direct relation\nbetween the di\u000berential conductance and the local spin-\nmoment is expected to make a signi\fcant impact on the\npotential capabilities in using SP-STM, and to extract\nquantitative information about local spin moments.\nWe also point out that despite the present di\u000eculties\nto experimentally record the time-dependent component\nin the conductance in a high GHz or even THz regime,\nit is important to observe that the dynamical component\nto the conductance opens new possibilities for the gen-\neration of high frequency ac currents and/or voltages.\nThus, by using local spin moments adsorbed onto metal-lic surface, such a scenario requires at least one mag-\nnetic electrode which produces a local magnetic \feld that\ncan interact with the local spin moment and, thus, pro-\nvide a time-dependent net contribution to the tunneling\ncurrent, and conductance. Having in mind anisotropy\n\felds from the substrate acting on the spin moment,\nit would be possible to generate spin-dependent ac cur-\nrents/voltages in the THz regime, since the anisotropy\n\felds may be of the order of meV.\nIt is important to note that our description goes be-\nyond the treatment reported in e.g. Refs. 23,24, since\nwe also include e\u000bects from the local spin-spin interac-\ntion between de-localized spin built up by the tunneling\nelectrons, and the localized spin of the adatom. These\nspin-spin interactions are included already in the tunnel-\ning matrix element, c.f. Eq. (1), and describe that the\ntunneling electrons of di\u000berent spins are subject to dif-\nferent tunneling barriers, i.e. spin-dependent tunneling\nbarriers.\nThe theory presented here for obtaining quantitative\nspin-resolved information in a SP-STM experiment pro-\nvides an alternative to the method used in Ref. 26, where\na mean-\feld model based on the Weiss molecular \feld39\nwas used. Future work, primarily of experimental nature,\nwill judge which approach is the most reliable one.\nAcknowledgments\nJ.F. and O.E. thank the Swedish Research Council\n(VR) and the Royal Swedish Academy of Sciences\n(KVA), and A.V.B. thanks US DOE, for \fnancial\nsupport. J.F. thanks LANL for hospitality during his\nvisit in 2008. Special thanks to A. Bergman, S. Bl ugel,\nL. Nordstr om, B. Gy or\u000by, and W. Wulfhekel for valuable\ndiscussions, and to M. Bode and R. 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Lett. 103,\n176601 (2009)."    },    {        "title": "1004.1352v2.Spin_charge_and_spin_orbital_coupling_effects_on_spin_dynamics_in_ferromagnetic_manganites.pdf",        "content": "arXiv:1004.1352v2  [cond-mat.str-el]  12 May 2010Spin-charge and spin-orbital coupling eﬀects\non spin dynamics in ferromagnetic manganites\nDheeraj Kumar Singh, Bhaskar Kamble, and Avinash Singh∗\nDepartment of Physics, Indian Institute of Technology Kanpur\nCorrelation-induced spin-charge and spin-orbital coupli ng eﬀects on spin dynam-\nics in ferromagnetic manganites are calculated with realis tic parameters in order\nto provide a quantitative comparison with experimental res ults for spin stiﬀness,\nmagnon dispersion, magnon damping, anomalous zone-bounda ry magnon softening,\nand Curie temperature. The role of orbital degeneracy, orbi tal ordering, and orbital\ncorrelations on spin dynamics in diﬀerent doping regimes is h ighlighted.\nPACS numbers: 75.30.Ds,71.27.+a,75.10.Lp,71.10.Fd2\nI. INTRODUCTION\nThe role of charge and orbital ﬂuctuations on magnetic couplings an d excitations is\nof strong current interest in view of the several zone-boundary anomalies observed in\nspin-wave excitation measurements in the metallic ferromagnetic ph ase of colossal mag-\nnetoresistive manganites.1–7The presence of short-range dynamical orbital ﬂuctuations\nhas been suggested in neutron scattering studies of ferromagne tic metallic manganite\nLa1−x(Ca1−ySry)xMnO3.7These observations areof crucial importancefor a quantitative u n-\nderstanding of the carrier-induced spin-spin interactions, magno n excitations, and magnon\ndamping, and have highlighted possible limitations of existing theoretic al approaches.\nFor example, the observed magnon dispersion in the Γ-X direction sh ows signiﬁcant\nsoftening near the zone boundary, indicating non-Heisenberg beh aviour usually modeled by\nincluding a fourth neighbour interaction term J4, and highlighting the limitation of the\ndouble-exchange model. Similarly, the prediction of magnon-phonon coupling as the origin\nof magnon damping2and of disorder as the origin of zone-boundary anomalous softenin g8\nhave been questioned in recent experiments.4–7Furthermore, the dramatic diﬀerence in the\nsensitivity of long-wavelength and zone-boundary magnon modes o n the density of mobile\ncharge carriers has emerged as one of the most puzzling feature. Observed for a ﬁnite range\nof carrier concentrations, while the spin stiﬀness remains almost co nstant, the anomalous\nsoftening and broadening of the zone-boundary modes show subs tantial enhancement with\nincreasing hole concentration.4,5\nThe role of orbital degeneracy, orbital ordering and orbital corr elations on spins dynam-\nics in ferromagnetic manganites in the entire hole-doping range 0 < x<∼0.5 is still not\nfully understood due to lack of quantitative comparisons of theore tical calculations with\nexperimental results in manganites, as highlighted in the recent rev iew.5\nMost of the earlier investigations including correlation-induced O(1 /S) quantum cor-\nrections to the magnon spectrum were carried out in the strong-c oupling (double-exchange)\nlimitandcouldnotsatisfactorilyaccountfortheobservedanomalou szone-boundarymagnon\nsoftening.9Whereas, withtypicalparametervalues t∼0.2−0.5eVforthemobile egelectrons\nand 2J∼2 eV for their exchange coupling to the localized t2gcore spins, the intermediate-\ncoupling regime J∼Wappears to be more appropriate for the ferromagnetic mangantie s.10\nRecent theoreticalstudies intheintermediate-coupling regimeand including theCoulomb3\nrepulsion between band electrons have demonstrated some realist ic features such as doping\ndependent asymmetry of the ferromagnetic phase and enhanced magnon softening and non-\nHeisenberg behaviour, thereby highlighting the importance of corr elated motion of electrons\non spin dynamics.11,12However, these investigations do not include orbital degeneracy a nd\ninter-orbital Coulomb interaction, and do not provide any quantita tive comparisons with\nexperimental results on manganites. Earlier theoretical investiga tions of the role of orbital-\nlattice ﬂuctuations and correlations on magnetic couplings and excit ations have mostly been\nlimited to ferro orbital correlations,13and have not addressed the physically relevant stag-\ngered orbital correlations.\nA self-consistent investigation of the interplay between spin and or bital orderings has\nrecently carried out using a two-orbital ferromagnetic Kondo latt ice model (FKLM) includ-\ning the Jahn-Teller coupling,14and doping dependence of the calculated Curie temperature\nwas compared with experiments for diﬀerent ferromagnetic manga nites. Finite Jahn-Teller\ndistortion and orbital ordering were shown to be self-consistently generated at low doping.\nHowever, again only ferro orbital ordering was included in this analys is.\nRecent investigations of the correlated motion of electrons using a non-perturbative,\ninverse-degeneracy-expansion based, Goldstone-mode-prese rving approach have highlighted\ntheroleofspin-chargeandspin-orbitalcouplingeﬀectsonmagnet iccouplingsandexcitations\nin orbitally degenerate metallic ferromagnets.15–18While spin-charge coupling was indeed\nfound to yield strong magnon softening, damping, and non-Heisenb erg behaviour, it was\nonly on including orbital degeneracy, inter-orbital interaction, an d a new class of spin-\norbital coupling diagrams that low-energy staggered orbital ﬂuct uations, particularly with\nmomentum near ( π/2,π/2,0)corresponding to CE-type orbital correlations, were found18to\ngenericallyyieldstrongintrinsicallynon-Heisenberg(1 −cosq)2magnonselfenergycorrection\nin three dimensions, resulting in no spin stiﬀness reduction, but stro ngly suppressed zone-\nboundary magnon energies in the Γ-X direction.\nInthis paper we will apply these new results speciﬁcally to thecase of ferromagnetic man-\nganites. We will consider a two-orbital FKLM including intra- and inter -orbital Coulomb\ninteractions, with realistic values of hole bandwidth, lattice paramet er, and Hund’s coupling\netc. Such a quantitative comparison should provide, within a physica lly transparent the-\noretical approach, insight into the role of orbital ordering, orbita l degeneracy, and orbital\ncorrelations on spin dynamics in ferromagnetic manganites, seamles sly covering the entire4\nhole-doping range 0 < x<∼0.5 within a single theoretical framework.\nThe outline of the paper is as follows. Spin-charge coupling eﬀects on spin dynamics in\na single-band FKLM are presented in Section II, highlighting the stro ng diﬀerences from\nthe canonical double-exchange behaviour. Spin stiﬀness and spin- wave energies are then\nobtained in the orbitally degenerate state of a two-orbital FKLM in S ection III, and com-\npared with neutron-scattering results for the optimally ferromag netic manganites ( x≈0.3).\nInter-orbital interaction Vis included next andSection IV describes spin dynamics in theor-\nbitally ordered ferromagnetic state. The hole-doping ( x) dependence of the estimated Curie\ntemperature is compared with experimental results for two diﬀere nt cases corresponding to\nwide- and intermediate-band manganites. Finally, the role of ( π/2,π/2,0)-type orbital cor-\nrelations on zone-boundary magnon softening and instability of the FM state near x= 0.5\nare discussed in section V.\nII. SPIN-CHARGE COUPLING MAGNON SELF ENERGY\nThe interplay between itinerant carriers in a partially ﬁlled band and loc alized magnetic\nmoments is conventionally studied within the ferromagnetic Kondo lat tice model (FKLM):\nH=−t/summationdisplay\n/angbracketleftij/angbracketrightσa†\niσajσ−J/summationdisplay\niSi.σi (1)\ninvolving a local exchange interaction between the localized spins Siand itinerant electron\nspinsσi. Due to crystal-ﬁeld splitting of the degenerate Mn 3 dorbitals into egandt2glevels,\nthe partially ﬁlled band in ferromagnetic manganites corresponds to the mobile egelectrons\nwhereas the three Hund’s-coupled localized t2gelectrons yield the localized magnetic mo-\nments with spin quantum number S= 3/2.\nWe will ﬁrst consider the single-band FKLM, which is appropriate for t he orbitally de-\ngenerate state of the optimally doped ferromagnetic manganites n ear hole doping x≈0.3.\nAs the doped holes are shared equally by the two degenerate orbita ls, this hole doping cor-\nresponds to band ﬁlling n= (1−x)/2≈0.35 in each band. Orbital ordering and orbital\ncorrelations will be incorporated later within a two-orbital FKLM inclu ding intra-orbital\nand inter-orbital Coulomb interactions. Throughout, we will consid er a simple cubic lattice\nwith lattice parameter a.5\nβ αα\n(b)\nα\nβα\n(c)ββ\nαβ\n(a)α\nFIG. 1: The ﬁrst-order quantum corrections to the irreducib le particle-hole propagator φ(q,ω).\nThe ﬁrst-order magnon self energy for the single-band FKLM, res ulting from quantum\ncorrections to the irreducible particle-hole propagator φ(q,ω) shown in Fig. 1, was recently\nobtained as:17\nΣmagnon(q,ω) =J2(2S)/summationdisplay\nQ/integraldisplaydΩ\n2πi/parenleftBigg\n1\nΩ+ω0\nQ−iη/parenrightBigg/summationdisplay\nk/bracketleftBigg/parenleftBigg\n1\nǫ↑+\nk−q+Q−ǫ↑−\nk+ω−Ω−iη/parenrightBigg\n×1\n2S/parenleftBigg\n1−2JS\nǫ↓+\nk−q−ǫ↑−\nk+ω−iη/parenrightBigg2\n (2)\nwhereω0\nQrefers to the bare magnon energies, ǫσ\nk=ǫk−σJSare the exchange-split band\nenergies, and the superscripts + /−indicate particle ( ǫσ\nk> ǫF) and hole ( ǫσ\nk< ǫF) energies.\nIncorporating correlation-induced self-energy and vertex corr ections, the above magnon self\nenergy represents a spin-charge coupling between magnons and c harge excitations in the\npartially-ﬁlled majority-spin band, with the coupling vertex explicitly v anishing for q= 0,\nthus ensuring that the Goldstone mode is explicitly preserved. This r esult was obtained by\napplying the systematic inverse-degeneracy (1 /N) expansion scheme to a purely fermionic\nrepresentation for the FKLM which allows conventional many-body diagrammatic expan-\nsion.\nThe imaginary part of the magnon self energy in the strong-coupling limit:\n1\nπImΣmagnon(q,ω) =/parenleftbigg1\n2S/parenrightbigg2/summationdisplay\nQ/summationdisplay\nk(ǫk−q−ǫk+ω)2δ(ǫ↑+\nk−q+Q−ǫ↑−\nk+ω+ω0\nQ) (3)\nyields ﬁnite magnondamping and linewidth at zero temperature, arisin g frommagnondecay\ninto intermediate magnon states accompanied with majority-spin ch arge excitations. On\nthe other hand, at the classical level (random phase approximatio n), magnon damping is6\n 0 0.01 0.02 0.03 0.04 0.05\n 0 0.2 0.4 0.6 0.8  1D (0)\nnS=3/2W=12t(a)J/W=1/6\n 1/4\n 1/2\n 1\n 0 0.01 0.02 0.03 0.04 0.05\n 0 0.2 0.4 0.6 0.8  1D\nn(b)\nFIG. 2: Comparison of the bare (a) and renormalized (b) spin s tiﬀness for the single-band FKLM,\nshowing the substantial spin stiﬀness reduction due to the sp in-charge coupling eﬀect.\npossible only due to decay into the Stoner continuum, and is therefo re completely absent in\nthe intermediate and strong coupling regimes where the magnon spe ctrum lies well within\nthe Stoner gap.\nThe small- qbehaviour of the magnon self energy in Eq. (2) yields the ﬁrst-orde r quantum\ncorrection to spin-stiﬀness in ddimensions:\nD(1)= Σ(1)(q)/q2=1\nd(2S)2/summationdisplay\nQ,k(∇ǫk)2\nǫ↑+\nk−q+Q−ǫ↑−\nk+ω0\nQ(4)\nwhich is down by a factor (1 /2S) relative to the bare (classical) spin stiﬀness:\nD(0)=ω(0)\nq/q2=1\nd(2S)/bracketleftbigg1\n2/angbracketleft∇2ǫk/angbracketright−/angbracketleft(∇ǫk)2/angbracketright\n2JS/bracketrightbigg\n. (5)\nThe two competing terms above of order tandt2/Jcorrespond to delocalization energy\nloss and exchange energy gain upon spin twisting, and determine the overall stability of\nthe ferromagnetic state against long wavelength ﬂuctuations. Th e spin stiﬀness quantum\ncorrection D(1)is only weakly dependent on J(through ω0\nQ), and involves only an exchange\ncontribution, which is suppressed for electronic spectral distribu tions characterized by dom-\ninant saddle-point behaviour ( ∇ǫk= 0), resulting in enhanced spin stiﬀness D=D(0)−D(1)\nand ferromagnetic stability.\nIn the double-exchange limit ( J→ ∞), only the delocalization part of the classical spin\nstiﬀness survives. This contribution has a particle-hole symmetric p arabolic band-ﬁlling\ndependence. Its behaviour with electron ﬁlling nis identical to that with hole doping\nx, vanishing at both ends n= 0 and n= 1 and increasing symmetrically with added\nelectrons or holes, leading to the conventional understanding with in the DE model that7\n 0 0.1 0.2 0.3 0.4 0.5Magnon Energy and Damping\nqJ =0.7W\nS = 3/2\nΧ Μ R Γn = 0.35\nΓω0\nq\nωq\nΓqx10\nFIG. 3: The bare and renormalized magnon energies and magnon damping for the single-band\nFKLM. The magnon damping linewidths are nearly one-tenth of the renormalized magnon energies\nin the Γ-X direction.\nferromagnetism increases with added carriers. However, both th e classical and quantum\nexchange contributions involving the ( ∇ǫk)2terms in Eqs. (4) and (5) break this particle-\nhole symmetry, and the stiﬀness actually vanishes at n <1 before the band is completely\nﬁlled.\nSince the spin stiﬀness quantum correction involves only an exchang e contribution, in-\ncludingquantumcorrectionisthereforeeﬀectively equivalent toen hancing thebareexchange\ncontribution by decreasing J. Fig. 2 shows the renormalized spin stiﬀness D=D(0)−D(1)\nin units of ta2, evaluated from Eqs. (4) and (5) for diﬀerent values of J. For band ﬁllings\nbelown= 0.5, the renormalized spin stiﬀness for J∼Wis indeed seen to be close to the\nbare spin stiﬀness for J/t= 4.\nFigure 3 shows a comparison of the bare and renormalized magnon dis persions in units of\nt, andalso themagnondamping obtained forthe single-band FKLM. Be sides the substantial\nreductionoftherenormalizedmagnonenergies, wealsoﬁndthatint heintermediatecoupling\nregime (J∼W), themagnon self energy becomes increasingly non-Heisenberg like , resulting\ninωX< ωR/3, as indeed observed. The magnon linewidth Γ qobtained from the imaginary\npart of the magnon self energy increases with momentum q, but the ratio Γ q/ωqis found to\nremain nearly constant at ∼1/10upto the zone boundaryin theΓ-Xdirection, inagreement\nwith magnon linewidth measurements.7In the X-M, M-R, and R-Γ directions, we ﬁnd the\nratio Γ q/ωqto be smaller than 0.1.8\n 0 50 100 150 200\n 0 0.2 0.4 0.6 0.8  1D  (meV Å2)\nnJAF=2.8meV\nS=3/2t=240meV J/W=1/2\n 1\n 2\nFIG. 4: The renormalized spin stiﬀness calculated for a two-o rbital model, including an AF contri-\nbution due to the Mn superexchange. For J≈W, the calculated values are close to the measured\nstiﬀness ∼170 meV ˚A2for the wide-band compound La 1−xSrxMnO3.\nThespin-chargecouplingeﬀect onmagnonexcitationscanbereadily extended tothetwo-\nband FKLM involving interaction term −J/summationtext\niSi.(σiα+σiβ) in Eq. (1). Fig. 4 shows the\nrenormalized spin stiﬀness for this two-band model obtained by dou bling the spin stiﬀness\nresultD=D(0)−D(1)from Eqs. (4,5), corresponding to the independent contributions\nof the two degenerate egorbitals. Here we have taken t= 240meV and lattice parameter\na= 3.87˚A for manganites. Corresponding to the AF superexchange intera ction between Mn\nspins, an AF contribution to spin stiﬀness D(0)\nAF=zJAFSa2/6 was also subtracted from the\nabove result with JAF= 2.8meV and lattice coordination z= 6.\nNear optimal band ﬁlling n≈0.35 (hole doping x= 1−2n≈0.3) for ferromagnetic man-\nganites, the calculated spin stiﬀness for J≈Wis in close agreement with the experimentally\nmeasured values ∼170 meV ˚A2for the wide-band compound La 1−xSrxMnO3. The strong\nsuppression near n=0.5 (x=0) seen above with decreasing Jcould also be a contributing\nfactor towards destabilizing ferromagnetism in manganites as xapproaches zero, in addition\nto orbital correlations.\nFig. 5 shows a comparison of the calculated magnon dispersion for th e two-band FKLM\nalong diﬀerent symmetry directions in the Brillouin zone with experimen tal data for the\nnarrow-band compound La 0.7Ca0.3MnO3obtained from neutron scattering measurements.5\nThe renormalized magnon energy was obtained using ωq= 2(ω0\nq−Σq)−zJAFS(1−γq). A\nrelatively smaller hopping term t= 180meV (bandwidth W= 12t≈2eV) was taken in the\ncalculation for this narrow-band compound. Ab-initio calculations als o yield an estimated9\n 0 20 40 60 80 100 120ωq  (meV)\nqJ =0.7W\nS = 3/2\nΧ Μ R Γ Γn = 0.35\nJAF = 2.1meVt=180meV experimental\ncalculated\nFIG. 5: Comparison of calculated renormalized magnon energ y dispersion with experimental data\nfor the narrow-band compound La 0.7Ca0.3MnO3from neutron scattering measurements.5\nbandwidth W= 2eV for La 1−xCaxMnO3.19The renormalized magnon energies are broadly\nin good agreement with the neutron scattering measurements. In cluding the zone-boundary\nmagnon softening resulting from the spin-orbital coupling eﬀect dis cussed later will improve\nthe agreement near the X point.\n 0 0.5 1 1.5 2\n 0 50 100 150 200 250 300 350 400〈Sz〉\nT (K)JAF=2.1meVS=3/2J=0.7W\nt=180meVn=0.35\nFIG. 6: Temperature dependence of magnetization calculate d using the self-consistent Callen\nscheme with the same set of parameters as above. The calculat ed Curie temperature is close\nto the measured values for narrow-band manganites.\nFrom the renormalized magnon energies obtained as above, the ﬁnit e-temperature spin\ndynamics was investigated using the self-consistent Callen scheme in which the magnetiza-\ntion for a quantum spin- Sferromagnet:20\n/angbracketleftSz/angbracketright=(S−Φ)(1+Φ)2S+1+(S+1+Φ)Φ2S+1\n(1+Φ)2S+1−Φ2S+1, (6)10\nwhere the boson (magnon) occupation number:\nΦ =1\nN/summationdisplay\nq1\neβ˜ωq−1(7)\nin terms of the thermally renormalized magnon energies:\n˜ωq=ωq/angbracketleftSz/angbracketright/S (8)\nFigure6showsthetemperaturedependence ofmagnetizationobt ainedbyselfconsistently\nsolving the coupled set of equations (6)-(8). With the same set of p arameters as in Fig. 5\nfor the magnon dispersion ﬁt, the calculated Curie temperature ( ∼280K) is close to the\nmeasured value ∼250K for LCMO.\nAsorbitalcorrelationsarerelativelyunimportantattheseﬁllings, t heorbitallydegenerate\nFKLM calculation presented here provides a good description of fer romagnetic properties.\nOrbital ordering and ﬂuctuations due to inter-orbital interaction and Jahn-Teller distortion\nbecome important near quarter ﬁlling ( x∼0) and near x∼0.5, and therefore must be\nincluded as both strongly suppress ferromagnetism. Eﬀects of or bital ordering and orbital\nﬂuctuations on spin dynamics are discussed below.\nIII. ORBITALLY ORDERED FERROMAGNETIC STATE\nWeakly doped manganites such as La 1−xSrxMnO3and La 1−xCaxMnO3exhibit an or-\nbitally ordered state for x<∼0.2, as inferred from x-ray diﬀraction and neutron scattering\nexperiments.7,21In this section, we will therefore consider spin dynamics in an orbitally or-\ndered ferromagnet with staggered orbital ordering, and examine the behaviour of the Curie\ntemperature with the onset of orbital ordering in the low-doping re gime. We therefore\nconsider a two-orbital correlated FKLM:\nH=−t/summationdisplay\n/angbracketleftij/angbracketrightσµa†\niσµajσµ−J/summationdisplay\niµSiµ.σiµ+U/summationdisplay\niµniµ↑niµ↓+V/summationdisplay\niniαniβ (9)\ncorrespondingtothetwo egorbitalsµ=α,βpersite, andincludeintra-orbital( U)andinter-\norbital(V)Coulombinteractions. Withincreasing V, theorbitallydegenerate ferromagnetic\nstate becomes unstable towards an orbitally ordered ferromagne tic state with staggered\norbital ordering near half-ﬁlling (in the majority-spin band). In the pseudo-spin space of the\ntwo orbitals, the orbitally-ordered state is exactly analagous to th e antiferromagnetic state11\nof the Hubbard model with staggered spin ordering.22For simplicity, we consider staggered\norbital ordering in all three directions, although the end compound LaMnO 3exhibits only\nplanar staggered orbital ordering.\nThe Jahn-Teller-phononic term is also considered to be important in m anganites, espe-\ncially in the low and intermediate doping range.23However, the inter-orbital Coulomb inter-\naction hasbeen suggested to bemuch stronger thanthe electron -phonon coupling in order to\naccount for the observed insulating behaviour in undoped manganit es above the Jahn-Teller\ntransition and the bond length changes below it.24,25Generally, Coulombic and Jahn-Teller-\nphononic approaches for manganites have been shown to be qualita tively similar.26Indeed,\na mean-ﬁeld treatment of the Jahn-Teller term14yields an electronic exchange-ﬁeld term in\norbital space proportional to the orbital magnetization /angbracketleftniσα−niσβ/angbracketright, exactly as would be\nobtained from the inter-orbital interaction term.\nWe consider an orbitally-ordered ferromagnetic state with stagge red orbital ordering:\n/angbracketleftniα↑/angbracketrightA=/angbracketleftniβ↑/angbracketrightB=n+M/2\n/angbracketleftniβ↑/angbracketrightA=/angbracketleftniα↑/angbracketrightB=n−M/2 (10)\nwhere the staggered orbital order Mcharacterizes the density modulation on the two sub-\nlattices A and B. For simplicity, we consider a saturated (half-metallic ) ferromagnetic state\nwith empty spin- ↓bands:/angbracketleftniα↓/angbracketright=/angbracketleftniβ↓/angbracketright= 0. The ferromagnetic ordering is chosen to be\nin the ˆzdirection.\nThe eﬀective spin couplings and magnon energies are again determine d from the particle-\nhole propagator, now evaluated in the orbitally ordered state. The magnon energies are\nobtained as:\nωq=J2(2S)[λ(0)−λ(q)] (11)\nwhereλ(q) is the maximum eigenvalue of the transverse spin propagator:\nχ−+(q,ω) =/summationdisplay\nµ=α,β[χ0\nµ(q,ω)]\n1−U[χ0\nµ(q,ω)](12)\nin terms of the bare particle-hole propagators [ χ0\nµ(q,ω)] for the two orbitals µ, which are\n2×2 matrices in the two-sublattice basis, with [ χ0\nβ]AA/BB= [χ0\nα]BB/AAfollowing from the\norbital-sublattice symmetry in the orbitally-ordered state. The ba re particle-hole propaga-12\ntors are obtained by integrating out the fermions in the orbitally ord ered ferromagnetic state\ndescribed by the eﬀective single-particle Hamiltonian matrix:\nH0\nµσ(k) = Γσ\n1 0\n0 1\n+\n−µσ∆σǫk\nǫkµσ∆σ\n (13)\nwithintheself-consistent ﬁeld(Hartree-Fock)approximation. He reΓ↑=−JS+VnandΓ↓=\nJS+(U+V)narethe orbitallyindependent ﬁelds with energy diﬀerence Γ ↓−Γ↑=Un+2JS\ncorrespondingtotheusualexchangebandsplitting, and∆ ↑=VM/2and∆ ↓= (U−V)M/2\naretheself-consistently determined exchange ﬁelds correspond ing to theorbital ordering M.\nThe spin ( σ) and orbital ( µ) indices in the matrix are + and −for spins ↑and↓, and orbitals\nαandβ, respectively.\nThe eigenvalues and eigenvectors of the above Hamiltonian matrix yie ld the bare-level\nband-electron energies and amplitudes for orbital µand spin σ:\nE0\nkµσ= Γσ±/radicalBig\n∆2σ+ǫ2\nk\na2\nkµσ⊖=1\n2/parenleftBigg\n1+µσ∆σ/radicalbig\n∆2σ+ǫ2\nk/parenrightBigg\n=b2\nkµσ⊕\nb2\nkµσ⊖=1\n2/parenleftBigg\n1−µσ∆σ/radicalbig\n∆2σ+ǫ2\nk/parenrightBigg\n=a2\nkµσ⊕ (14)\nwhere⊕and⊖refer to the two eigenvalue branches ( ±). Orbital ordering splits the electron\nbands with energy gaps 2∆ ↑=VMand 2∆ ↓= (U−V)Mfor the two spins. At quarter\nﬁlling (n= 1/2), the spin- ↑lower band ⊖iscompletely ﬁlled andtheupper band ⊕isempty,\nyielding a ferromagnetic insulator, and a metal-insulator transition o ccurs when orbital\nordering melts, either driven by band overlap with decreasing Vor when the temperature\nexceedstheorbitalmeltingtemperature. Inoursaturatedferr omagneticstatewith /angbracketleftni↓/angbracketright= 0,\nboth branches of the spin- ↓electron band are above the Fermi energy.\nThe similarity of the orbitally-ordered state with the antiferromagn etic state of the Hub-\nbard model results not only in similar expressions for quasiparticle en ergies and amplitudes\nas above, but also in an identical self-consistency condition:\n1\nV=/summationdisplay\nk↑⊖1\n2/radicalBig\n∆2\n↑+ǫ2\nk(15)\nwhich implicitly determines the magnitude of the orbital order m= 2∆↑/Vas a function of\nthe eﬀective interaction strength V. When only nearest-neighbor hopping is present, orbital13\n 0 50 100 150 200 250 300 350 400\n 0 0.1 0.2 0.3 0.4 0.5Tc (K)\ndoping (x)(a)\norbitally\norderedt=180 meV\norbitally\ndegenerateV=6\nV=0\nexpt\n 0 50 100 150 200 250 300 350 400\n 0 0.1 0.2 0.3 0.4 0.5Tc (K)\ndoping (x)(b)orbitally\norderedt=240 meV\norbitally\ndegenerateV=5\nV=0\nexpt\nFIG. 7: Comparison of the doping ( x) dependence of calculated Tcwith experiments for the\ntwo compounds (a) La 1−xCaxMnO3and (b) La 1−xSrxMnO3. Staggered orbital ordering sharply\nsuppress ferromagnetism due to band narrowing and reduced e lectronic delocalization.\nordering at quarter ﬁlling sets in for any positive Vdue to Fermi-surface nesting, whereas a\nﬁnitecriticalinteractionstrengthisrequiredwhenfrustratingne xt-nearest-neighbor hopping\nterms are included.\nFor ﬁnite doping xaway from quarter ﬁlling, the staggered orbital order parameter mde-\ncreases rapidly and vanishes at a critical doping value xc. Although long-range orbital order\nmay be susceptible to orbital ﬂuctuations at ﬁnite doping in the same way as the staggered\nspin ordering is in the doped antiferromagnet,31this HF result does provide a measure of the\nlocal orbital order. The pseudogap structure associated with sh ort-range orbital order in the\nmetallic phase has been observed in recent high-resolution scanning tunneling microscopy\nand spectroscopy measurements of La 0.7Sr0.3MnO3and La 0.625Ca0.375MnO3thin ﬁlms.27\nForJ∼WandU∼W, the Hund’s coupling contribution 2 JSto the exchange band\nsplitting typically dominates over the intra-orbital (Hubbard) inter action contribution Um.\nConsequently, the calculated magnon energies depend rather wea kly onU, and we have\ntherefore dropped it for simplicity in the calculations presented belo w.\nFigure 7 shows a comparison of the doping dependence of the calcula ted tran-\nsition temperature Tcwith experiments for the two compounds La 1−xCaxMnO3and\nLa1−xSrxMnO3.28,29Assuming a Heisenberg form ωq=zJS(1−γq) for the magnon spec-\ntrum, the transition temperature Tc= (1/3)zJS(S+ 1)//summationtext\nq(1−γq)−1= (5/6)ωXwas\ncalculated approximately from the magnon energy at the X point ( π,0,0) obtained from14\nEq. (11). An AF superexchange interaction JAFbetween the neighboring Mn core spins was\nalso included which correspondingly reduces the magnon energy. St aggered orbital ordering\nwas found to persist up to x∼0.25, as indicated by the dashed vertical lines, and is clearly\nseen to sharply suppress ferromagnetism due to band narrowing a nd reduced electronic de-\nlocalization. In the orbitally degenerate state obtained beyond this critical doping value,\nthe calculated Curie temperature is indicated by V= 0. As earlier, we have taken hop-\nping energies t= 180meV and t= 240meV for the narrow-band and wide-band compounds\nLa1−xCaxMnO3and La 1−xSrxMnO3, corresponding to bandwidths ∼2eV and ∼3eV, re-\nspectively. Other parameters are V/t= 6 and 5, J AF= 2.8meV and 2.1meV for the two\ncompounds. Also, J/t= 4 in both cases, as the bare magnon energy in the orbitally degen-\nerate case then approximately corresponds to the renormalized m agnon energy (Fig. 2) for\nJ∼Wonincluding thespin-charge coupling eﬀect asdiscussed insection II .The anomalous\nzone-boundary magnon softening observed in narrow-band comp ounds will sharply reduce\nTcasxapproaches 0.5.\nThe above analogy with the antiferromagnet can be readily extende d to inter-orbital\nﬂuctuations and orbital waves (orbiton) in analogy with transvers e-spin ﬂuctuations and\nspinwaves.22Thisanalogythendirectly yieldstheorbitonexcitationspectrum zJS/radicalbig1−γ2q\nin terms of the orbital exchange coupling J ≡4t2/Vin the strong-coupling limit, quantum\nreduction in the orbital order Mdue to quantum orbital ﬂuctuations, orbital order-disorder\ntransition at Tc∼(1/3)zJS(S+1)//summationtext\nq(1−γ2\nq)−1/2in the strong-coupling limit ( V≫t)\nand orbital-order melting induced metal-insulator transition in the we ak-coupling limit.\nSimilarly, hole motion in an orbitally ordered ferromagnet will lead to scr ambling of the\nlocal orbital order, string of broken orbital bonds, incoherent h ole spectral function and\nquasiparticle and band-gap renormalizations due to multiple orbiton e mission-absorption\nprocesses, in analogy with corresponding results for the AF.30Insights into the stability of\ntheorbitally-orderedstateatﬁnitedopingawayfromquarterﬁlling cansimilarlybeobtained\nfromcorrespondingresults forthedopedantiferromagnet.31However, ifrealisticinter-orbital\nhoppingterms tαβareincluded, thecontinuousorbital-rotationsymmetryisbroken, resulting\nin a gapped orbiton spectrum, making the orbitally ordered state int rinsically diﬀerent.15\nIV. CE-TYPE ORBITAL CORRELATIONS AND ANOMALOUS MAGNON\nSOFTENING\nThe orbitally degenerate ferromagnetic metallic state of manganite s also typically be-\ncomes unstable near hole doping x= 0.5 due to onset of CE-type orbital correlations. The\nrole of such planar staggered orbital correlations with momentum n ear (π/2,π/2,0) on spin\ndynamics was recently investigated for a two-orbital Hubbard mod el:\nH=−t/summationdisplay\n/angbracketleftij/angbracketrightσ(a†\niασajασ+a†\niβσajβσ)+U/summationdisplay\ni(niα↑niα↓+niβ↑niβ↓)+/summationdisplay\niVniαniβ (16)\nincluding intra and inter orbital Coulomb interactions UandV. In the saturated ferromag-\nnetic state (band ﬁlling nequal to magnetization m), electron correlation eﬀects were incor-\nporated within the framework of a non-perturbative Goldstone-m ode-preserving approach.18\nThe magnon self energy due to coupling between spin and orbital ﬂuc tuations [Fig. 8] was\nshown to generically yield strong intrinsically non-Heisenberg (1 −cosq)2momentum depen-\ndence, implying no spin stiﬀness reduction but strongly suppressed zone-boundary magnon\nenergies in the Γ-X direction. In this section we will extend these res ults to the two-orbital\ncorrelated FKLM as in Eq. (9) with similar manganite parameters as us ed in previous\nsections.\nThecorrelation-inducedspin-orbital coupling magnonself energy f orthetwo-orbitalHub-\nbard model was obtained as:\nΣsp−orb(q)≈m2/angbracketleft[Γsp−orb(q)]2/angbracketrightQ′,Ω′\nΩspin+Ωorb−iη(17)\nwhere Ω orband Ω spinrepresent characteristic orbital and spin ﬂuctuation energy sca les, the\nangular brackets /angbracketleft /angbracketrightrefer to averaging over the orbital ﬂuctuation modes Q′. The spin-\norbital interaction vertex [Γ sp−orb(q)], represented diagrammatically in Fig. 8, was shown to\nexplicitly vanish at momentum q= 0 in accordance with the Goldstone mode requirement,\nand yields the dominant qdependence of the magnon self energy.\nAlthough the same overall structure is obtained for the correlate d FKLM, there are\ntwo essential diﬀerences. First, the magnon self energy correct ion for the FKLM is given by\nΣ(n)\nmagnon=J2(2S)φ(n)intermsofthecorrepondingquantumcorrection φ(n)totheirreducible\nparticle-hole propagator (instead of U2mφ(n)as for the Hubbard model). Second, since the\nbare particle-hole propagator χ0for the mobile electrons in the FKLM involves the total16\nU U α α↑\n↓V Vβαβ\n+ + ≡↑ ↑ ↑ ↑\n↓ ↓ ↓ ↓ ↓V\nV\nVU(a)↑ ↑↑\n(b)↑Γ Γ\nΓ\nFIG. 8: The spin-orbital coupling diagrams for the irreduci ble particle-hole propagator (a) can be\nrepresented in terms of a spin-orbital interaction vertex Γ sp−orb, the three diagrammatic contri-\nbutions to which are shown in (b) involving three-fermion ve rtices. The missing fourth diagram\nvanishes because of the assumption of complete polarizatio n.\nexchange splitting Um+2JS, the spin-ﬂuctuation propagator χ0/(1−Uχ0) corresponding\nto theUladders as in Fig. 8 yields a gapped spectrum with characteristic gap e nergy∼2JS\n(instead of the O( t) magnon energy scale for the Hubbard model).\nConsequently, theﬁrst-orderspin-orbitalcouplingmagnonselfe nergiesforthetwomodels\nare approximately related by:\nΣFKLM\nsp−orb(q) = ΣHM\nsp−orb(q)×/parenleftbiggJ22S\nU2m/parenrightbigg/parenleftbiggt\n2JS/parenrightbigg\n. (18)\nForU= 20t,m≈0.3(astakenintheHubbardModelcalculation18)andJ≈W, therelative\nfactor is about 1/10. For simplicity, we have therefore obtained th e spin-orbital coupling\nmagnon self energy for the FKLM from the earlier Hubbard model re sult by multiplying by\nthis relative factor.\nDoubling the magnon self energy to account for the independent co ntributions of the two\ndegenerate egorbitals, the renormalized magnon energy ωq=ω0\nq−Σsp−orb(q) for the two-\norbital correlated FKLM is shown in Fig. 9 for diﬀerent hole dopings. H ereω0\nqis the bare\nmagnon energy for the two-band FKLM including the negative contr ibutionzJAFS(1−γq)\nof the superexchange interaction JAF= 2.1meV between Mn core spins. This bare magnon\nenergy was evaluated for J= 4 as it then approximately corresponds to the renormalized17\n 0 5 10 15 20 25 30 35\n 0 0.5 1 1.5 2 2.5 3ωq (meV)\nqV/t = 4\nJ ≈ W\nt = 180 meVωq 0\nx=0.0\n0.2\n0.3\n0.4\n0.5\nFIG. 9: Renormalized magnon energies ωq=ω0\nq−Σsp−orb(q) including the spin-orbital coupling\nmagnon self energy, plotted along the Γ-X direction for diﬀer ent hole dopings, showing the anoma-\nlous zone-boundary magnon softening due to coupling of spin excitations with CE-type, period 4 a,\nplanar orbital correlations with momentum modes near ( π/2,π/2,0).\nmagnon energy (Fig. 2) for J≈Won including the spin-charge coupling eﬀect as discussed\nin section II.\nThe above analysis clearly shows the importance of CE-type stagge red orbital correla-\ntions on the observed anomalous zone-boundary magnon softenin g. Indeed, a sharp onset\nof CE-type orbital correlations at momentum ( π/2,π/2,0) has been observed in neutron\nscattering studies near the Curie temperature,7which gradually diminishes in intensity and\nsharpness with decreasing Ca concentration in La 1−x(CaySr1−y)xMnO3crystals. Interest-\ningly, this behaviour is exactly similar to the resistivity temperature p roﬁle of these crystals\nwith varying Ca concentration.32Taken in conjunction with our spin-orbital coupling theo-\nretical result of strong zone-boundary magnon softening produ ced by such CE-type orbital\ncorrelations,18this sharp onset of orbital correlations near Tcwould imply a sudden magnon\nsoftening and collapse of the ferromagnetic state. As the spin-or bital coupling magnon self\nenergy∼(1−cosq)2does not renormalize the spin stiﬀness, this behaviour could also ac-\ncount for the ﬁnite spin stiﬀness observed just below Tcin the low-bandwidth materials,\nwhich has so far been a puzzling feature.\nIn the earlier Hubbard model calculation, the spin-orbital interact ion term [Γ sp−orb]2was\ncalculated for the orbital ﬂuctuation mode Q′= (π/2,π/2,0) and an averaging factor of18\n1/10 was included, as obtained on averaging over all orbital ﬂuctua tion modes assuming\na ﬂat distribution. As ﬂuctuation modes near ( π/2,π/2,0) become increasingly prominent\ndue to onset of CE type combined charge-orbital correlations nea r half doping (stabilized\nby an inter-site Coulomb interaction V′ninj), the enhanced weight of such staggered orbital\ncorrelations will substantially increase the magnon self energy and h ence the anomalous\nmagnon softening as xapproaches 0.5.\nV. CONCLUSIONS\nVarious aspects of spin dynamics in ferromagnetic manganites were theoretically investi-\ngated in terms of a realistic two-orbital FKLM including intra- and inte r-orbital Coulomb\ninteractions, staggered orbital ordering, and orbital correlatio ns. For the same set of man-\nganite parameters, the calculations are in close agreement with rec ent neutron scattering\nexperiments onspinstiﬀness, magnondispersion, magnonlinewidth, Curietemperature, and\nanomalous magnon softening. The set of manganite parameters ta ken were lattice parame-\ntera= 3.87˚A, hopping terms t= 240meV and t= 180meV (corresponding to bandwidths\nW≈3eV and ≈2eV, respectively) for the wide-band (LSMO) and narrow-band (L CMO)\ncompounds, Hund’s coupling J≈W, inter-orbital Coulomb repulsion V/t≈5, and Mn spin\nsuperexchange interaction JAF≈2.5meV.\nIn the orbitally degenerate ground state appropriate for the opt imally doped ferromag-\nnetic manganites with x∼0.3, the spin-charge coupling magnon self energy for the FKLM\nwas shown to yield substantial suppression of ferromagnetism in th e physically relevant ﬁ-\nniteJregime. Incorporating this magnon energy renormalization, we obt ained spin stiﬀness\nD∼150meV˚A2, Curie temperatures Tc∼300K, and the ratio of the magnon linewidth\nto magnon energy Γ q/ωq∼1/10 for magnon modes upto the zone boundary in the Γ-X\ndirection, and the calculated magnon dispersion ﬁtted well with rece nt neutron scattering\ndata for La 1−xCaxMnO3.\nThe onset of orbital ordering at low doping ( x<∼0.25) due to the inter-orbital interaction\nVwas found to sharply suppress Tcdue to band narrowing eﬀects associated with the\nstaggered orbital order. The doping dependence of the calculate d Curie temperature for\ntwo bandwidth cases corresponding to narrow-band and wide-ban d compounds was in close\nagreement with the observed behaviour of Tcfor LCMO and LSMO.19\nFinally, the role of CE-type planar orbital ﬂuctuations with momentu m near (π/2,π/2,0)\nwas investigated on spin dynamics in the orbitally degenerate ferrom agnet near half doping\n(x∼0.5). The magnon self energy due to correlation-induced spin-orbita l coupling arising\nfrom inter-orbital interaction and ﬂuctuation was evaluated for t he correlated FKLM from\ntheearlier Hubbardmodel result. Forthesamesetofmanganitepa rameters, thespin-orbital\ncoupling magnon self energy with U∼Wwas found to yield anomalous zone-boundary\nmagnon softening of the same magnitude as observed in neutron sc attering experiments.\nOnset of the CE-type charge-orbital correlations and the spin-o rbital coupling are therefore\nevidently responsible for the instability of the ferromagnetic state near half doping.\n∗Electronic address: avinas@iitk.ac.in\n1H. Y. Hwang, P. Dai, S-W. Cheong, G. Aeppli, D. A. Tennant, and H. A. Mook, Phys. Rev.\nLett.80, 1316 (1998).\n2P. Dai, H. Y. Hwang, J. Zhang, J. A. Fernandez-Baca, S.-W. Che ong, C. Kloc, Y. Tomioka,\nand Y. Tokura, Phys. Rev. B 61, 9553 (2000).\n3T. Chatterji, L. P. Regnault, and W. Schmidt, Phys. Rev. B 66, 214408 (2002).\n4F. 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Kilian, Phys. Rev. B 61, 3494 (2000).\n14M. Stier and W. Nolting, Phys. Rev. B 75, 144409 (2007).\n15A. Singh, Phys. Rev. B 74, 224437 (2006).\n16S. Pandey and A. Singh, Phys. Rev. B 75, 064412 (2007); ibid.76, 104437 (2007); ibid.78,\n014414 (2008).\n17S. Pandey, S. K. Das, B. Kamble, S. Ghosh, D. K. Singh, R. Ray, a nd A. Singh, Phys. Rev. B\n77, 134447 (2008).\n18D. K. Singh, B. Kamble, and A. Singh, Phys. Rev. B 81, 064430 (2010).\n19G. Trimarchi and N. Binggeli, Phys. Rev. B 71, 035101 (2005).\n20H. B. Callen, Phys. Rev. 130, 890 (1963).\n21M. Pissas, I. Margiolaki, G. Papavassiliou, D. Stamopoulos , and D. Argyriou, Phys. Rev. B 72,\n064425 (2005).\n22A. Singh and Z. Teˇ sanovi´ c, Phys. Rev. B 41, 614 (1990); A. Singh, Phys. Rev. B 43, 3617\n(1991); A. Singh, cond-mat/9802047 (1998).\n23A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett.74, 5144 (1995).\n24P. Benedetti and R. Zeyher, Phys. Rev. B 59, 9923 (1999).\n25S. Okamoto, S. Ishihara, and S. Maekawa, Phys. Rev. B 65, 144403 (2002).\n26T. Hotta, A. L. Malvezzi, and E. Dagotto, Phys. Rev. B 62, 9432 (2000).\n27U. R. Singh, A. K. Gupta, G. Sheet, V. Chandrasekhar, H. W. Jan g, and C.-B Eom, Appl.\nPhys. Lett. 93, 212503 (2008); U. R. Singh, Ph.D. Thesis, IIT Kanpur (2009) .\n28Y. Tokura and Y. Tomioka, J. Magn. Magn. Mater. 200, 1 (1999).\n29S.-W. Cheong and H. Hwang, Ferromagnetism vs. Charge/Orbital Ordering in Mixed-Valen t\nManganites , chapter in “Colossal Magnetoresistive Oxides”, edited by Y. Tokura (Gordon and\nBreach, Monographs in Condensed Matter Science, 1998).\n30P. Srivastava and A. Singh, Phys. Rev. B 70, 115103 (2004).\n31A. Singh and H. Ghosh, Phys. Rev. B 65, 134414 (2002).\n32Y. Tomioka, A. Asamitsu, and Y. Tokura, Phys. Rev. B 63, 024421 (2000)."    },    {        "title": "1704.07758v2.Unified_Treatment_of_Spin_Torques_using_a_Coupled_Magnetisation_Dynamics_and_Three_Dimensional_Spin_Current_Solver.pdf",        "content": "1 \n Unified treatment of s pin torques using a c oupled magnetisation \ndynamics and t hree-dimensional spin current solver  \n \nSerban Lepadatu* \nJeremiah Horrocks  Institute for Mathematics, Physics and Astronomy, University of Central \nLancashire, Preston PR1 2HE, U.K.  \nAbstract  \n \nA three -dimensional spin current solver based on a generalised spin drift -diffusion \ndescription , including the spin Hall effect, is integra ted with a magnetisation dynamics  \nsolver . The resulting model is shown to si multaneously reproduce the spin -orbit  torques \ngenerated using  the spin Hall effect, spin pumping torques generated by magnetisation \ndynamics in multilayers, as well as the spin transfer torques acting on magnetisation  regions \nwith spatial gradients, whilst field-like and spin -like torques are reproduced in a spin valve \ngeometry. Two approaches to modelling interfaces are analysed, one based on the spin \nmixing conductance and the other based on continuity of spin currents where the spin  \ndephasing length governs the  absorption of transverse spin  components . In both cases \nanalytical formulas  are derived for the spin-orbit  torques in a heavy metal / ferromagnet  \nbilayer geometry, showi ng in general both field -like and damping -like torques  are generated. \nThe limitations of the analytical approach are discussed , showing that even in a simple \nbilayer geometry, due to the non -uniformity of the spin currents, a full three -dimensional \ntreatme nt is required.  Finally the model is applied to the quantitative analysis of the spin Hall \nangle in Pt by reproducing published experimental data on the ferromagnetic resonance \nlinewidth in the bilayer geometry . \n \n* SLepadatu@uclan.ac.uk  2 \n I. Introduction  \n \nThe study of spin torques is currently of great interest due to applications to magnetic \ndevices, including race -track memory [ 1] and magnetic tunnel junction devices for memory \nand spin logic applications [ 2]. Spin transport in magnetic multilayers is a c omplex and \nintensely researched topic, with many  sources  of spin torques identified, ranging from bulk \nspin transport phenomena  including the spin Hall effect  (SHE) [ 3], to interfacial Rashba -type \nspin transfer phenomena  [4] including inverse spin galvanic  effects [ 5] and intrinsic spin -\norbit torques originating from the Berry phase [ 6]. The usual  approach to  analysing the effect \nof various types of spin torques on the magnet isation structure and dynamics is based on \nintroducing  separate analytical formulat ions for the spin torques within a magnetisation \ndynamics model. As devices become more complex, involving several layers and multiple \nsources of spin torques, more advanced approaches  are required capable of resolving the non -\nlocal nature of spin transpor t and non -uniformity of spin currents, which  lead to a complex \nspatial dependence of the spin torques that cannot be fully accounted for within an analytical \nformulation. An alternative approach describes t he flow of charges and spins using a  drift-\ndiffusion model  [7-10]. The incorporation of a drift -diffusion formulation  within a \nmicromagnetics model  is currently of great interest [11-16]. An important source of spin \ncurrents which needs to be included within a three-dimensional model of magnetisation \ndynamics is the SHE . This effect was first predicted by Dyakonov and Perel [ 3], where due to \nspin-orbit interaction [ 17] an electrical curre nt results in transverse  flow of spins with \npolarisation perpendicular to both the charge and spin current directions [18]. The SHE was \ndemonstrated using a number of techniques, including field-swept ferromagnetic resonance \n(FMR ) [19,20], spin torque FMR [ 21-23], optical FMR [24], time -resolved optical techniques \n[25] and electrical methods [26]. The SHE -generated spin polarisation results in  a torque 3 \n when injected in a ferromagnetic layer [27,28], which makes this effect very important for \nmagnetic devices, resulting in motion of domain walls  [29] with important applications to \nsynthetic antiferromagnetic domain wall devices  [30,31]. The reverse effect also exists where \na spin current, typically generat ed through spin pumping from  a ferromagnetic layer, results \nin a transverse displacement of charges through the inverse SHE [ 32-34]. An important \nparameter that characterizes both the SHE and its inverse is the spin Hall angle. This has been \nmeasured in various heavy metals – for reviews see Refs. [ 35,36] – but remains a topic of \ncontention; in particular for Pt there is a large d isparity in report ed results, spanning a range \nof around 30 times from ~0.004 to over 0.1.  The spin pumping effect itself also generates a \nspin torque on a dynamically excited magnetisat ion texture due to loss of spin angular \nmomentum [ 37]. \nHere the SHE and spin pumping effect are incorporated within a three -dimensional \ngeneralised drift -diffusion model , and coupled to a magnetisation dynamics solver. The \nimplementation of the model is pre sented in Section II and shown to self -consistently \nreproduce a number of spin torques within the same description, namely the SHE -generated \nspin-orbit  torques (SOT), torques due to loss of spin ang ular momentum via spin pumping, \nspin transfer torques (STT ) [38] arising in the presence of magnetisation structures with \nspatial gradients, as well as the field -like and Slonczewski spin torque s in spin valve \ngeometries [ 39,40]. The SOT in a bilayer geometry is analysed in some detail in Section III, \nobtaining analytical expressions  for two approaches to modelling spin transfer between \nlayers: one based on the spin mixing conductance at the interface [ 41], and the other based on \ncontinuity of spin currents, with the absorption of transverse spin components governed by \nthe spin dephasing length ; the limitations of the analytical approach  are also discussed . \nFinally the model is applied to the analysis of FMR linewidth in Section IV, obtaining 4 \n estimations of the spin Hall angle in Pt based on both the spin mixing conductance and spin \ndephasing length absorption approaches.  \n \nII. Spin Drift -Diffusion Model  and Implementation  \n \n The flow of charge s and spin s in a multi -layered structure can be described  as [18,7]: \n nDeDeDe\nBe SHA\nBe D C    S mS E J\n  \n(1)  \n \n  nDeD nDeeB\nSHA e e DB\nS    Eε S m E J  \n\n  \n(2)  \n \n \nHere the convention used by Dyakonov [ 18] has been adopted, where JS is a rank -2 tensor \nsuch that JSij signifies the flow of the j component of spin polarisation  in the direction i.. JC is \nthe usual electrical current density and JS is the spin polarisation current density , but will be \nimproperly referred to as the spin current for brevity; JS can be converted to spin current by \nmultiplication with \nB2/ . Equation  (1) contains the usual Ohm’s law term, where  is the \nelectrical conductivity,  as well as a term due to the giant magneto -resistance contribution \narising in cur rent perpendicular to the plane (CPP -GMR)  stacks [7,8], where  De is the \nelectron diffusion constant,  D is the diffusion spin polarisation , m is the magnetisation \ndirection and S is the spin accumulation.  The last two terms in equation  (1) are the inverse \nspin Hall effect, where SHA is the spin Hall angle,  and diffusion of charges due to charge \ndensity  gradients, where n is the volume charge density ; note  = n, where  is the electron \nmobility. Equation  (2) contains three spin current contributions : i) the flow of spins carried \nby a spin -polarised  charge current  in a magnetic layer , either due to an external electric field \nE or diffusion of char ges, where  is the charge  current spin polarisation , ii) diffusion of \nspins due to local spin accumulation g radients, and iii) spin current generated by the SHE , 5 \n where  is the rank -3 unit anti -symmetric tensor . Due to the complexity of the problem the \nimplementation of equations  (1) and (2) is split into two parts:  here we concentrate only on \nthe direct SHE which is responsible for generating spin to rques on a ferromagnetic layer. The \nimplementation of the inverse SHE will be addressed in a  forthcoming publication; without \nthis term charge density  gradients may be ignored and equation  (1) is curl -free, thus the usual \nrelations .Jc = 0 and E = -V apply, with V being the electrical potential.  \nThe exchange  interaction between the spin accumulation and local magnetic moment \nresults in a torque on the magnetisation . Only the transverse component of the spin \naccumulation (meaning transverse to the magnetisation direction) generates a torque. In order \nto conserve total spin angular momentum the transverse spin components are quickly \nabsorbed, thus the relaxation of the longitudinal and transverse spin accumulation s are \ngoverned by different length scales . The decay of the longitudinal spin accumulation is a \ndiffusive process governed by the spin-flip length  sf, whilst the relaxation of the transverse \nspin accumulation is a ballistic process occurring on a much shorter length scale.  One of \nthese length scales is the exchange rotation length  \nJDe J / , where J is the exchange \ninteraction energy  strength  [7,8]. Another important length scale is the spin dephasing length  \nwhich governs the decay of transverse spin accumulation components.  This is given by  \nL J ll/\n, where l and lL are the spin coherence and spin precession lengths \nrespectively  [9]. The equation of mo tion for spin accumulation  is now given  by: \n\n\n\n\n \n2 2 2.\n mS m mS SJS\nJ sfe SDt\n  \n(3)  \n \n \nThe torque on the magnetisation is obtained using the same arguments given in Ref. [ 8], by \nconsidering the conservation of total spin angular momentum ; thus in the steady state where \nS/t = 0, this is obtained from the divergence of the spin current as:  6 \n \nSm m Sm TS 2 2\n e\nJe D D  \n(4)  \n \n \nThe equation of motion for magnetisation is now a modified Landau -Lifshitz -Gilbert (LLG) \nequation containing the additional total spin torque as : \nSTmm Hmm\nSeffMt t1 \n  \n(5)  \n \n \nHere \ne0 , where  \n/B e g  is the electron gyromagnetic ratio,  Ms is the saturation \nmagnetisation, and Heff contains all the usual effective field contributions, typically including \ndemagnetising, direct exchange and applied field contributions .  \nThe response time -scales of m and S are separated typically by 3 orders of magnitude \n(ps vs fs time -scales respectively) thus equations  (3) and (5) may be evaluated separately. \nExplicitly , the following computational procedure  has been implemented in the finite -\ndifference micromagnetics -oriented software Boris [ 42,43]. Using the relati ons .Jc = 0 and \nE = -V, the following Poisson equation is obtained from equation  (1): \nmS  ..2\nBe D eD VV\n\n  \n(6)  \n \n \nIn general the conductivity is allowed to vary within the same material  even for uniform \ncharge density , for example due to anisotropic MR in magnetic layers [ 43], however it is not \nincluded in this work.  Equation  (6) is evaluated for a given spin accumulation  and fixed \npotential boundary conditions on two electrodes. Using the calculated  electrical potential, the \ncharge current density is obtained using equation  (1) and substituted in equation  (2) to obtain  \nthe spin current density as:  \n 7 \n \nCB\nSHA e e CB\nSeD DeεJ S m mS m J JD       \n(7)  \n \n \nFinally from equations  (3) and (7) the spin accumulation equation of motion is rewritten in \nterms of the charge current density as: \n    \n\n\n\n\n   \n2 2 22. . . .\n \n  \nmS m mS SSεJ mS mm mS m JS\nD\nJ sfe eCB\nSHA e CB\nD DeDe t\n  \n(8)  \n \n \nEquation  (8) is solved to obtain the steady state spin accumulation by setting  S/t = 0. For \nthe spatial discretization a multi -level multi -grid method is used  [44], with equation  (5) \nevaluated on a coarse mesh, whilst equations  (6) and (8) are evaluated on a  sufficiently  \nrefined sub -mesh ; all meshes use rectangular prism cells with the z cellsize  independent of \nthe xy-plane cellsize.  \nFor a multi -layered geometry it is important to consider both the  interface  and mesh  \nboundary conditions.  Boundary conditions for evaluating differential operators are derived \nfrom the physically motivated requirements that both the charge  and spin currents \nperpendicular to a mesh boundary  not containing an electrode are zero: JC.n = 0 and JS.n = 0 \n[17], where n is the boundary  normal. In this case we obtain the following Neumann \nboundary conditions from equations  (1) and (2): \n nεJ nSn\n.e.0.\nB SHA\nC\neDV\n\n  \n(9)  \n \n \nFor boundaries containing an electrode V is specified on the boundary thus V.n is also \nprescribed. The spin current perpendicular to an electrode is not zero in general  and \nelectrode -containing boundaries need special consideration to ensure physically valid results. 8 \n One general principle is to define the problem geometry and electrical contacts such that the \nspin accumulation gradients no rmal to the electrodes are zero (in particular it may be \nnecessary to allow the spin accumulation itself to decay to zero by including in the model a \nsufficiently large part of the electrical contacts) – again  we can use ( S).n = 0; for magnetic \nregions th is furthe r requires the magnetisation be uniform around the electrodes.   \nIn the transparent interface limit where specular scattering  can be neglected , values of \nV and S at the interface  cells can be derived by enforcing  the continuity of  both JC.n and JS.n \n[8,45]. In this picture  the absorption of transverse spin components is governed by the length -\nscales J and φ. An alternative approach is that of magnetoelectric circuit theory [ 41], where \nthe absorption of transverse spin components is confined to the interface  and modelled via the \ncomplex spin mixing cond uctance  G. The boundary conditions  for the charge and spin \ncurrents  at a normal metal ( N) / ferromagnet (F)  interface are written as:  \n\n  \n  m mmV nJV m V m m nJ nJmV nJ nJ\nV G G G GeG GeG G V G G\nSB\nF SS SB\nF S N SS F C N C\n   \n   \n. .Im Re2. .. . .\n\n  \n(10)  \n \n \nHere V is the potential drop across the N/F interface (V  = VF – VN) and VS is the spin \nchemical potential drop, where \nS VB e S e D/ / , and G, G are interface conductances \nfor the majority and minor ity spin carriers respectively. Equation (10) together with e quations \n(1) and (7) are used to calculate the potential and spin accumulation either side of the \nboundary; the transverse spin current absorbed at the interface then gives rise to a torque \nwhich may be included in the magnetic cells at the interface with cellsize dh, in addition to \nany other torques resulting from transverse spin accumulation in Eq. (4), as: \n 9 \n \n  S S\nhB erface\nS G GedgV m V m m T    Im Reint  \n(11)  \n \n \nThe Onsager reciprocal process to absorption of transverse spin currents is the generation of \nspin currents via dynamical magnetisation processes, e.g. magnetisation precession, known as \nspin pumping [ 37,46]. This is given in e quation (12), where g = (h/e2)G, and may be \nincluded on the N side of e quation (10) when  calculatin g the spin chemical potential  drop. \n\n\n  \ntgtgB pump\nSm mm J Im Re2\n  \n(12)  \n \n \nThe implemented model is now applied to a N/F bilayer geometry, similar to that used \nin FMR experiments [ 20,21]. The diffusion spin polarisation , D, is set to zero for this \ngeometry – for completeness the effect of spin torques in a CPP -GMR stack is addressed in \nAppendix A. For the N and F layer s parameters associated with Pt and Ni80Fe20 (Py) are used  \nas in Ref. [20]; SHA is initially set to 0.1.  Additionally for metals De = 10-2 m2/s [47]; from \nequation  (4) it appears the spin torque is proportional to De, however the spin accumulation is \ninversely proportional to De, thus if the two metal layers have similar diffusion constants the \nspin torqu e is independent of De. Using a non -adiabaticity parameter  = 0.04 for Py [ 48], J \n 0.8 nm is determined using the relation \n2 2/sf J , and\nL J ll/   0.6 nm is \nfurther obtained by using the values l = 0.9  nm and lL = 1.4 nm from  Ref. [ 9]. The problem \ngeometry is shown in Figure 1 .  \n 10 \n  \nFigure 1 . Spin accumulation and spin currents in a bilayer geometry.  The bilayer consists of N \n(10 nm) / F (10 nm) layers with electrical contacts at the x-axis ends of the geometry. The computed \nspin accumulation S and perpendicular spin current Jsz are rendered  for uniform magnetisation along \nthe -y direction. The width of the N layer is 160 nm (the rendered plots use different display scaling \nfactors along the z and xy-plane directions respectively for clarity). The decay of the y components of \nS and Jsz are shown for the longitudinal mode (magnetis ation along -y) and transverse mode \n(magnetisation along -x). The transverse components decay on a much shorter length scale governed \nby the spin dephasing length, which defines a narrow spin transfer region where the spin torque is \nexerted on the magnetis ation. For comparison, the transverse spin current is also shown for an \ninterface where the absorption of transverse spin components is modelled via the spin mixing \nconductance G. \n \nAs an initial test t he magnetisation is set along the –y direction so the  spin torque on \nthe F  layer is zero. As shown in Figure 1, t he spin accumulation in the N layer follows the \nright -hand rule around the charge current direction and decays in the F layer  as governed b y \n11 \n \nF\nsf. The SHE -generated spin polarisa tion is perpendicular to both the charge and spin current \ndirections and decays when injected in the F layer, reaching zero at the edges as expected.  \nEquations  (3) and (7) may be solved analytically to obtain the following expression for the \nvalue of spin polarisat ion at the interface in the longitudinal configuration , where dN and dF \nare the thickness values of the N and F layers respectively , and the quantities N and F are \ndefined as  \nN\nsfN\nsf Nd tanh N  / / , \nF\nsfF\nsf Dd tanh F  / / : \n\n\n\n\n\n\n\n \n\nFN\n) cosh(dJeJN\nsf NcxB\nSHA szy 1\n/11\n  \n(13)  \n \n \nIII. Spin Torques  \n \n When the magnetisation is not aligned with the y-axis, due to conservation of total \nspin angular momentum, the absorption of transverse spin components results in a spin \ntorque. For interfaces modelled  using magnetoelectric circuit theory the absorption occurs \npurely at the interface resulting in the interfa cial torque of e quation (11). This sets any \ntransverse spi n components to zero on the F side. For interfaces where the spin current is \ncontinuous, the generation of the  spin torque  in equation (4) is accompanied by a rapid, but \ngradual, absorption of transverse spin components.  These cases are  exemplified in Figure 1 \nwhere , for the transverse configuration in the continuous case , the injected spin polarisation \nrapidly decays within a spin transfer region , dependent principall y on φ. For thin F  layers the \ndirect exchange interaction acts to keep the magnetisation constant along the z direction and \nthus in both cases we may define locally in the xy-plane a net, o r average, spin torque on the \nF layer . First the continuous interface case is analysed. It is known that  both field -like (FL) \nand (DL) torques act on the F layer [ 28]. The total spin torque may be decomposed  into these \ncomponents as shown in equation (14), where < S> is the spin accumulation averaged along 12 \n the z direction, \ny zxσ ˆ ˆˆ , f is a factor dependent on the various length scales, and r is \nthe ratio of the FL and DL torque magnitudes.  \n  FL DL S T Tσmσm m S m m S m T   rdfJ D D\nFszy e\nJe\n2 2\n \n  \n(14)  \n \n \nExpressions for f and r may be derived in this bilayer geometry by solving equations (3) and \n(7) in the transverse configuration  and equating coefficients in equa tion (14). By introducing \nthe length scales  , , +,  defined by  the relations \n2 2 2/1 /1 /1 F\nsf , \n4 4 4/1 /1 /1J\n, \n2 2 2/1 /1 /1   , \n2 2 2/1 /1 /1    and further defining \n    2/2// /1 / /1        i d tanh i CF\n, we obtain : \n \nFor example with the values given above for the N and F layers, f = 1.36 and r = 0.03. In a \nsimpler analytical formulation, as used e.g. in Ref. [ 20], a DL torque  is obtained from \nballistic spin transfer at the interface [ 39] as \nσm m T  F zy DL d/Js . Note that this torque \nis similar to the DL torque of e quation (14). Another approach to calculating interfacial spin \ntorques is through the  spin mixing conductance as in e quation (11). Using the boundary \nconditions of e quation (10), the spin torque acting on the F layer is now obtained as:  \n ,σmσm m TS  G\nFszy GrdJf\n \nwhere\n \n\n,~ ~~ ~Re\n12 22\nGIm GRe NG G N\nFNfG\n\n\n\n\n\n\n    \n(16)  \n \n\n\n\n\n\n\n\n\n2 21\nJGIm GRe\nFNf\n , \n\n\n\n\n\n\n\n\n\n2 2 2 2Re Re\nJ JGIm G GIm Gr  \n, \nwhere \n   2 2 2 22\n/ /11\nJi CIm CRe NC CNG\n  \n  \n   \n(15)  \n 13 \n \n\n2~ ~Re~Im\nG G NG NrG\n\n \n \nHere \n/ 2~G G , noting the above expressions are equivalent to  those obtained in Ref. \n[10]. \n \nFigure 2 . Spin torques in a bilayer geometry with layers of equal width.  Spin torques calculated \nfor a transverse domain wall in the bilayer geometry are shown for (a) SOT only , shown in the middle \nof the track for a wide (200 nm) and narrow (100 nm) domain wall width, (b) SOT only , shown at the \nleft and right edges of the track, and (c) both SOT and STT enabled. The spin torques are shown as \nobtained f rom numerical calculations (solid lines), as well as analytical formulas (solid discs ). The \ndifferences in the torques obtained at the edges and the middle of the track arise due to the non -\nuniformity of the spin current and spin accumulation as indicated by the rendering of S at the top of \nthe figure.  \n \n14 \n Thus the two approaches result in qualitatively identical descriptions of the SHE -generated \nspin torques, and we note they even have identical limits: for the continuous case, taking the \nlimit φ0 (thus the spin torque is generated purely at the interface in this limit) results in an \nidentical torque to that obtained in the limit Re{G}, namely \n0FLT and \n σm m T    ) cosh(d JeN\nsf N cx B SHA DL   / /11 /\n. A quantitative comparison is given \nin Section IV.  \nThe DL and FL torques with constant coefficients obtained above may of course be \nadded to the LLG equation . This approach however does suffer from serious limitations. \nFirstly, due to the non -local nature of spin transport it can become i ntractable to obtain \nexpressions for the torque coefficients  in more complex geometries involving several layers, \nsuch as synthetic antiferromagnetic racetrack designs [ 30,31]. An even more serious \nlimitation comes from the implicit assumption used in the above analysis, namely that the \nspin current incident on the N/F interface is uniform  (for uniform magnetisation) . For the \ngeometry in Figure 1 this is a good approximation, however in general the charge and spin \ncurrents can be non -uniform and also the geometry thickness can vary  for more complex \nthree -dimensional devices , which would result in a complicated spatial dependence of the DL \nand FL torques. To reinforce this point  the simple bilayer geometry is analysed again, but this \ntime the F and N layers have the same width. Due to the rotation of the spin accumulation \naround the charge current direction, as seen in Figure 2, the spin current incident on the F \ninterface is no longer uniform, resulting in a variation  of the DL and FL torques across  the \ntrack.  Figure 2 shows results obtained using the continuous interface, however qualitatively \nidentical results are obtained using  G. First, the total torque with only SHE  enable d is \nobtained in the middle of the track for a transverse domain wall, shown in Figure 2 (a). In this \ncase equation  (14) holds and a good agreement is obtained between the analytical formulas  \nand numerical results . In Figure 2 (a) the torques are calculated for two domain wall widths, 15 \n 200 nm and 100 nm. A slight discrepancy arises for the narrower domain wall due to three -\ndimensional diffusion effects  not captured by the analytical description of equation  (15), \nhowever this effect is small. A much more significant discrepancy arises at the edges of the \nwire, see Figure 2 (b), where the torques are completely different showing  significant FL \ncomponents . \nFor regions with magnetisation gradients STT also act on the magnetisation , given by \nZhang and Li [ 38] as additional terms to the normalised LLG equation:  \n\n211. .\n\n\ne MPB\nSCJumu m mu\n  \n(17)  \n \n \nHere  P is the current spin polarisation , P = (nn)/(n+n) with n and n being the majority \nand minority conduction electron density of states respectively . These torques are obtained \nusing the drift -diffusion equations in the absence of the spin dephasing length by using the \nvalid approximation S  0, see e.g. Ref. [ 16], noting β = P. When the spin dephasing \nlength is included , the approximation S  0 is no longer valid and the non -adiabaticity \nparameter is modified as shown in Ref. [ 9]. For typical domain wall widths in Py however , \nequation  (17) remains a good approximation with  \n2 2/sf J . This is verified in Figure 2 (c) \nwhere both the SOT and STT are enabled. In general the full three -dimensional treatment \nwith the spin torque calculated using the generalised drift -diffusion equations is superior to \nthe approach of incorporating analytical representations of the differen t torques in the LLG \nequation. The computational time i s typically doubled, which is an acceptable cost given the \nincreased accuracy, subtlety and depth of physical effects  which can be modelled, as well as \nthe convenience of a self-consistent approach to modelling spin torques.  \n 16 \n IV. F erromagnetic Resonance for Spin Hall Effect Bilayers  \n \nIt is well known that the DL torque in N/F bilayers modifies the linewidth obtained from \nFMR measurements. This is investigated here using the geometry shown in Figure 3, \nincluding contributions from SHE, Oersted field and spin pumping. Field -swept FMR peaks \nare simulated for bias field along the –y direction and r.f. field along the x-axis as detailed in \nthe Methods section – typical calculated FMR peaks are shown in Figure 3. First, the FMR \npeaks are simulated using the boundary conditions of equation (10), with spin pumping also \nincluded using G = 1015 + i1014 (S/m2) appropriate for Pt/Py interfaces [ 49]. \n \n  \nFigure 3 . Ferromagnetic resonance peaks in a spin Hall effect bilayer.  FMR peaks are shown for \nPt/Py bilayers  at 20 GHz, showing simulated  FMR absorption peaks with fitted Lorentzian peak \nfunctions for different charge current density va lues in the Pt  layer. The bias field and r .f. field \nconfiguration, as well as the charge and z direction spin currents , including the pumped and SHE -\ngenerate d spin currents,  with the resulting Oersted field, are shown in the rendered images at the top.  \n17 \n For FMR simulations with an applied current the resonance field is shifted due to a \ncombination of the Oersted field and the field resulting from FL component o f the spin \ntorque, namely ( rf/MsdF)Jszy, as shown by the dotted lines  in Figure 4(a), in agreement with \nthe values extracted from  the FMR peaks (solid squares). Spin pumping results in a resonance \nfield shift independent of the current density  (but dependent on the frequency)  as seen in \nFigure 4(a), in agreement with the shift predicted due to the change in effective gyromagnetic \nratio [ 50] – see Methods section.  \nThe change in damping due to the DL torque is shown in Figure 4(b), calculated as a  \nfunction of current density in the Pt layer for 3 different frequencies, 10, 20 and 40 GHz. Spin \npumping results in a significant increase in damping of 0.0055, where the base Gilbert \ndamping is set to 0.01, in agreement with the expected increase for a diffusive system [ 50]. \nThe damping increase due to spin pumping is constant within the fitting uncertainty both with \nfrequency and current density. Thus by taking  the difference in damping for currents with \nopposite direction, the resultant change 2 SHE is solely due to the SHE. As shown in Ref. \n[20] the change in damping is approximately inversely proportional to the r.f. frequency and \ndirectly proportional to the strength of the DL torque, given by:  \n \nFor the higher frequencies a good agreement is obtained between numerical calculations and \nequation (18). For the lower 10 GHz frequency, close to the frequency used in Ref. [ 20], this \nrelation is no longer accurate and the numerical results must be used instead.  \nFsszy\nSHEdMfJ\n  \n(18)  \n 18 \n  \nFigure 4 . Change in resonance field and damping as a function of current density.  The c hange in \nFMR peak properties as a function of current density in the Pt layer are obtained, showing  (a) \nresonance field for 20 GHz frequency with the contributions of the Oersted field and FL term  \nidentified  by the dotted lines , both with and without spin pumping, and (b) change in effective \ndamping due to DL torque calculated for 10, 20 and 40 GHz  frequency . The dotted lines are obtained \nfrom equation  (18), noting the plot represents  2SHE. For both parts t he error bars represent the \nLorentzian peak fitting uncertainties.  \n \nThe experimentally measured change in damping from Ref. [ 20] may be reproduced with the \nmodel introduced here by using SHA as a fitting factor.  The results are shown in Figure 5 as a \nfunction of  the spin diffusion length in Pt, and for two extremes of the spin diffusion length \nin Py, noting \nD sf sdl  1  [7]. With the shorter diffusion length in Py a good \n19 \n agreement is obtained between the continuous interface calculations and those with a spin \nmixing conductance with Re{G} = 5×1015 S/m2. This value is significantly larger than the \naccepted value for Pt/Py interfaces, which is typically Re{G}  1015 S/m2 [49]. Repeating \nthe calculations with this lowe r value results in SHA in the range 0.08 – 0.1, comparable to \nthat obtained in Ref. [ 20]. A quantitative agreement with the continuous interface calculations \nmay again be obtained by using a longer diffusion length in Py of 12 nm [ 47] as shown in \nFigure 5. Note  = 0.04 is kept fixed and φ now takes on larger values in t he range 2.5 – 4.5 \nnm obtained from \nL J ll/  [9]. \n \n \nFigure 5 . Calculated spin Hall angle.  Experimental results in Ref. [ 20] are reproduced as a function \nof spin diffusion  length for Pt , for both the continuous and spin mixing conductance interface \ncalculations. For the continuous interface , results for two diffusion lengths in Py are shown as \nindicated, where φ varies in the range  0.3 – 1 nm in 0.1 nm steps for t he shorter diffusion length and \n2.5 – 4.5 nm in 0.5 nm steps for the longer diffusion length. In both cas es the non -adiabaticity \nparameter is set to 0.04.  \n \n It should be noted that for both approaches the drift -diffusion model is an \napproximation to the stronger Boltzmann semiclassical approximation [ 10]. For the spin \n20 \n mixing conductance interface no direct dependence on the transport parameters in the F layer \nexists, whilst results obtained with the Boltzmann equation show the torques have a marked \ndependence on the F spin diffusion  length [ 10]. On the other hand the spin torque magnitudes \ncalculated with the continuous interface do increase with the spin diffusion length, in a \nqualitative agreement with the Boltzmann equation approach. However , in the continuous \ninterface case the validity of the drift -diffusion formalism is limited to cases where the \ntransverse spin relaxation  time is greater than the momentum  relaxation time. Whilst the \nBoltzmann equation approach is more powerful, also allowing for inclusion of current -in-\nplane transport effects [ 51], in addition to the CPP effects modelled via the present drift -\ndiffusion approach, the computational cost is much greater [ 52] whi ch may currently \npreclude an efficient integration within a three -dimensional model of magnetisation \ndynamics. For CPP transport a  hybrid approach may be possible, where the spin torques are \ncalculated using the Boltzmann equation for a static magnetisatio n configuration, and \nappropriate correction factors introduced for the spin tor que magnitudes in the drift -diffusion \nmodel. In this way the advantages of a three -dimensional approach to including spin transport \neffects within a magnetisation dynamics simul ation is maintained ; the investigation of this \npossibility is left for further work.  \n \nV. Conclusions  \n \n A three -dimensional spin current solver based on the generalised drift -diffusion \ndescription, including the spin Hall effect  and spin pumping , was implemented within a \nthree -dimensional magnetisation dynamics  formulation. This model was shown to self -\nconsistently reproduce a number of spin torques in CPP geometries and single ferromagnetic \nlayers. Two approaches to modelling interfaces between normal  metals and fer romagnets 21 \n were  investigated, one based on th e spin mixing conductance and the other based on \ncontinuity of spin currents. Both approaches are in qualitative agreement, showing the SHE -\ngenerated spin torques  contain both field -like and dampin g-like components in general. A \nquantitative comparison between the two approaches  was made by calculating the spin Hall \nangle in Pt from published FMR data. Whilst the spin mixing conductance approach does not \ndirectly take into account the spin diffusion  length in the ferromagnet, the two approaches \nwere shown to be in approximate agreement  for published  transport parameters. Finally, \nanalytical approaches to including spin torque terms in calculations are restricted only to \nspecial cases where the spin c urrents incident on the metal / ferromagnet interface are \nuniform. In general this is not the case, as shown even for a simple bilayer geometry, and the \nfull three -dimensional spin current solver approach is more appropriate.  It is hoped this \napproach to m odelling spin torques will lead to a better understanding of experimentally \nobtained spin torque -driven magnetisation dynamics.  \n \n \n \n \n \n \n \n \n \n \n 22 \n Appendix A  – Spin Torques in CPP -GMR Stacks  \n \nIt is well known  that both field-like and spin -like torque s act on the layer s of a CPP -\nGMR stac k [7]. These torques are of the form am×(m×mF) + bm×mF, where m is the local \nmagnetisation direction of the free layer, and mF is the magn etisation direction of the fixed \nlayer. For a macrospin approximation these torques may be added to the LLG equation with \nappropriate values for the coupling constants a and b [53]. In general however these \ncoefficients depend on the spin accumulation and h ave a spatial dependence.  Moreover the \nspin accumulation is important in understanding the magneto -resistance of the CPP -GMR  \nstack [ 54]. For completeness the model implemented here is tested in a simple spin valve , \nshowing simultaneou s reproduction of both the spin torque switching effect, as well as the \nmagneto -resistance effect.  The full micromagnetics model is used, including demagnetising \nand direct exchange contributions, coupled to the three -dimensional spin current solver.  The \nCPP-GMR stack consists of the  layering N (59 nm) / F (5 nm) / N (2 nm) / F (3 nm) / N (59 \nnm). The thicker F layer is the fixed magnetic layer where the magnetisation is kept fixed \nalong the x direction, and the thinner F la yer is the free magnetic layer.  This is shown in \nFigure A1, where two extreme configurations are distinguished:  the anti -parallel \nconfiguration (AP) where the magneto -resistance is the highest, and the parallel configuration \n(P) where the magneto -resistance is the lowest. The stack is ell iptical in shape with 160 nm × \n40 nm dimensions, and the electrodes are placed at the z-axis ends of the structure. Stair -step \nboundary corrections are applied to the elliptical shape to correct for the finite difference \nartefacts on the demagnetising fiel d [42,55]. The outer N leads are purposely extended (for \nsimplicity here they are simply extended along the z-axis, but more complicated contact \ngeometries are possible) to allow the spin accumulation to decay to zero – it is important to \ninclude in the model enough of the contacting electrical leads since only then can the 23 \n boundary condition (S).n = 0 be applied  correctly. The resulting x components of the spin \naccumulation for the AP and P states are shown in Figure A1b. The same material parameters \nfor the N and F layers used in the main text are applied here using the spin mixing \nconductance interface approach  (but with SHA = 0); additionally D = 0.9  [47]. The results \nare shown in Figure A1, where the magnetisation of the free layer is switched from the AP to \nthe P state using a charge current density along the z direction of 1012 A/m2 (electrons flow \nfrom the fixed to the free layer), and back to the AP st ate using a charge current density of \n+1012 A/m2. For simplicity the layers are not surface -exchange coupled, but interact only \nthrough the demagnetising field and the spin torque. As expected the resistance switches from \na high state (AP) to a low state ( P), and back to the original state. Since the demagnetising \nfield preferentially acts to keep the layers in the AP state, the switching process is slower \nfrom the AP to the P state than vice -versa, however once the P state is achieved the shape \nanisotropy of the ellipse stabilises this configuration.  24 \n  \nFigure A1 . Current -induced magnetisation switching in a spin valve.  Spin torque switching in an \nelliptically shaped CPP -GMR stack, showing  (a) normalised x component of magnetisation (solid \nline) and stack re sistance (dotted line) showing switching from the AP state ( using Jcz = 1012 A/m2) \nto the P state and back ( using Jcz = +1012 A/m2) to the AP state,  (b) x component of spin accumulation \nfor the AP and P configurations.  \n \n \n \n \n25 \n Appendix B  – Methods  \n \n All simulations were done using the multi -physics micromagnetics -oriented software \nBoris  [42,43] written by the author. The software is written mainly in C++ with all \ncomputational routines a vailable for both CPU and GPU computations using the CUDA C \nframework. For small problem sizes but highly repetitive computations, such as the FMR \nsimulations, CPU routines are more efficient. To optimise  the computational speed the most \nexpensive routines  have been written directly in assembly language using the SIMD AVX \ninstruction set. In particular taking advantage of the larger AVX registers allowing 2 FFTs to \nbe computed simultaneously on each processor core, custom interleaved FFT routines h ave \nbeen implemented using the r adix 4 algorithm (this was found to be more efficient than the \nmore common split radix algorithm). For larger problem sizes, such as the CPP -GMR stack, \nGPU routines are increasingly more efficient  due to massive parallelisation . For more \ncomplex simulations  and simulation chains, such as the FMR simulations, instead of console -\nbased user control the compiled program is controlled using local or remote Python scripts \nwith a communication protocol implemented over network sockets.  \nFor the FMR simulations, for each bias field value the magnetisation precession is \nallowed to reach steady state before extracting the oscillation amplitude. At the end of the \nbias field sequence t he Lorentzian peak function F( x) = y0 + S [H + A (x-H0)] / [4( x-H0)2 + \nH2] – this formula contains both symmetric and asymmetric components, however the fitted \nformula contains virtually only the symmetric component ( A  0) as expected – is \nautomatically fitted using the Levenberg -Marquardt algorithm . Typical fitted  FMR peaks \nobtained at 20 GHz frequency are shown in Figure 3, where the quoted charge current density \nis the average value in the N layer. To obtain a peak representative of the FMR power \nabsorption the oscillation amplitude is squared: the resulting peak  is described very well by 26 \n the Lorentzian peak function from which the damping value can be extracted as  = H/2 \n[56], where H is the full -width half -maximum linewidth and  is the angular frequency. \nThe zero -current FMR peaks have a resonance field H0 close to that predicted by the Kittel \nformula [ 56]  = ([H0 + (Ny – Nx)Ms][H0 + (Nz – Nx)Ms])0.5, where Nx = Ny = 0.113 and Nz = \n0.774 are demagnetising factors calculated for the Py rectangle; note, an exact agreement \nwith this formula cannot be expected since it only strictly applies to ellipsoidal shapes.  Spin \npumping results in a change in the effective gyromagnetic ratio and  effective damping. For an \nideal spin sink these can be expressed in terms of the spin mixing conductance, see equations \n(59) and (60)  in Ref. [ 50]. For example a t 20 GHz the predicted reduction in resonance field \nis ~430 A/m, comparable to the value obtained from simulations of ~480 A/m. Similarly the \npredicted increase in damping for an ideal spin sink is 0.0079, consistent with the lower value \nobtained from simu lations of 0.0055, expected for a non -ideal diffusive spin sink.  \nThe LLG equation was solved using  the 4th order Runge -Kutta method with fixed \ntime-step of 0.1 ps.  Equation  (6) is solved using  a custom FFT -based Poisson solver . \nEquation  (8) is solved using  an alternating direct ion scheme  [44]. The cellsizes used are 5 nm \nfor the LLG equation  (refined to 1 nm in the z direction for the CPP -GMR stack) with the \nspin current sol ver sub -mesh refined along the z-axis to 0.125 nm.  \n \n \n \n \n \n \n 27 \n References  \n \n1. Parkin, S. & Yang, S. -H. Memory on the racetrack. Nature Nano.  10, 195 -198 (2015).  \n2. Liu, L., Pai, C. -F., Li, Y., Tseng, H. W., Ralph, D. C. & Buhrman, R. A. 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Inverse spin -Hall effect voltage generation by nonlinear spin -wave excitation. \nPhys. Rev. B  93, 064408 (2016).  \n33. Nakayama, H. et al. Geometry dependence on inverse spin Hall effect induced by spin pumping in \nNi81Fe19/Pt films. Phys. Rev. B  85, 144408 (2012).  \n34. Kumar, A. et al. Spin transfer torque ferromagnetic resonance induced spin pumping in the Fe/Pd \nbilayer system. Phys. Rev. B  95, 064406 (2017).  \n35. Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall effects. Rev. \nMod. Phys.  87, 1213 –1260 (2015).  \n36. Hoffmann, A. Spin Hall Effects in Metals. IEEE Trans. Mag.  49, 5172 –5193 (2013).  \n37. Brataas, A., Tserkovnyak, Y., Bauer, G.E.W., Kelly, P.J. Spin pumping and spin transfer. In: \nMaekawa, S., Valenzuela,  S.O., Saitoh, E. & Kimura, T. (Eds.) Spin Current. Oxford University \nPress , 87 – 135 (2012).  \n38. Zhang, S. & Li, Z. 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Identification of transverse spin currents in \nnoncollinear magnetic structures. Phys. Rev. Lett.  93, 256602 (2004).  \n46. Tserkovnyak, Y., Brataas, A. &  Bauer, G.E.W. Spin pumping and magnetization dynamics in \nmetallic multilayers. Phys. Rev. B  66, 224403 (2002).  \n47. Bass, J. & Pratt, W. P. Jr. Spin -diffusion lengths in metals and alloys, and spin -flipping at \nmetal/metal interfaces: an experimentalist’s cr itical review. J. Phys. Condens. Matter  19, 183201 \n(2007).  \n48. Lepadatu, S. et al. Experimental determination of spin -transfer torque nonadiabaticity parameter \nand spin polarization in permalloy. Phys. Rev. B  79, 094402 (2009).  \n49. Ando, K. et al. Inverse sp in-Hall effect induced by spin pumping in metallic system. J. Appl. Phys.  \n109, 103913 (2011).  \n50. Tserkovnyak, Y., Brataas, A., Bauer, G.E.W. & Halperin, B.I. Nonlocal Magnetization Dynamics \nin Ferromagnetic Hybrid Nanostructures. Rev. Mod. Phys. 77, 1375 –1421 (2005).  \n 31 \n  \n51. 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"    },    {        "title": "1807.09672v2.Nuclear_spin_dynamics_influenced_and_detected_by_electron_spin_polarization_in_CdTe_CdMgTe_quantum_wells.pdf",        "content": "arXiv:1807.09672v2  [cond-mat.mes-hall]  3 Apr 2019Nuclear spin dynamics inﬂuenced and detected by electron sp in polarization in\nCdTe/(Cd,Mg)Te quantum wells\nE. Evers,1,∗T. Kazimierczuk,1,2F. Mertens,1,†D. R. Yakovlev,1,3\nG. Karczewski,4T. Wojtowicz,4,5J. Kossut,4M. Bayer,6,3and A. Greilich6\n1Experimentelle Physik 2, Technische Universit¨ at Dortmun d, D-44221 Dortmund, Germany\n2Institute of Experimental Physics, Faculty of Physics,\nUniversity of Warsaw, PL-02093 Warsaw, Poland\n3Ioﬀe Institute, Russian Academy of Sciences, 194021 St.Pete rsburg, Russia\n4Institute of Physics, Polish Academy of Sciences, PL-02668 Warsaw, Poland\n5International Research Centre MagTop, PL-02668 Warsaw, Po land\n6Experimentelle Physik 2, Technische Universit¨ at Dortmun d, 44221 Dortmund, Germany\nNuclear spin coherence and relaxation dynamics of all const ituent isotopes of an n-doped\nCdTe/(Cd,Mg)Te quantum well structure are studied employi ng optically detected nuclear mag-\nnetic resonance. Using time-resolved pump-probe Faraday e llipticity, we generate and detect the\ncoherent spin dynamics of the resident electrons. The photo generated electron spin polarization is\ntransferred into the nuclear spin system, which becomes pol arized and acts back on the electron\nspins as the Overhauser ﬁeld. Under the inﬂuence of resonant radio frequency pulses, we trace the\ncoherent spin dynamics of the nuclear isotopes111Cd,113Cd, and125Te. We measure nuclear Rabi\noscillations, the inhomogeneous dephasing time T∗\n2, the spin coherence time T2, and the longitudinal\nrelaxation time T1. Furthermore, we investigate the inﬂuence of the laser exci tation and the corre-\nsponding electron spin polarization on the nuclear spin rel axation time and ﬁnd a weak extension\nof this time induced by interaction with the electron spins.\nI. INTRODUCTION\nThe nuclear spin system of semiconductors is of par-\nticular interest as it exhibits a long coherence time as\na result of a reduced interaction with the surrounding.\nConsequently, the direct manipulation of nuclear spins\nis only possible in a slow manner and, as an additional\ndrawback, exhibits only a very low light-matter interac-\ntion. Electron spins, however, can be manipulated much\nmore eﬃciently by light1. Their spin acts on the nu-\nclear spin system allowing for an electron-mediated op-\ntical nuclear spin polarization, both in bulk semiconduc-\ntors2–4and in low-dimensional structures5–8. The nu-\nclear spin system, in turn, is a prominent source for the\nelectron spin dephasing9so that mapping out the nu-\nclear spin dynamics is a key for understanding the elec-\ntron spin time evolution. Letting the electron spin de-\ntect the nuclear spin dynamics oﬀers the possibility to\nstudy a small number of nuclear spins in the localiza-\ntion volume of electrons in a quantum well (QW) struc-\nture5. Additionally, one might get insight into the dy-\nnamics of diﬀerent isotopes. At low magnetic ﬁelds, op-\ntically detected nuclear magnetic resonance (ODNMR)\ncan be applied by measuring the photoluminescence po-\nlarization10, whereas,athigherﬁelds, time-resolvedFara-\nday rotation using a pump-probe technique can be car-\nried out11,12. Furthermore, the use of radio-frequency\n(RF) pulses enables one to study the nuclear spin de-\nphasing and coherence times12,13. In this letter, we re-\nport on a series of pulsed ODNMR experiments in a\nlow perturbation regime. In contrast to Ref. [12], where\nGaAs/(Al,Ga)As QWs were investigated, we observe the\ndynamics of a small nuclear polarization ( ≤6%) in the\nabsence of a pumped electron spin polarization. Further-more, in CdTe/(Cd,Mg)Te QW, all isotopes with non\nzero nuclear spins exhibit a total spin of 1/2, while in\nGaAs/(Al,Ga)As all isotopes have spins 3/2. We can,\ntherefore, exclude the nuclear quadrupole-induced spin\nrelaxation, which plays an important role in the nuclear\nspin dynamics7. Additionally, we investigate the impact\nofthe electronspin polarizationon the nuclearrelaxation\ndynamics.\nII. EXPERIMENTAL DETAILS\nExperiments were performed with a pump-probe\ntechnique using a picosecond Ti:Sapphire laser with\n1.5ps pulse duration operating at a repetition rate of\n75.75MHz. The polarization of the pump beam was\nmodulatedbetween σ+andσ−usingaphotoelasticmod-\nulator at a frequency of 50kHz. The probe was linearly\npolarized and overlapped with the pump at the sample.\nBoth beams were focused to spots with similar diameters\nof about 40 µm and had equal laser powers of 0.6mW\n(power density 48W/cm2). The photon energy was ﬁxed\nat 1.5986eV (wavelength 775.6nm), which corresponds\nto the trion transition, see the black arrow in the inset to\nthe right in Fig. 1(a). After passing through the sample,\nthe Faraday ellipticity (FE) of the probe was tested us-\ning an optical bridge consisting ofa quarter-waveplate, a\nWollaston prism, and a balanced photoreceiver. The FE\nwas measured using a lock-in ampliﬁer at the modulator\nfrequency as a function of the path diﬀerence between\nthe pump and probe beams. We use the FE instead of\nthe commonly used Faraday rotation, as close to the res-\nonance the spectroscopic response for the ellipticity is\nstronger14.2\nThe studied sample (#031901C) consists of a\nCdTe/Cd 0.78Mg0.22Te QW heterostructure grown by\nmolecular-beam epitaxy on a (100) oriented GaAs sub-\nstrate and separated from it by a CdTe/Cd 0.78Mg0.22Te\nsuperlattice grownon athick 2 µm Cd0.78Mg0.22Te buﬀer\nlayer. The heterostructure has ﬁve periods, each of them\nconsists of a 20-nm-thick CdTe QW and a 110-nm-thick\nCd0.78Mg0.22Te barrier. An additional110-nm-thickbar-\nrier was grown on top of this layer sequence to reduce\nthe contribution of surface charges. The barriers were\nmodulation doped with iodine donors. Electrons from\nthe barrier donors, being collected in the QWs, provide\na two-dimensional electron gas with a density of about\nne= 1.1×1010cm−2. Weak localization of the resident\nelectrons at the interface ﬂuctuations leads to the ap-\npearance of a trion line in the emission spectra, see inset\nto Fig. 1(a). The trion states (T) are formed by a res-\nident electron and an optically created exciton, and are\nenergetically shifted to the lower energy by the binding\nenergy from the free excition transition (X), the higher\nenergy PL line15. To facilitate the transmission experi-\nments, theGaAssubstratewaschemicallyremoved. Dur-\ning measurements, the sample wasmounted inside abath\ncryostat at temperature T= 1.5K. The gfactor of the\nresident electrons ge=−1.64±0.02was determined from\nthe Larmor precession frequency in an external mag-\nnetic ﬁeld of 1T15,16. Detailed information on the co-\nherent electron spin properties in this sample was pub-\nlished in Refs. [15, 17, and 18]. Extended information\non the electron-nuclear spin interaction and the polariza-\ntion of nuclear spins by optical excitation can be found\nin Ref. [19]. This reference compares several systems\nwith diﬀerent localization volumes of the resident elec-\ntrons and considers a case of ODNMR, which is quite\ndiﬀerent from the presented case, where the nuclear po-\nlarization is studied far from the NMR.\nTo directly manipulate the nuclei, the sample holder\nwas equipped with a copper coil of about 1mm in di-\nameter made out of 10 turns of wire. For the sake of\nbroadbandoperation, the coilwasdirectlyconnectedto a\ncoaxialcablewithout impedance matching. It wasdriven\nby an RF signal synthesized by a function generator and\nrouted through a 100W pulsed RF ampliﬁer. The coil\nwas placed directly on the sample surface with its open-\ning oriented along the optical axis to apply an RF ﬁeld\nwith its magnetic component orthogonal to the external\nstatic magnetic ﬁeld. The sample was excited through\nthe opening in the center of the coil.\nFigure 1 shows the typical spin dynamics at B= 1T.\nThe pump pulse at zero delay induces a net electron spin\npolarization along the optical axis zparallel to the sam-\nple growth axis. In a transverse magnetic ﬁeld Bxthe\nelectron spins undergo a Larmor precession with the fre-\nquencyωe=geµBBx//planckover2pi1≈114.5rad/ns atB= 1T,\nwhereµBis the Bohr magneton and /planckover2pi1the reduced\nPlanck constant. The precession is observed as oscilla-\ntory changes of the ellipticity signal. The amplitude of\nthese oscillations decays due to inhomogeneous dephas-/s57 /s49 /s48 /s49 /s49 /s49 /s50 /s49 /s51 /s49 /s49 /s49 \n/s67/s100/s49 /s49 /s51 \n/s67/s100/s80 /s104/s97/s115 /s101/s32/s115 /s104/s105/s102/s116\n/s82 /s97/s100/s105/s111 /s32/s102 /s114/s101 /s113/s117/s101 /s110/s99 /s121 /s32/s40/s77 /s72 /s122/s41/s49 /s50 /s53 \n/s84/s101 /s40/s98 /s41\n/s66 /s32/s61 /s32/s49 /s32/s84 /s45 /s49 /s46/s53 /s45 /s49 /s46/s48 /s45 /s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45 /s53 /s48 /s53 /s49 /s48 /s49 /s53 /s50 /s48 /s69/s108/s108/s105/s112/s116/s105/s99/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115 /s41\n/s68 /s101 /s108/s97/s121 /s32/s40/s110/s115 /s41/s32/s119 /s105/s116/s104 /s111 /s117 /s116/s32/s82 /s70 \n/s32/s119 /s105/s116/s104 /s32/s82 /s70 /s40/s97/s41/s45 /s49/s46 /s56 /s45 /s49/s46 /s55\n/s68 /s101/s108/s97/s121 /s32/s40/s110 /s115/s41/s69 /s108/s108/s105/s112 /s116/s105/s99/s105/s116/s121 \n/s49/s46 /s53/s57/s53 /s49/s46 /s54/s48/s48/s88 \n/s69 /s110 /s101/s114/s103 /s121 /s32/s40/s101/s86 /s41/s80 /s76 /s84 \nFIG. 1. (a) Dynamics of the Faraday ellipticity of the probe\nbeam polarization after passing through the sample versus\ntime delay relative to the pump pulse measured at B= 1T.\nThe left inset presents a zoom of the data around a delay of\n−1.7ns (i.e. 11 .5ns after the previous pump pulse). The red\ncurve corresponds to the measurement in the presence of RF,\naﬀecting the nuclear polarization. The right inset shows th e\nphotoluminescence spectrum (PL) exhibiting trion (T) and\nexciton (X) emission lines. The arrow marks the position of\nthe laser for the pump-probe experiment. (b) Phase shift as\na function of applied continuous wave RF. Sharp resonances\ncorrespond to the NMR conditions of diﬀerent isotopes at\nB= 1T at T= 1.5K. The line is a guide to the eye.\ning of the electron spin ensemble caused by the electron\ng-factor dispersion15.\nGenerally, the observedLarmorfrequencyis deﬁnedby\nthe common action ofthe external magnetic ﬁeld and the\nOverhauser ﬁeld of nuclear spins. Their contribution is\nevidenced by measuring the pump-probe spectra in the\npresence of an RF ﬁeld. Once the RF frequency fRF\nmatches the resonance conditions fRF=frfor a speciﬁc\nisotope with a resonance frequency fr, the nuclear spins\nbecome depolarized and the Overhauser ﬁeld reduced.\nThe result of such an experiment is plotted in Fig. 1(a)\nwhere the red curve is taken under RF ﬁeld radiationand\nthe blue curve without it. The main diﬀerence is a siz-\nable shift of the electron precession frequency (left inset\nin Fig. 1(a)), which indicates that without the RF the\nnuclear spins were partially polarized along the external\nmagnetic ﬁeld, aﬀecting the electron spin precession. We\nevaluate the frequency shift by taking a short excerpt of\nthe spectrum roughly 11ns after the pump-probe coin-\ncidence and determine the phase shift of the oscillation\nwith RF in comparison to the case without RF, see the3\nleft inset in Fig. 1(a). We use this phase shift as an indi-\ncator to measure the eﬀect of the RF on the nuclear spin\nsystem.\nMonitoring the frequency (or equivalently the phase of\nthe oscillation for a given part of the pump-probe de-\nlay) of the electron Larmor precession allows us to as-\nsess the contributions of individual isotopes separately.\nThe phase of the oscillation was determined by perform-\ning a short pump-probe scan around -1.7ns covering the\nrangeofasingleelectron-spinprecessionperiod ataﬁxed\nmagnetic ﬁeld. Figure 1(b) shows the resonances of sev-\neral nuclear isotopes measured by applying a continuous\nsweep of RF at B= 1T. For this measurement, the func-\ntion generator was connected directly to the copper coil\nwithout using the ampliﬁer. The resonance frequency\nfrfor a NMR can be calculated as fr=|γ|B/(2π).\nThe values of the gyromagnetic ratio γfor the111Cd\n(−56.9[Tµs]−1),113Cd (−59.6[Tµs]−1) and125Te iso-\ntopes (−85.1[Tµs]−1) lead to: fr[MHz] = 9 .06B[T],\n9.48B[T] and 13 .52B[T], respectively20. Negative val-\nues ofγare related to the direction of the nuclear ﬁeld\nprecession with respect to the external ﬁeld. As seen in\nFig. 1(a), for the negative sign, the Overhauser ﬁeld is\noriented anti-parallel to the external ﬁeld, and the Lar-\nmor frequency is reduced for polarized nuclei (i.e., the\nsignal without RF).\nThe asymmetry of the peaks in Fig. 1(b) is a result of\na too short time interval between subsequent RF steps,\nwhich did not allow the nuclei to reach a steady-state\npolarization, see below for the measurement of the lon-\ngitudinal spin relaxation time T1. The direction of the\nscan was from low to high frequencies. It is important\nto mention, that all isotopes of CdTe have nuclear spin\nI= 1/2, so no quadrupole eﬀects have to be considered.\nThis simpliﬁes the interpretation of the results and leads\nto a strong spectroscopic response, as all nuclei of the\nsame isotope have the same Zeeman splitting.\nIn the experiments presented below, we ﬁx the mag-\nnetic ﬁeld at 1T, the RF at resonant frequencies, and ap-\nply RF pulses. The following measurement cycle is used:\nThe laser pulses polarize the electrons for ﬁve minutes\nfor theT1measurements or forty seconds for all other\nmeasurements. Then we choose one of two scenarios: (i)\napply RF and continue to apply the laser radiation, or\n(ii) apply RF, but block the laser radiation during RF\napplication. Afterward, the laser radiation is unblocked,\nand the phase shift of the electron precession is read out\nwith the RF switched oﬀ. Then the cycle is repeated for\nthe next data point.\nResonant RF pulses allow us to provide a coherent nu-\nclear spin control of speciﬁc isotopes, where the action\nof the RF can be considered as an action of an eﬀective\nmagnetic ﬁeld along the z-axis, which produces a coher-\nent rotation of the nuclear spin in the xy-plane. The\ndependence of the electron precession phase shift on the\nRF pulse parameters is presented in Fig. 2(a) for the\n111Cd isotope at fr= 9.06MHz. Clear oscillatory be-\nhavior is observed as a function of the pulse duration,which is an unequivocal signature of Rabi oscillations of\nthe addressed nuclear spins. The period of oscillations\nallows us to determine the pulse duration correspond-\ning toπ/2 andπrotations and to characterize the value\nof the eﬀective magnetic ﬁeld produced by the RF-coil.\nSo, for a full 2 πRabi period with an applied voltage of\n650mV, it requires a 8.6 µs pulse, which corresponds to\nBeﬀ= 2π/(8.6[µs] 0.0569[mT µs]−1)= 12.8mT.\nIII. RESULTS\nFigure 2(b) demonstrates the Ramsey fringe experi-\nment21and the corresponding inhomogeneous spin de-\nphasing time T∗\n2of the111Cd isotope. The sequence for\nthis experiment consists of two π/2 pulses separated by\na delay τ12. The ﬁrst pulse rotates the nuclear spins\nout of the x-axis to the y-axis, where they start to pre-\ncess around the x-axis in yz-plane. The second pulse\nbringsthenuclearspinsbacktothe x-axis. Dependingon\nthe phase accumulated over the time τ12, the maximally\nachieved spin polarization of the ensemble decreases. As-\nsuminganexponentialdecayofthe oscillationamplitude,\nwe ﬁt the value of T∗\n2,E= 144µs for the studied111Cd\nisotope (red curve in Fig. 1(b)). A Gaussian decay deliv-\ners a dephasing time of T∗\n2,G= 122µs (blue dashed curve\nin Fig. 2(b)). Based on the results, the exponential ﬁt\ngives a better description.\nFigure 2(c) shows the next sequence, the spin Hahn-\necho22. The ﬁrst and the third pulse have an area of\nπ/2 and serves for the same purpose as in the previous\ncase. The ﬁrst pulse rotates the nuclear polarization by\nπ/2 to the yz-plane, and the third pulse rotates it back\nto the original orientation along the x-axis. The second\npulse of that sequence has an area of πand is delayed\nby approximately τ13/2 after the ﬁrst pulse. The role\nof the second pulse is to invert the nuclear polarization\nby 180◦, which reverses the dephasing of the ensemble\nand allows to cancel the eﬀect of the Larmor frequency\nspread. By comparing the eﬀect of the ﬁrst (Fig. 2(b))\nand the second sequence (Fig. 2(c)) we see that the intro-\nduction of the πpulse considerablyextends the timescale\nover which a coherent nuclear precession is observed. In\nthat case, we extract a coherence time of T2= 2.5ms\nusing an exponential ﬁt.\nTo complete the characterization of the nuclear spin\ndynamics, we provide the measurement of the longitudi-\nnalnuclearspinrelaxationtime T1. Here, asingle π-pulse\nis used to invert the nuclear spin polarization along the\nx-axisafter initial optical pumping ofthe nuclearsystem.\nIt causes a sudden phase shift of the electron precession\nfollowed by a slow recovery on the timescale of T1. Fig-\nure2(d) showsthe correspondingmeasurementwhich ex-\nhibitsalongitudinalspinrelaxationtimeof T1= 104s. It\ndemonstrates that the condition T1≫T2is valid in this\nsample, and the spin temperature of the nuclei reaches\nits equilibrium much faster than the energy transfers to\nthe lattice. Furthermore, this experiment allows us to4\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49\n/s48 /s53 /s49/s48/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s84/s42\n/s50/s44/s71/s32/s61/s32/s49/s50/s50/s32/s181/s115/s70/s114/s105/s110/s103/s101/s32/s97/s109/s112/s108/s46/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s32/s40/s181/s115/s41/s84/s42\n/s50/s44/s69/s32/s61/s32/s49/s52/s52/s32/s181/s115/s40/s98/s41\n/s49/s51/s49/s50/s69/s99/s104/s111/s32/s97/s109/s112/s108/s46/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s40/s109/s115/s41/s84\n/s50/s32/s61/s32/s50/s46/s53/s32/s109/s115/s40/s99/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s80/s104/s97/s115/s32/s115/s104/s105/s102/s116/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84/s105/s109/s101/s32/s97/s102/s116/s101/s114/s32 /s45/s112/s117/s108/s115/s101/s32/s40/s115/s41/s84\n/s49/s32/s61/s32/s49/s48/s52/s32/s115/s40/s100/s41/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s84\n/s82/s97/s98/s105/s32/s61/s32/s56/s46/s54/s32/s181/s115\n/s49/s49/s49\n/s67/s100/s80/s104/s97/s115/s101/s32/s115/s104/s105/s102/s116/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s80/s117/s108/s115/s101/s32/s100/s117/s114/s97/s116/s105/s111/s110/s32/s40/s181/s115/s41/s40/s97/s41\nFIG. 2. Coherent control of the111Cd isotope at B= 1T\nandfr= 9.06MHz. (a) Phase shift of the electron precession\ninduced by an RF pulse. The open circles correspond to a\n650mV pulse with varied duration and the red curve is a sine\nﬁt with a Rabi period of 8 .6µs. (b) Decay of the Ramsey\nfringes amplitude. The red line corresponds to an exponen-\ntial decay with a time constant of 144 µs, whereas the blue\ndashed curve corresponds to Gaussian ﬁt with a decay con-\nstant of 122 µs. (c) Decay of the spin-echo amplitude. The\nline is an exponential decay ﬁt featuring a time constant of\n2.5ms. (d) Decay of the phase shift with recurring nuclear\npolarization. The line is an exponential decay ﬁt with a time\nconstant of 104s. The measurements in panels (b)-(d) are\ngivenfor RFapplication while thelaser radiation was block ed.\nThe sketches show the corresponding pulse sequences used in\nthe experiments.\nextract the degree of nuclear polarization from the phase\nshift of the electron precession right after the application\nof the RF pulse. We determine the change of the nu-\nclear ﬁeld BNacting on the electron spin to be 1 .5mT\nwhen manipulating111Cd isotopes by a single π-pulse.\nThis value corresponds to 6% of the maximal nuclearTABLEI.Relaxationtimesforallnonzerospinisotopespeci es\nin the CdTe QW. Values are given without and with illumina-\ntion, corresponding to the nuclear spin dynamics in scenari o\n(ii)/scenario (i).\nIsotope T∗\n2[µs]T2[ms]T1[s]\n111Cd 144/366 2.5/2.9 104/173\n113Cd 191/446 2.6/4.6 116/169\n125Te 200/200 2.6/3.8 130/138\nspin polarization of111Cd, which can be estimated as\nBN/BN,max= 1.5[mT]/25.25[mT] = 0 .06, see Ref. [19].\nThe ﬁeld BN,maxis the maximal ﬁeld induced by a com-\npletely polarized nuclear spin of111Cd [19]. Other iso-\ntopesgivesimilarpolarizationvalues: 5.5%for113Cdand\n9.7%for125Te. AllmeasurementspresentedinFigs.2(b)-\n(d) were realized with the second scenario, where the\nRF pulses were applied, while no optical excitation of\nthe sample was present. These measurements were per-\nformed for111Cd,113Cd, and125Te and the times are\nsummarized in Tab. I.\nOne noticeable feature of the times of the diﬀerent iso-\ntopes measured in the dark condition (without laser ra-\ndiation) is their similarity to each other. It cannot be\nrelated to a nuclear spin diﬀusion, which happens due\nto a dipole-dipole interaction of the nuclei of a particu-\nlar isotope. In the case of CdTe QWs, the electrons are\nweakly localized and, therefore, do not produce a strong\ninhomogeneity of the Knight ﬁeld. Correspondingly, the\nnuclearﬁeld distribution only showsa small inhomogene-\nity which weaklyaﬀects the nuclearspin diﬀusion23. Fur-\nthermore, allnonzeronuclearspinisotopeshavespin 1/2,\nwhich simpliﬁes the spin diﬀusion due to the absence of\nquadrupole splittings. However, at a magnetic ﬁeld of\n1T, the spin diﬀusion can happen purely within one type\nof isotopes, as the diﬀerence in the Zeeman splitting of\ndiﬀerent isotopes is much bigger than the local ﬁeld of\nthe nuclei. Another option to couple diﬀerent types of\nisotopes and nuclei of one isotope with each other is the\npossibility to use the electron spin as a mediator in a ﬂip-\nﬂop process of two nuclear spins. This process is shown\nto be eﬃcient in quantum dots24,25and can be applied\nto the present case as well.\nThe previously described measurements were done\nwithoutsampleilluminationduringtheapplicationofRF\nﬁelds. We repeated them with ongoing optical electron-\nspin orientation during the RF sequences. These exper-\niments were performed for all involved times. The times\nare also presented in Tab. I after each slash. In all exper-\niments, the presence of a pump-probe excitation during\nRF action does not lead to a shortening of the nuclear\nspin relaxation times. On the contrary, some times are\nprolonged under illumination. This is a striking result.\nUsing the results discussed in Ref. [25], we can assume\nthat the T1,eof electrons is anticorrelated with the T2\nof nuclei. For higher excitation power, the spin lifetime\n(T1,e) of electrons becomes reduced (see e.g. Ref. [23] for\nZnSe:F or Ref. [26] for (In,Ga)As/GaAs quantum dots),5\nand therefore one could expect an increase of the nu-\nclearT2time, which is consistent with our results. The\nchanges of T2of nuclei in our case are not as dramatic\nas in quantum dots, due to a much weaker localization\nof electrons and therefore reduced electron-nuclear in-\nteraction, see Ref. [25]. Furthermore, Refs. [25 and 27]\nreport on a direct correlation between T1,eof electron\nspins and T1of nuclei. Similar dependence was observed\nin Ref. [28], where the authors compared the longitudi-\nnal polarization dynamics in the dark scenario to the dy-\nnamics in the brightscenariofor asystem with a strongly\nlocalized donor electrons and reported a reduction of the\nnuclearT1time by three orders of magnitude under illu-\nmination. The result, that in our case, the nuclei T1time\nslightly increases under the illumination is still puzzling\nto us and requires further investigations.IV. CONCLUSIONS\nTo conclude, we present a comprehensive characteri-\nzation of the nuclear spin dynamics in CdTe/(Cd,Mg)Te\nQWs. We use ODNMR based technique on a pump-\nprobe setup and manipulate the nuclear polarization di-\nrectly by using RF pulses. The characteristic time scales\nare similar between all isotopes and demonstrate a weak\nstabilizing eﬀect of the laser illumination.\nV. ACKNOWLEDGEMENTS\nWe acknowledge the ﬁnancial support by Deutsche\nForschungsgemeinschaft in the frame of the ICRC TRR\n160 (Projects A1 and A6) and Russian Science Founda-\ntion (Grant No. 14-42-00015).\n∗Corresponding author: eiko.evers@tu-dortmund.de\n†Present address: Experimentelle Physik6, Technische Uni-\nversit¨ at Dortmund, D-44221 Dortmund, Germany.\n1A. J. Ramsay, Semiconductor Science and Technology 25,\n103001 (2010).\n2G. Lampel, Phys. Rev. Lett. 20, 491 (1968).\n3A. I. Ekimov and V. I. Safarov, Journal of Experimental\nand Theoretical Physics Letters 15, 179 (1972).\n4F. Meier, B. P. Zakharchenya, ed., Optical Orientation\n(North-Holland, Amsterdam, 1984).\n5G. P. Flinn, R. T. Harley, M. J. Snelling, A. C. Tropper,\nand T. M. Kerr, Semiconductor Science and Technology 5,\n533 (1990).\n6S. E. Barrett, R. Tycko, L. N. Pfeiﬀer, and K. W. West,\nPhys. Rev. Lett. 72, 1368 (1994).\n7B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin,\nP. Maletinsky, A. H¨ ogele, and A. 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Warbur-\nton, Nature Nanotechnology 11, 885 (2016).\n26E. A. Zhukov, E. Kirstein, D. S. Smirnov, D. R. Yakovlev,\nM. M. Glazov, D. Reuter, A. D. Wieck, M. Bayer,and\nA. Greilich, Phys. Rev. B 98, 121304 (2018).\n27A. Greilich, A. Shabaev, D. R. Yakovlev, Al. L. Efros,\nI. A. Yugova, D. Reuter, A. D. Wieck, and M. Bayer, Sci-\nence317, 1896–1899 (2007).\n28F. Heisterkamp, E. Kirstein, A. Greilich, E. A. Zhukov,\nT. Kazimierczuk, D.R.Yakovlev, A.Pawlis, andM. Bayer,\nPhys. Rev. B 93, 081409 (2016)."    },    {        "title": "0807.1580v2.Spin_squeezing_in_a_bimodal_condensate__spatial_dynamics_and_particle_losses.pdf",        "content": "arXiv:0807.1580v2  [quant-ph]  19 Dec 2008EPJ manuscript No.\n(will be inserted by the editor)\nSpin squeezing in a bimodal condensate: spatial dynamics an d\nparticle losses\nYun Li1,2, P. Treutlein3, J. Reichel1and A. Sinatra1\n1Laboratoire Kastler Brossel, ENS, UPMC and CNRS, 24 rue Lhom ond, 75231 Paris Cedex 05, France\n2State Key Laboratory of Precision Spectroscopy, Departmen t of Physics, East China Normal University, Shanghai 200062 ,\nChina\n3Max-Planck-Institut f¨ ur Quantenoptik and Fakult¨ at f¨ ur Physik der Ludwig-Maximilians-Universit¨ at, Schellings trasse 4, 80799\nM¨ unchen, Germany\nReceived: date / Revised version: date\nAbstract. Wepropose ananalytical methodtostudytheentangledspati al andspindynamicsof interacting\nbimodal Bose-Einstein condensates. We show that at particu lar times during the evolution spatial and spin\ndynamics disentangle and the spin squeezing can be predicte d by a simple two-mode model. We calculate\nthe maximum spin squeezing achievable in experimentally re levant situations with Sodium or Rubidium\nbimodal condensates, including the eﬀect of the dynamics an d of one, two and three-body losses.\nPACS.PACS-03.75.Gg Entanglement and decoherence in Bose-Einst ein condensates – PACS-42.50.Dv\nQuantum state engineering and measurements – PACS-03.75.K k Dynamic properties of condensates; col-\nlective and hydrodynamic excitations, superﬂuid ﬂow – PACS -03.75.Mn Multicomponent condensates;\nspinor condensates\n1 Introduction\nIn atomic systems eﬀective spins are collective variables\nthat can be deﬁned in terms of orthogonal bosonic modes.\nIn this paper the two modes we consider are two diﬀerent\ninternal states of the atoms in a bimodal Bose-Einstein\ncondensate. States with a large ﬁrst order coherence be-\ntween the two modes, that is with a large mean value\nof the eﬀective spin component in the equatorial plane\nof the Bloch sphere, can still diﬀer by their spin ﬂuctua-\ntions. For an uncorrelated ensemble of atoms, the quan-\ntum noise is evenly distributed among the spin compo-\nnents orthogonalto the mean spin. Howeverquantum cor-\nrelations can redistribute this noise and reduce the vari-\nance of one spin quadrature with respect to the uncorre-\nlated case, achieving spin squeezing [1,2]. Spin-squeezed\nstates are multi-particle entangled states that have prac-\ntical interest in atom interferometry, and high precision\nspectroscopy [3]. Quantum entanglement to improve the\nprecision of spectroscopic measurements has already been\nused with trapped ions [4] and it could be used in atomic\nclockswherethe standardquantum limit hasalreadybeen\nreached [5].\nA promising all-atomic route to create spin squeez-\ning in bimodal condensates, proposed in [6], relies on the\nKerr-typenon linearitydue toelastic interactionsbetween\natoms. Quite analogously to what happens to a coherent\nstate in a nonlinear Kerr medium in optics [7], an initial\n“phasestate”orcoherentspinstate,wherealltheeﬀectivespinspoint atthe samedirection,dynamicallyevolvesinto\na correlatedspin-squeezedstate. A straightforwardwayto\nproduce the initial phase state in a bimodal condensate is\nto start with one atomic condensate in a given internal\nstateaand perform a π/2-pulse coupling coherently the\ninternalstate atoasecondinternalstate b[8].However,as\nthe strength of the interactions between two atoms a−a,\nb−banda−bare in general diﬀerent, the change in the\nmean ﬁeld energy excites the spatial dynamics of the con-\ndensatewavefunctions. In the evolutionsubsequent tothe\npulse, the spin dynamics creating squeezing and the spa-\ntial dynamics are entangled [6,9,10,11] and occur on the\nsame time scale set by an eﬀective interaction parameter\nχ. This makes it a priori more diﬃcult to obtain simple\nanalytical results.\nIn this paper we develop a simple formalism which al-\nlows us to calculate analytically or semi analytically the\neﬀect of the spatial dynamics on spin squeezing. In Sec-\ntion2wepresentourdynamicmodel. Usingourtreatment\nwe show that at particular times in the evolution the spa-\ntial dynamics and the spin dynamics disentangle and the\ndynamical model gives the same results as a simple two-\nmodemodel.Wealsoidentifyconﬁgurationsofparameters\nin which the simple two mode-model is a good approxi-\nmation at all times. Restricting to a two-mode model, in\nSection 3 we generalizeour analyticalresults of[12] on op-\ntimal spin squeezing in presence of particle losses to the\ncase of overlapping and non-symmetric condensates.2 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\nIn Sections 4 and 5, we apply our treatment to cases of\npractical interest. We ﬁrst consider a bimodal87Rb con-\ndensate. Rb is one of the most common atoms in BEC\nexperiments and it is a good candidate for atomic clocks\nusing trapped atoms on a chip [13]. Restricting to states\nwhich are equally aﬀected by a magnetic ﬁeld to ﬁrst or-\nder, the most common choices are |F= 1,m=−1∝an}bracketri}htand\n|F= 2,m= 1∝an}bracketri}htwhich can be magnetically trapped, or\n|F= 1,m= 1∝an}bracketri}htand|F= 2,m=−1∝an}bracketri}htthatmustbetrapped\noptically but for which there exists a low-ﬁeld Feshbach\nresonance which can be used to reduce the inter-species\nscattering length [14,15]. Indeed a particular feature of\nthese Rb states is that the three s-wave scattering lengths\ncharacterizing interactions between a−a,b−banda−b\natoms are very close to each other. A consequence is that\nthe squeezing dynamics is very slow when the two conden-\nsatesoverlap.Theinter-speciesFeshbachresonancecanbe\nused to overcomethis problem and speed up the dynamics\n[15].\nIn schemes involving the |F= 2,m=±1∝an}bracketri}htof Rubid-\nium, the main limit to the maximum squeezing achievable\nis set by the large two-body losses rate in these states. As\na second case of experimental interest we then consider\nNa atoms in the |F= 1,mF=±1∝an}bracketri}htstates [6]. Although\ntheses states have opposite shifts in a magnetic ﬁeld, they\npresent the advantage of negligible two-body losses. Using\nour analytical optimization procedure, we calculate the\nmaximum squeezing achievable in this system including\nthe eﬀect of spatial dynamics and particle losses.\nIn Section 5 we examine a diﬀerent scenario for Rb\ncondensates in which, instead of changing the scattering\nlength, one would spatially separate the two condensates\nafter the mixing π/2 pulse and hold them separately dur-\ning a well chosen squeezing time. An interesting feature\nof this scheme is that the squeezing dynamics acts only\nwhen the clouds are spatially separated and it freezes out\nwhen the two clouds are put back together so that one\ncould prepare a spin squeezed state and then keep it for\na certain time [12]. State-selective potentials for87Rb in\n|F= 1,m=−1∝an}bracketri}htand|F= 2,m= 1∝an}bracketri}ht[13] have recently\nbeenimplementedonanatomchip,andsuchschemecould\nbe of experimental interest.\n2 Dynamical spin squeezing model\nIn this section we develop and compare dynamical models\nfor spin squeezing. No losses will be taken into account in\nthis section.\n2.1 State evolution\nWe consider the model Hamiltonian\nH=/integraldisplay\nd3r/summationdisplay\nε=a,b/bracketleftbigg\nˆψ†\nεhεˆψε+1\n2gεεˆψ†\nεˆψ†\nεˆψεˆψε/bracketrightbigg\n+gabˆψ†\nbˆψ†\naˆψaˆψb (1)wherehεis the one-body hamiltonian including kinetic\nenergy and external trapping potential\nhε=−/planckover2pi12∆\n2m+Uext\nε(r). (2)\nThe interactions constants gεε′are related to the corre-\nspondings-wave scattering lengths gεε′= 4π/planckover2pi12aεε′/M\ncharacterizing a cold collision between an atom in state\nεwith an atom in state ε′(ε,ε′=a,b), andMis the\nmass of one atom.\nWe assume that we start from a condensate with N\natoms in the internal state a; the stationary wavefunction\nofthecondensateis φ0(r).Afteraπ/2pulse,aphasestate\nis created, which is our initial state:\n|Ψ(0)∝an}bracketri}ht=1√\nN!/bracketleftBig\nCaa†\n|φ0/an}bracketri}ht+Cbb†\n|φ0/an}bracketri}ht/bracketrightBigN\n|0∝an}bracketri}ht(3)\nwhereCa,Cbaremixing coeﬃcients with |Ca|2+|Cb|2= 1\nand the operator a†\n|φ0/an}bracketri}htcreates a particle in the internal\nstateawith wave function φ0. To describe the entangled\nevolution of the spin dynamics and the external dynamics\nof the wave functions, it is convenient to introduce Fock\nstates with well deﬁned number of particles in |a∝an}bracketri}htand\n|b∝an}bracketri}ht, these numbers being preserved during time evolution\nsubsequent to the mixing pulse. Expanded over the Fock\nstates, the initial state (3) reads:\n|Ψ(0)∝an}bracketri}ht=N/summationdisplay\nNa=0/parenleftbiggN!\nNa!Nb!/parenrightbigg1/2\nCNaaCNb\nb|Na:φ0,Nb:φ0∝an}bracketri}ht,\n(4)\nwhereNb=N−Na, and\n|Na:φa,Nb:φb∝an}bracketri}ht=/bracketleftBig\na†\n|φa(Na,Nb)/an}bracketri}ht/bracketrightBigNa\n√Na!/bracketleftBig\nb†\n|φb(Na,Nb)/an}bracketri}ht/bracketrightBigNb\n√Nb!|0∝an}bracketri}ht.\n(5)\nWithin an Hartee-Fock type ansatz for the N-body\nstate vector, we calculate the evolution of each Fock state\nin (4). We get [9]:\n|Na:φ0,Nb:φ0∝an}bracketri}ht →e−iA(Na,Nb;t)//planckover2pi1\n×|Na:φa(Na,Nb;t),Nb:φb(Na,Nb;t)∝an}bracketri}ht,(6)\nwhereφa(Na,Nb;t) andφb(Na,Nb;t) are solutions of the\ncoupled Gross-Pitaevskii equations:\ni/planckover2pi1∂tφε=/bracketleftbig\nhε+(Nε−1)gεε|φε|2+N′\nεgεε′|φε′|2/bracketrightbig\nφε(7)\nhere with the initial conditions\nφa(0) =φb(0) =φ0, (8)\nand the time dependent phase factor Asolves:\nd\ndtA(Na,Nb;t) =−/summationdisplay\nε=a,bNε(Nε−1)gεε\n2/integraldisplay\nd3r|φε|4\n−NaNbgab/integraldisplay\nd3r|φa|2|φb|2.(9)Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 3\nWith this treatment we fully include the quantum dynam-\nics of the two condensate modes aandb, as one does for\nthe simple two modes model, but also including the spa-\ntial dynamics of the two modes and their dependence on\nthe number of particles. The approximation we make is\nto neglect all the other modes orthogonal to the conden-\nsates which would be populated thermally. An alternative\nmethod is to use a number conserving Bogoliubov theory\nthat explicitly includes the operators of the condensates\nas in [10]. In that case all the modes are present but the\nmodes orthogonal to the condensates are treated in a lin-\nearized way. In [10], the author compares the number con-\nserving Bogoliubov approach to our approach using many\nGross-Piaevskii equations, also used in [6], and he ﬁnds\nvery similar result for the spin squeezing. He also ﬁnds\nthat within the Bogoliuobov approximation the thermally\nexcited modes strictly do not aﬀect the squeezing in the\nscheme we considerhere. If the number conservingBogoli-\nubovhastheadvantageofbeingsystematic,ourapproach,\nsupplementedwithafurtherapproximation(themodulus-\nphase approximation introduced in Sect. 2.3) allows us to\nget some insight and obtain simple analytical results.\n2.2 Calculation of spin squeezing\nThe eﬀective collectivespin ofatwo-componentsBEC can\nberepresentedontheBlochsphereasshowninFig.1(Top).\nFormally, we introduce three spin operators in terms of\nﬁeld operators [6]\nSx=1\n2/integraltext\nd3r[ˆψ†\nb(r)ˆψa(r)+ˆψ†\na(r)ˆψb(r)],(10)\nSy=i\n2/integraltext\nd3r[ˆψ†\nb(r)ˆψa(r)−ˆψ†\na(r)ˆψb(r)],(11)\nSz=1\n2/integraltext\nd3r[ˆψ†\na(r)ˆψa(r)−ˆψ†\nb(r)ˆψb(r)].(12)\nDeﬁnitions (10)-(12) explicitly take into account the spa-\ntial wave functions of the condensate and depend in par-\nticular on the overlap between the two modes.\nReferring to the Fig.1(Top) we introduce the polar an-\nglesϑandϕgiving the direction z′of the mean spin; ϑde-\ntermines the relative mean atom number in the two inter-\nnal states, cos ϑ=|Ca|2−|Cb|2, while the azimuthal angle\nϕcorresponds to the relative phase between the compo-\nnents,ϕ= arg(C∗\naCb).\nThe minimal variance of the spin in the plane ( x′,y′)\northogonal to the mean spin ∆S2\n⊥,min, represented in Fig.\n1(Bottom), is given by\n∆S2\n⊥,min=1\n2(∆S2\nx′+∆S2\ny′−|∆S2\n−|) (13)\nwhere we introduced\nS−=Sx′−iSy′. (14)\nThe degree of squeezing is then quantiﬁed by the pa-\nrameter [6,3]\nξ2=N∆S2\n⊥,min\n∝an}bracketle{tS∝an}bracketri}ht2, (15)x'y\nxz\ny'z'\nϑ \nϕ〈\rS〉\r\nx'y'\nFig. 1.(Top) average spin. (Bottom) variance of the the spin\ncomponents in the plane orthogonal to the mean spin.\nwhere∝an}bracketle{tS∝an}bracketri}htis the length of the average spin.\nWhen expressed in the original frame of reference, the\nminimal variance in the orthogonal plane is:\n∆S2\n⊥,min=1\n2(cos2ϑcos2ϕ+sin2ϕ)∆S2\nx\n+1\n2(cos2ϑsin2ϕ+cos2ϕ)∆S2\ny\n+1\n2sin2ϑ∆S2\nz−1\n4sin2ϑsin2ϕ∆xy\n−1\n4sin2ϑcosϕ∆zx−1\n4sin2ϑsinϕ∆yz\n−1\n2/radicalbig\n˜A2+˜B2 (16)\nwhere\n˜A= (sin2ϕ−cos2ϑcos2ϕ)∆S2\nx\n+ (cos2ϕ−cos2ϑsin2ϕ)∆S2\ny−sin2ϑ∆S2\nz\n−1\n2(1+cos2ϑ)sin2ϕ∆xy+1\n2sin2ϑcosϕ∆zx\n+1\n2sin2ϑsinϕ∆yz; (17)\n˜B= cosϑsin2ϕ(∆S2\nx−∆S2\ny)−cosϑcos2ϕ∆xy\n−sinϑsinϕ∆zx+sinϑcosϕ∆yz; (18)\nand where we introduced the correlations\n∆ij=∝an}bracketle{tSiSj+SjSi∝an}bracketri}ht−2∝an}bracketle{tSi∝an}bracketri}ht∝an}bracketle{tSj∝an}bracketri}ht, i∝ne}ationslash=j=x,y,z.(19)\nThe spin squeezing is then calculated in terms of averages\nof ﬁeld operators products, with the state of the system4 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\nat timet, obtained by evolving equation (4) with equation\n(6). To calculate the averages one needs to compute the\naction of the ﬁeld operators ˆψaˆψbon the Fock states (5)\n[16],\nˆψa(r)|Na:φa(Na,Nb),Nb:φb(Na,Nb)∝an}bracketri}ht\n=φa(Na,Nb,r)/radicalbig\nNa\n× |Na−1 :φa(Na,Nb),Nb:φb(Na,Nb)∝an}bracketri}ht,(20)\nˆψb(r)|Na:φa(Na,Nb),Nb:φb(Na,Nb)∝an}bracketri}ht\n=φb(Na,Nb,r)/radicalbig\nNb\n× |Na:φa(Na,Nb),Nb−1 :φb(Na,Nb)∝an}bracketri}ht.(21)\nThe explicit expressions of the averages needed to calcu-\nlate the spin squeezing parameter are given in Appendix\nA. These quantum averages correspond to an initial state\nwith a well-deﬁned number of particles N. In case of ﬂuc-\ntuations in the total number of particleswhere the density\nmatrix of the system is a statistical mixture of states with\na diﬀerent number of particles, a further averaging of N\nover a probability distribution P(N) is needed [9,17].\n2.3 Dynamical modulus-phase approach\nIn principle, equations (7)-(9) can be solved numerically\nforeachFockstateinthesumequation(4),andthesqueez-\ning can be computed as explained in the previous section.\nHowever, for a large number of atoms and especially in\nthree dimensions and in the absence of particular symme-\ntries (e.g. spherical symmetry) this can be a very heavy\nnumerical task. To overcome this diﬃculty, in order to\ndevelop an analytical approach, we can exploit the fact\nthat for large Nin the initial state (4) the distributions of\nthe number of atoms NaandNbare very peaked around\ntheir average values with a typical width of order√\nN.\nMoreover, assuming that possible ﬂuctuations in the total\nnumber of particles are described by a distribution P(N)\nhaving a width much smaller than the average of the total\nnumber of particles ¯N, we can limit to NaandNbclose\nto¯Na=|Ca|2¯Nand¯Nb=|Cb|2¯N. We then split the\ncondensate wave function into modulus and phase\nφε=|φε|exp(iθε)ε=a,b, (22)\nand we assume that the variation of the modulus over the\ndistribution of Nεcan be neglected while we approximate\nthe variation of the phase by a linear expansion around\n¯Nε[9]. The approximate condensate wave functions read\nφε(Na,Nb)≃¯φεexp\ni/summationdisplay\nε′=a,b(Nε′−¯Nε′)(∂Nε′θε)¯Na,¯Nb\n\n(23)\nwhere¯φε≡φε(Na=¯Na,Nb=¯Nb).\nThe modulus phase approximation takes into account,\nin an approximate way, the dependence of the condensate\nwave functions on the number of particles. It is preciselythiseﬀectthatisresponsibleofentanglementbetweenspa-\ntial dynamics and spin dynamics.\nAs explained in Appendix B, all the relevant averages\nneeded to calculate spin squeezing can then be expressed\nin terms of ¯φεand of three time and position dependent\nquantities:\nχd(r) =1\n2[(∂Na−∂Nb)(θa−θb)]¯Na,¯Nb,(24)\nχs(r) =1\n2[(∂Na+∂Nb)(θa−θb)]¯Na,¯Nb,(25)\nχ0(r) =1\n2[(∂Na−∂Nb)(θa+θb)]¯Na,¯Nb.(26)\nIn some cases (see Sect. 2.4) these quantities can be ex-\nplicitly calculated analytically. To calculate the squeezing\nin the general case, it is suﬃcient to evolve a few coupled\nGross-Pitaevskii equations (7) for diﬀerent values of Na,\nNb, to calculate numerically the derivatives of the phases\nappearing in (24)-(26). Although we do not expect a per-\nfect quantitative agreement with the full numerical model\nfor all values of parameters, we will see that the analytical\nmodel catches the main features and allowsus to interpret\nsimply the results.\nIn the particular case of stationary wave functions of\nthe condensates, the parameters χd,χsandχ0become\nspace-independent:\nχst\nd=−[(∂Na−∂Nb)(µa−µb)]¯Na,¯Nb\n2/planckover2pi1t(27)\nχst\ns=−[(∂Na+∂Nb)(µa−µb)]¯Na,¯Nb\n2/planckover2pi1t(28)\nχst\n0=χst\ns. (29)\nIn this case we recover a simple two-mode model. Equa-\ntions (27)-(28) will be used in section 3. In that contest\nwe will rename χst\nd/t=−χandχst\ns/t=−˜χto shorten the\nnotations.\nTo test our modulus-phase dynamical model, in Fig. 2,\nwe consider a situation in which the external dynamics is\nsigniﬁcantly excited after the π/2 pulse which populates\nthe stateb. Parameters correspond to a bimodal Rb con-\ndensate in |F= 1,mF= 1∝an}bracketri}htand|F= 2,mF=−1∝an}bracketri}htwith\n¯Na=¯Nb= 5×104and where a Feshbach resonance is\nused to reduce aabby about 10% with respect to its bare\nvalue [14,15]. The considered harmonic trap is very steep\nω= 2π×2 kHz. In the ﬁgure we compare our modulus-\nphase approach (dashed line) with the full numerical so-\nlution (solid line) and with a stationary calculation using\n(27)-(28) (dash-dotted line) which is equivalent to a two-\nmode model. The oscillation of the squeezing parameter\nin the two dynamical calculations (dashed line and solid\nline) are due to the fact that the sudden change in the\nmean-ﬁeld causes oscillations in the wave functions whose\namplitude and the frequency are diﬀerent for each Fock\nstate. From the ﬁgure, we ﬁnd that our modulus-phase\napproach obtained integrating 5 Gross-Pitaevskii equa-\ntions (dashed line) reproduces the main characteristics of\nthe full numerical simulation using 3000 Fock states (solid\nline). The stationarytwomode model on the other hand isPlease give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 5\n0  0.51  1.52  2.53  3.54  4.510−410−310−210−1100\nt [ms]ξ2Two−mode stationary\nModulus−phase 5 GPE\nFull numerical simulation\nFig. 2.Spinsqueezingas afunction oftime. Comparison ofthe\nmodulus-phase model (red dashed line) with a full numerical\ncalculation with 3000 Fock states (blue solid line) and with a\nstationary two-mode model (violet dash-dotted line). Spat ial\ndynamics is strongly excited after the π/2-pulse populating a\nsecond internal state. ω= 2π×2 kHz,¯Na=¯Nb= 5×104,\nm=87 a.m.u., aaa= 100.44rB,abb= 95.47rB,aab= 88.28rB.\nNo particle losses. rBis the Bohr radius.\nnot a good approximation in this case. Only for some par-\nticulartimes the threecurvesalmosttouch.At these times\nthe wave functions of all the Fock states almost overlap\nand, as we will show in our analytical treatment, spatial\ndynamics and spin dynamics disentangle.\nIn Fig.3 we move to a shallow trap and less atoms. We\nnote that in this case both the modulus-phase curve and\nthe numerical simulation are very close to the stationary\ntwo-mode model which is then a good approximation at\nall times.\n2.4 Squeezing in the breathe-together solution\nIn this section we restrict to a spherically symmetric har-\nmonic potential Uext=mω2r2/2 identical for the two\ninternal sates. For values of the inter particle scattering\nlengths such that\naab<aaa,abb (30)\nand for a particular choice of the mixing angle such that\nthe mean ﬁeld seen by the two condensates with ¯Naand\n¯Nbparticles is the same:\n¯Nagaa+¯Nbgab=¯Nbgbb+¯Nagab≡¯Ng, (31)\nthewavefunctions ¯φaand¯φbsolvethesameGross-Pitaevskii\nequation. In the Thomas-Fermi limit, the wave functions\n¯φaand¯φbshare the same scaling solution ¯φ[18,19] and\n“breathe-together” [9].\n¯φa=¯φb=¯φ(r,t)≡e−iη(t)\nL3/2(t)eimr2˙L(t)/2/planckover2pi1L(t)¯φ0(r/L(t))\n(32)00.10.20.30.40.510−310−210−1100\nt [s]ξ2Two−mode stationary\nModulus−phase 5 GPE\nFull numerical simulation\n0 0.1 0.2 0.3 0.4 0.5-101\nt [s]/K6A×π \nFig. 3.(Top) spin squeezing as a function of time in a case in\nwhich the spatial dynamics is weakly excited. Blue solid lin e:\nfull numerical calculation with 1000 Fock states. Red dashe d\nline: modulus-phase model. Violet dash-dotted line: stati on-\nary two-mode model. (Bottom) angle giving the direction of\nthe mean spin projection in the equatorial plane of the Bloch\nsphere. Parameters: ω= 2π×42.6 Hz,¯Na=¯Nb= 1×104,\nm=87 a.m.u., aaa= 100.44rB,abb= 95.47rB,aab= 88.28rB.\nNo particle losses. rBis the Bohr radius.\nwith\n˙η=g\ngaa¯µ\nL3/planckover2pi1(33)\nd2L\ndt2=g\ngaaω2\nL4−ω2L; (34)\n¯φ0(r) =/parenleftbigg15\n8πR3\n0/parenrightbigg1/2/bracketleftbigg\n1−r2\nR2\n0/bracketrightbigg1/2\n(35)\n¯µis the chemical potential of the stationary condensate\nbeforetheπ/2 pulse, when all the Natoms are in state a,\nandR0=/radicalbig\n2¯µ/mω2is the corresponding Thomas-Fermi\nradius. The initial conditions for (34) are L(0) = 1 and\n˙L(0) = 0.\nNote that the scaling solution identical for the two\nmodesaandbis valid only for Na=¯Na,Nb=¯Nband\ndoes not apply to all the wave functions φa(Na,Nb) and\nφb(Na,Nb) in the expansion equation (4). Nevertheless,\nan advantage of choosing the mixing angle in order to sat-\nisfy the breathe-together condition equation (31), is that\nthe mean spin has no drift velocity. In Fig.4 we calculate\nthe spin squeezing (Top) and the angle ϕgiving the direc-\ntion of the mean spin projectionon the equatorialplane of\nthe Bloch sphere (Bottom), for the same parameters as in\nFig.3 except for the mixing angle that we now choose sat-6 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\nisfying equation (31) while in Fig.3 we had ¯Na=¯Nb. Note\nthatϕpractically does not evolve. The maximum amount\nof squeezing is lower in the breathe-together conﬁguration\nthan in the even-mixing case (see also [11]). However, as\nwe will see in the next section, this conclusion does not\nhold when particle losses are taken into account.\n00.10.20.30.40.510−210−1100101102\nt [s]ξ2Two−mode stationary\nModulus−phase 5 GPE\nFull numerical simulation\n0 0.1 0.2 0.3 0.4 0.5-101\nt [s]/K6A×π \nFig. 4.(Top) spin squeezing in breathe-together conditions as\na function of time. Blue solid line: full numerical calculat ion.\nRed dashed line: modulus-phase model. Violet dash-dotted\nline: stationary two-mode model. (Bottom) angle giving the\ndirection of the mean spin projection on the equatorial plan e\nof the Bloch sphere. Parameters: ω= 2π×42.6 Hz,¯Na= 7432,\n¯Nb= 12568,m=87 a.m.u., aaa= 100.44rB,abb= 95.47rB,\naab= 88.28rB. No particle losses. rBis the Bohr radius.\nBy linearization of φa(Na,Nb) andφb(Na,Nb) around\nthe breathe-together solution ¯φand using classical hydro-\ndynamics, it is even possible to calculate analytically the\nparameters χdandχsrelevant for the squeezing dynamics\n[9]. One obtains:\nχd(r,t) =−1\n2/planckover2pi1/parenleftbigg2\n5¯µ\nN/parenrightbigggaa+gbb−2gab\ngaa\n×/braceleftBigg/integraldisplayt\n0dt′\nL3(t′)+5\n2ImB(t)\nΩ5/bracketleftBigg/parenleftbiggr\nL(t)R0/parenrightbigg2\n−3\n5/bracketrightBigg/bracerightBigg\n(36)\nχs(r,t) =/parenleftbig\n|Cb|2−|Ca|2/parenrightbig\nχd=χ0(r,t). (37)\nwith\nΩ5=/parenleftbigg¯Na¯Nb\nN2gaa+gbb−2gab\ngaa/parenrightbigg1/2\n51/2ω(38)and where Im B(t) is solution of the diﬀerential equations\ni˙A=Ω5\nL2(t)B (39)\ni˙B=Ω5\nL3(t)A (40)\nto be solved together with equation (34), with initial con-\nditionsA(0) =B(0) = 1. In practice, when we expand the\ncondensate wave functions around the breathe-together\nsolution equation (32) as in [9], we encounter the hydro-\ndynamics operator S[20]\nS[α]≡ −Ngaa\nMdiv[¯φ2\n0gradα]. (41)\nThe deviation of the relative phase and the relative den-\nsity from the breathe-together solution expand over two\neigenmodes of S: A zero-energy mode which grows lin-\nearly in time and gives the dominant features of phase\ndynamics and squeezing (integral term in the curly brack-\nets in Eq.(36)), and a breathing mode of frequency Ω5\nwhich is responsible for the oscillations of the squeezing\nparameter. The fact that in breathe-together conditions\nand within the modulus-phase approximation χs=χ0is\nshown in Appendix C.\nWe give an example corresponding to strongly oscil-\nlating wave functions in Fig.5 where we compare the spin\nsqueezing from the analytical theory with a numerical\nsimulation. In the analytical formula, the entanglement\nbetween spatial degrees of freedom and spin dynamics is\napparent as χdequation (36) is position dependent. The\npoints in which the dynamical curve (dotted line) touches\nthe stationary two mode curve (dash-dotted line) corre-\nspond to Im B= 0 (see the bottom curve) where space\nand spin dynamics are disentangled. We note however\nthat the validity conditions of classicalhydrodynamics are\nmore stringent for a mixture of condensates with rather\nclose scattering lengths than for a single condensate [9].\nWe checked numerically that in order for equation (36)\nto correctly predict the frequency of the oscillations in\nthe squeezing parameter, we have to enter deeply in the\nThomas-Fermi regime.\n2.5 “Extracted” spin squeezing\nAs we pointed out, the deﬁnitions equations (10)-(12) ex-\nplicitlyinclude the spatialoverlapbetween thetwomodes.\nHere we give an alternative deﬁnition that can be used al-\nways, whether or not the modes overlap. To this aim, we\nintroduce the time-dependent operators\n˜a=/integraldisplay\nd3r¯φ∗\na(r,t)ˆψa(r), (42)\n˜b=/integraldisplay\nd3r¯φ∗\nb(r,t)ˆψb(r), (43)\nwhere¯φε(r,t) is the solution of Gross-Pitaevskii equation\n(7) for mode εwith¯Na,¯Nbparticles. We then introducePlease give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 7\n0  0.20.40.60.81  1.21.41.610−410−310−210−1100\nt [ms]ξ2Two−mode stationary\nAnalytical solution\nModulus−phase 5 GPE\nFull numerical sumulation\n00.2 0.4 0.6 0.8 1.0  1.2 1.4 1.6-202\nt [ms]Im\nFig. 5. (Top) test of the analytical formula equation (36) in\nthe deep Thomas-Fermi regime. Spin squeezing as a function\nof time. Blue solid line: full numerical calculation. Red da shed\nline: modulus-phase model. Black dotted line: analytical c urve\nusing equation (36). Violet dash-dotted line: stationary t wo-\nmodemodelusing(27)-(28).(Bottom)functionIm B(t).Spatial\nand spin dynamics disentangle when Im B(t) = 0. Parameters:\n¯Na=¯Nb= 5×105,ω= 2π×2 kHz,m=87 a.m.u., aaa=abb=\n0.3aho,aab= 0.24aho.ahois the harmonic oscillator length:\naho=p\n/planckover2pi1/Mω. No particle losses.\nthe spin operators:\n˜Sx=1\n2(˜b†˜a+˜a†˜b), (44)\n˜Sy=i\n2(˜b†˜a−˜a†˜b), (45)\n˜Sz=1\n2(˜a†˜a−˜b†˜b). (46)\nIn the new deﬁnition of spin squeezing calculated by the\nspin operators deﬁned in equations (44)-(46), which we\ncall the “extracted” spin squeezing, we still take into ac-\ncount entanglement between external motion and spin dy-\nnamics, but we give up the information about the over-\nlap between the two modes. In Appendix D, we give the\nquantum averages useful to calculate the extracted spin\nsqueezing within the modulus-phase approach described\nin Section 2.3. We will use this extracted spin squeezing\nin Section 5.\nComparing the expressions given in Appendix D with\nthose of Appendix F (in the absence of losses), one real-\nizes that in the stationary case, where χd,χsandχ0are\nspace independent, the extracted spin squeezing dynami-\ncal model reduces to a two-mode model that we study in\ndetail in the next section.3 Two-mode model with Particle losses\nIn this section we generalize our results of [12] to possibly\noverlapping and non-symmetric condensates. In subsec-\ntion 3.1 we address the general case, while in subsection\n3.2 we restrict to symmetric condensates and perform an-\nalytically an optimization of the squeezing with respect\nto the trap frequency and number of atoms. In the whole\nsection, as in [12], we will limit to a two-mode stationary\nmodel and we do not address dynamical issues.\n3.1 Spin squeezing in presence of losses\nWe consider a two-component Bose-Einstein condensate\ninitiallypreparedinaphasestate,thatiswith welldeﬁned\nrelative phase between the two components,\n|Ψ(0)∝an}bracketri}ht=|ϕ∝an}bracketri}ht ≡/parenleftbig\n|Ca|e−iϕ/2a†+|Cb|eiϕ/2b†/parenrightbigN\n√\nN!|0∝an}bracketri}ht.(47)\nWhen expanded over Fock states, the state (47) shows bi-\nnomial coeﬃcients which, for large N, are peaked around\nthe average number of particles in aandb,¯Naand¯Nb. In\nthe same spirit as the “modulus-phase” approximation of\nsubsection 2.3, we can use this fact to expand the Hamil-\ntonian of the system to the second order around ¯Naand\n¯Nb\nH0≃E(¯Na,¯Nb)+/summationdisplay\nε=a,bµε(ˆNε−¯Nε)+1\n2∂Nεµε(ˆNε−¯Nε)2\n+1\n2(∂Nbµa+∂Naµb)(ˆNa−¯Na)(ˆNb−¯Nb) (48)\nwhere the chemical potentials µεand all the derivatives\nofµεshould be evaluated in ¯Naand¯Nb. We can write\nH0=fˆN+/planckover2pi1vˆN(ˆNa−ˆNb)+/planckover2pi1χ\n4(ˆNa−ˆNb)2(49)\nwith\nvˆN=1\n2/planckover2pi1/bracketleftBig\n(µa−µb)−/planckover2pi1χ(¯Na−¯Nb)+/planckover2pi1˜χ(ˆN−¯N)/bracketrightBig\n(50)\nχ=1\n2/planckover2pi1(∂Naµa+∂Nbµb−∂Nbµa−∂Naµb)¯Na,¯Nb(51)\n˜χ=1\n2/planckover2pi1(∂Naµa−∂Nbµb)¯Na,¯Nb. (52)\nThe function fof the total number of particles, ˆN=\nˆNa+ˆNb, commutes with the density operator of the sys-\ntem and can be omitted. The second term in equation\n(49) proportional to Szdescribes a rotation of the aver-\nage spin vector around the zaxis with velocity vˆN. The\nthird term proportional to S2\nzprovides the nonlinearity\nresponsible for spin squeezing. It also provides a second\ncontribution to the drift of the relative phase between the\ntwo condensates in the case ¯Na∝ne}ationslash=¯Nb.\nIn presence of losses, the evolution is ruled by a master\nequation for the density operator ρof the system. In the8 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\ninteraction picture with respect to H0, with one, two, and\nthree-body losses, we have:\nd˜ρ\ndt=3/summationdisplay\nm=1/summationdisplay\nε=a,bγ(m)\nε/bracketleftbigg\ncm\nε˜ρc†m\nε−1\n2{c†m\nεcm\nε,˜ρ}/bracketrightbigg\n+γab/bracketleftbigg\ncacb˜ρc†\nac†\nb−1\n2{c†\nac†\nbcacb,˜ρ}/bracketrightbigg\n(53)\nwhere ˜ρ=eiH0t//planckover2pi1ρe−iH0t//planckover2pi1,ca=eiH0t//planckover2pi1ae−iH0t//planckover2pi1, and\nsimilarly for b,\nγ(m)\nε=K(ε)\nm\nm/integraldisplay\nd3r|¯φε(r)|2m, (54)\nγab=Kab\n2/integraldisplay\nd3r|¯φa(r)|2|¯φb(r)|2.(55)\nK(ε)\nmis them-body rate constant ( m= 1,2,3) and¯φε(r)\nis the condensate wave function for the εcomponent with\nNa=¯NaandNb=¯Nbparticles.Kabis the rate constant\nfor a two-body loss event in which two particles coming\nfrom diﬀerent components are lost at once.\nIn the Monte Carlowavefunction approach[21] we de-\nﬁne an eﬀective Hamiltonian Heﬀand the jump operators\nJ(m)\nε(J(2)\nab)\nHeﬀ=−i/planckover2pi1\n23/summationdisplay\nm=1/summationdisplay\nε=a,bγ(m)\nεc†m\nεcm\nε−i/planckover2pi1\n2γabc†\nac†\nbcacb; (56)\nJ(m)\nε=/radicalBig\nγ(m)\nεcm\nε, J(2)\nab=√γabcacb (57)\nWe assume that a small fraction of particles will be lost\nduring the evolution so that we can consider χ,γ(m)\nεand\nγabas constant parameters of the model. The state evolu-\ntion in a single quantum trajectory is a sequence of ran-\ndom quantum jumps at times tjand non-unitary Hamil-\ntonian evolutions of duration τj:\n|Ψ(t)∝an}bracketri}ht=e−iHeff(t−tk)//planckover2pi1J(mk)\nεk(tk)e−iHeffτk//planckover2pi1J(mk−1)\nεk−1(tk−1)\n...J(m1)\nε1(t1)e−iHeffτ1//planckover2pi1|Ψ(0)∝an}bracketri}ht, (58)\nwhere nowεj=a,borab. Application of a jump J(mj)\nεj(tj)\nto theN-particle phase state at tjyields\nJ(mj)\nεj(tj)|φ∝an}bracketri}htN∝ |φ+∆jtj∝an}bracketri}htN−mj, (59)\n∆j= 2˜χδεj,ab+(˜χ+χ)mjδεj,a+(˜χ−χ)mjδεj,b.(60)\nAfter a quantum jump, the phase state is changed into a\nnew phase state, with mparticle less and with the relative\nphasebetweenthetwomodesshowingarandomshift ∆jtj\nwith respect to the phase before the jump. Note that in\nthe symmetrical case ˜ χ= 0 and no random phase shift\noccurs in the case of a jump of ab. Indeed we will ﬁnd\nthat at short times in the symmetrical case theses kind of\ncrossedablosses are harmless to the the squeezing.\nIn presence of one-body losses only, also the eﬀective\nHamiltonian changes a phase state into another phasestate and we can calculate exactly the evolution of the\nstate vector analytically, as we did in [12] for symmetrical\ncondensates. When two and three-body losses enter into\nplay, we introduce a constant loss rate approximation [22]\nHeﬀ≃ −i/planckover2pi1\n23/summationdisplay\nm=1/summationdisplay\nε=a,bγ(m)\nε¯Nm\nε−i/planckover2pi1\n2γab¯Na¯Nb≡ −i/planckover2pi1\n2λ\n(61)\nvalid when a small fraction of particles is lost at the time\nat which the best squeezing is achieved. In this approxi-\nmation, the mean number of particles at time tis\n∝an}bracketle{tˆN∝an}bracketri}ht=N\n1−\n/summationdisplay\nε=a,b/summationdisplay\nmΓ(m)\nε+Γab\nt\n(62)\nΓ(m)\nε≡¯Nm−1\nεmγ(m)\nε;Γab=γab/radicalbig¯Na¯Nb(63)\nwhere for example Γ(m)\nεtis the fraction of lost particles\ndue tom-body losses in the εcondensate. Let us present\nthe evolution of a single quantum trajectory: Within the\nconstant loss rate approximation, we can move all the\njump operators in (58) to the right. We obtain:\n|Ψ(t)∝an}bracketri}ht=e−λt/2k/productdisplay\nj=1J(mj)\nεj(tj)|Ψ(0)∝an}bracketri}ht\n=e−λt/2e−iTk|αk||ϕ+βk∝an}bracketri}htN−N(k)(64)\nwhere\n|αk|2=k/productdisplay\nj=1\n\n/summationdisplay\nm′=1,2,3/summationdisplay\nε′=a,bδmj,m′δεj,ε′¯Nm′|Cε′|2m′γ(m′)\nε′\n+¯N2|Ca|2|Cb|2γabδεj,ab/bracerightbig\n(65)\nβk=k/summationdisplay\nj=1tj\n\n2˜χδεj,ab+/summationdisplay\nm′=1,2,3m′δmj,m′\n×[(χ+ ˜χ)δεj,a−(χ−˜χ)δεj,b]/bracerightbig\n(66)\nN(k) =k/summationdisplay\nj=1/summationdisplay\nm′=1,2,3m′δmj,m′(δεj,a+δεj,b)+2δεj,ab,(67)\nandTkis a phase which cancels out when taking the av-\nerages of the observables [23].\nThe expectation value of any observable ˆOis obtained\nby averaging over all possible stochastic realizations, that\nis all kinds, times and number of quantum jumps, each\ntrajectory being weighted by its probability\n∝an}bracketle{tˆO∝an}bracketri}ht=/summationdisplay\nk/integraldisplay\n0<t1<t2<···tk<tdt1dt2···dtk/summationdisplay\n{εj,mj}∝an}bracketle{tΨ(t)|ˆO|Ψ(t)∝an}bracketri}ht.(68)\nNotethat the singletrajectory(64) is not normalized.The\nprefactor will provide its correct “weight” in the average.\nWe report in Appendix E and F the averages needed\nto calculate the spin squeezing for one-body losses onlyPlease give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 9\n(exact solution) and for one, two and three-body losses\n(constant loss rate approximation) respectively. The ana-\nlytical results are expressed in terms of the parameters χ\nand ˜χdeﬁned in equations (51) and (52) respectively and\nof the drift velocity\nv=1\n2/planckover2pi1/bracketleftbig\n(µa−µb)−/planckover2pi1χ(¯Na−¯Nb)+/planckover2pi1˜χ(N−¯N)/bracketrightbig\n,(69)\nwhereNis the total initial number of atoms.\n3.2 Symmetrical case: optimization of spin squeezing\nIf we restrict to symmetrical condensates which may or\nmay not overlap, we can carry out analytically the opti-\nmizationofsqueezingin presenceoflosses.In the symmet-\nric case and constant loss rate approximation it turns out\nthat∆S2\nz=∝an}bracketle{tˆN∝an}bracketri}ht/4. This allows to express ξ2in a simple\nway:\nξ2=∝an}bracketle{ta†a∝an}bracketri}ht\n∝an}bracketle{tb†a∝an}bracketri}ht2/parenleftBig\n∝an}bracketle{ta†a∝an}bracketri}ht+˜A−/radicalbig\n˜A2+˜B2/parenrightBig\n,(70)\nwith\n˜A=1\n2Re/parenleftbig\n∝an}bracketle{tb†a†ab−b†b†aa∝an}bracketri}ht/parenrightbig\n(71)\n˜B= 2 Im/parenleftbig\n∝an}bracketle{tb†b†ba∝an}bracketri}ht/parenrightbig\n. (72)\nAn analytical expression for spin squeezing is calculated\nfrom (70) with\n∝an}bracketle{tb†a∝an}bracketri}ht=e−λt\n2cosN−1(χt)˜NF1 (73)\n˜A=e−λt\n8˜N(˜N−1)/bracketleftbig\nF0−F2cosN−2(2χt)/bracketrightbig\n(74)\n˜B=e−λt\n2cosN−2(χt)sin(χt)˜N(˜N−1)F1(75)\nwhere the operator ˜N= (N−∂σ) acts on the functions\nFβ(σ) = exp/bracketleftBigg3/summationdisplay\nm=12γ(m)teσmsin(mβχt)\nmβχtcosm(βχt)\n+γabte2σ\ncos2(βχt)/bracketrightbigg\n, (76)\nwithβ= 0,1,2 and all expressions should be evaluated in\nσ= ln¯N.\nWe wantnow to ﬁnd simple resultsfor the best squeez-\ning and the best squeezing time in the large Nlimit. In\nthe absence of losses [1] the best squeezing and the best\nsqueezing time in units of 1 /χscale asN−2/3. We then\nsetN=ε−3and rescale the time as χt=τε2. We expand\n(70) forε≪1uptoorder2included,keeping Γ(m)/χcon-\nstant. The key point is that in this expansion, for large N\nand short times, the crossed losses abdo not contribute.\nAs in [12], introducing the squeezing ξ2\n0(t) in the no-loss\ncase, we obtain:\nξ2(t) =ξ2\n0(t)/bracketleftbigg\n1+1\n3Γsqt\nξ2\n0(t)/bracketrightbigg\n. (77)with:\nΓsq=/summationdisplay\nmΓ(m)\nsqandΓ(m)\nsq=mΓ(m).(78)\nThe result (77) very simply accesses the impact of losses\non spin squeezing. First it shows that losses cannot be ne-\nglected as soon as the lost fraction of particles is of the\norder ofξ2\n0. Second it shows that in the limit N→ ∞\nandξ2\n0(tbest)→0, the squeezing in presence of losses is of\nthe order of the lost fraction of particles at the best time:\nξ2(tbest)∼Γsqtbest/3. This also sets the limits of validity\nof our constant loss rate approximation. For our approx-\nimation to be valid, the lost fraction of particle, hence\nsqueezing parameter at the best squeezing time, should\nbe small.\nFrom now on, the optimization of the squeezing in the\nlargeNlimit proceeds much as in the case of spatially\nseparated condensates [12]. The only diﬀerence is in the\nstationary wave functions in the Thomas-Fermi limit. For\noverlappingcondensatesweconsiderastablemixturewith\naab<aaa=abb, (79)\nand we introduce the sumanddiﬀerence of the intra and\ninter-species s-wave scattering lengths:\nas=aaa+aab (80)\nad=aaa−aab. (81)\nIn the symmetric case considered here we have\nµa=µb=1\n2/planckover2pi1¯ω/bracketleftbigg15\n2Nas\naho/bracketrightbigg2/5\n, (82)\nχ=23/532/5\n53/5/parenleftbigg/planckover2pi1\nM/parenrightbigg−1/5\n¯ω6/5N−3/5ad\na3/5\ns(83)\nΓ(1)=K1 (84)\nΓ(2)=152/5\n27/57π/parenleftbigg/planckover2pi1\nM/parenrightbigg−6/5\n¯ω6/5N2/5a−3/5\nsK2(85)\nΓ(3)=54/5\n219/531/57π2/parenleftbigg/planckover2pi1\nM/parenrightbigg−12/5\n¯ω12/5N4/5\na−6/5\nsK3, (86)\nwhereaho=/radicalbig\n/planckover2pi1/M¯ωis the harmonic oscillator length, ¯ ω\nis the geometric mean of the trap frequencies. We recover\nthe case of spatially separated condensates [12] setting\naab= 0 in (80)-(81).\nThe squeezing parameter for the best squeezing time\nξ2(tbest,¯ω) is minimized for an optimized trap frequency\n¯ωopt=219/1275/12π5/6\n151/3/planckover2pi1\nMa1/2\ns\nN1/3/parenleftbiggK1\nK3/parenrightbigg5/12\n.(87)\nNote however that this optimization concerns one- and\nthree-body losses only. The eﬀect of decoherence due two\ntwo-body losses quantiﬁed by the ratio Γ(2)/χis indepen-\ndent of the trap frequency.10 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\nOnce the trap frequency is optimized, ξ2(tbest,¯ωopt)\nis a decreasing function of N. The lower bound for ξ2,\nreached for N→ ∞is then\ninf\nt,¯ω,Nξ2=/bracketleftBigg\n5√\n3M\n28π/planckover2pi1/bracketrightBigg2/3/bracketleftBigg/radicalBigg\n7\n2K1K3\na2\nd+K2\nad/bracketrightBigg2/3\n.(88)\nAsimpleoutcomeofthisanalyticstudyisthat,forpos-\nitive scattering lengths aaa,aab, the maximum squeezing\nis obtained when aab= 0 that is for example for spatially\nseparated condensates. Another possibility is to use a Fes-\nchbach resonance to decrease the inter-species scattering\nlengthaab[14,15], knowing that the crossed a−blosses\ndo not harm the squeezing at short times.\n4 Results for overlapping condensates\nIn this and the next section we give practical examples of\napplicationoftheanalysisledin thetwoprevioussections.\n4.1 Feshbach resonance-tuned bimodal Rb BEC\nWe consider a bimodal Rb condensate in |F= 1,mF= 1∝an}bracketri}ht\nand|F= 2,mF=−1∝an}bracketri}htstates where the scattering length\naabis lowered by about 10% with respect to its bare value\nusing a Feshbach resonance [14,15].\nIn Fig.6 (Top) and (Bottom) we compare a situation\nin which the initial condensate is split evenly in the aand\nbcomponents to a situation in which the mixing is cho-\nsen in order to satisfy the “breathe-together” conditions\n(31). For the considered parameters, which are the same\nas Fig.3 and Fig.4, the spatial dynamics is not important\nand the two-mode model is a good approximation at all\ntimes.\nThe squeezing in presence of losses is calculated using\nour general results of Section 3.1 for asymmetric conden-\nsates. Although without losses the even splitting is more\nfavorable,withone,two,three-bodylossesresultsarecom-\nparableξ2≃6×10−2. We also show a curve obtained for\none and three-body losses only (dashed-line). It is clear\nthat for the considered Rb states the dominant contribu-\ntion for decoherence comes from the two-body losses in\ntheF= 2 state severely limiting the maximum amount\nof obtainable squeezing.\nIn the cases considered in Fig.6 asymmetric two-body\nlosses are very high, we therefore check the validity of\nthe constant loss rate approximation with an exact Monte\nCarlo wave function simulation. The main result is that\nthe constant loss rate approximation is accurate up to\nthe best squeezing time. A more complete discussion is\npresented in Appendix G.\n4.2 Bimodal BEC of Na atoms\nBy using two states in the lower hyperﬁne manifold, one\ncan greatly reduce two body losses. A possible example is0 0.05 0.1 0.15 0.210−310−210−1100\nt [s]ξ2Spin squeezing without losses\nSpin squeezing with 1,3−body losses\nSpin squeezing with 1,2,3−body losses\n0 0.05 0.1 0.15 0.210−210−1100\nt [s]ξ2Spin squeezing without losses\nSpin squeezing with 1,3−body losses\nSpin squeezing with 1,2,3−body losses\nFig. 6. Spin squeezing with and without losses in a bimodal\nRb condensate from the stationary two-mode model. A Fesh-\nbach resonance is used to reduce the inter-species scatteri ng\nlength. Violet dash-dotted line: without losses. Blue dash ed\nline: with one and three-body losses. Red solid line: with\none, two and three-body losses. (Top) with a 50 −50 mix-\ning of the two states: ¯Na=¯Nb= 104,χ= 5.367×10−3s−1,\n˜χ= 5.412×10−4s−1. (Bottom) in breathe-together conditions:\n¯Na= 7432, ¯Nb= 12568,χ= 5.392×10−3s−1, ˜χ= 1.386×\n10−3s−1. Other parameters: ω= 2π×42.6 Hz,m=87 a.m.u.,\naaa= 100.44rB,abb= 95.47rB,aab= 88.28rB,rBis the\nBohr radius. Particle losses: K(a)\n1=K(b)\n1= 0.01s−1,K(a)\n2= 0,\nK(b)\n2= 119×10−21m3s−1[24],K(ab)\n2= 78×10−21m3s−1[25],\nK(a)\n3= 6×10−42m6s−1[26].\nof using Na atoms in the |F= 1,mF=±1∝an}bracketri}htstates [6].\nIn Fig.7 we calculate the best obtainable squeezing with\nthese two states. Parameters are chosen according to our\noptimization procedure of Section 3.2. A large amount of\nsqueezingξ2= 1.9×10−3can be reached at the best\nsqueezing time.\nUsingourfullnumericalandourapproximateddynam-\nical approaches, (not shown) we checked that the two-\nmode model is an excellent approximation for these pa-\nrameters.Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 11\n00.04 0.08 0.12 0.1610−410−310−210−1100\nt [s]ξ2Spin squeezing without losses\nSpin squeezing with 1,3−body losses\nFig. 7.Spinsqueezingwith andwithoutlosses inabimodal Na\ncondensate from the stationary two-mode model. Violet dash -\ndotted line: without losses. Blue dashed line: with one and\nthree-body losses. Optimized parameters: ¯Na=¯Nb= 4×104\nω= 2π×183 Hz,m=23 a.m.u., aaa=abb= 51.89rB,aab=\n48.25rB,rBis the Bohr radius. χ= 5.517×10−3s−1, ˜χ= 0.\nParticle losses: K(a)\n1=K(b)\n1= 0.01s−1,K(a)\n2=K(b)\n2= 0,\nK(a)\n3=K(b)\n3= 2×10−42m6s−1[27].\n5 Dynamically separated Rb BEC\nIn this subsection we consider a bimodal Rb condensate\nin|F= 1,mF=−1∝an}bracketri}htand|F= 2,mF= 1∝an}bracketri}htstates. Rather\nthan using a Feshbach resonance to change gab, we con-\nsider the possibility of suddenly separating the two clouds\nright after the mixing π/2 pulse using state-selective po-\ntentials [28], and recombining them after a well chosen\ninteraction time. A related scheme using Bragg pulses in\nthe frameof atominterferometrywas proposedin [29]. We\nconsider disc shaped identical traps for the two states a\nandbwithωz> ωx,y≡ω⊥, that can be displaced inde-\npendently along the zaxes. In order to minimize center-\nof-mass excitation of the cloud, we use a triangular ramp\nfor the displacement velocity, as shown in Fig.8 (Bottom),\nwith total move-out time 2 τ= 4π/ωz[30]. In Fig.8 (Top)\nwe show the z-dependence of densities of the clouds, in-\ntegrated in the perpendicular xyplane, as the clouds are\nseparated and put back together after a given interaction\ntime.\nWe use our dynamical modulus-phase model in 3 di-\nmensions to calculate the spin squeezing in this scheme.\nAs the spatial overlap between the two clouds reduces\na lot as they are separated, in Fig.9 we calculate both\nthe spin squeezing obtained from the deﬁnitions (10)-(12)\nof spin operators (dashed line), and the “extracted spin\nsqueezing”introduced in Section 2.5based on the “instan-\ntaneous modes” (42)-(43) (solid line). The oscillations in\nthe dashed line are due to tiny residual center of mass os-\ncillations of the clouds that change periodically the small\noverlap between the two modes. They are absent in the\nextracted spin squeezing curve (solid line) as they do not\naﬀect the spin dynamics. When the clouds are put back\ntogetherandthe overlapbetweenthe modesislargeagain,\nthe spinsqueezingandthe extractedspin squeezingcurves\n0 4 8 12 16-202x 10-3\nt [ms]δz [m/s] /j46\nFig. 8. (Top)|φa(z,t)|2and|φb(z,t)|2in arbitrary units as\nthe clouds are separated and put back together after an inter -\naction time of about 15 ms. The harmonic potential for the\na-component does not move, while that for the b-component\nis shifted vertically with a speed ˙δz. The distance between the\ntwo trap centers when they are separated is δz= 4p\n/planckover2pi1/Mωz.\n(Bottom) variation in time of ˙δz. Parameters: ¯Na=¯Nb=\n5×104,ωx,y= 2π×2.31 Hz,ωz= 2π×1 kHz,m=87 a.m.u.,\naaa= 100.44rB,abb= 95.47rB,aab= 98.09rB,rBis the Bohr\nradius. No particle losses.\ngive close results (not identical as the overlap of the two\nclouds is not precisely one).\nIn Fig.10 (Top) we compare the extracted spin squeez-\ning curve of Fig.9 (solid line) with a two-mode stationary\ncalculation (dash-dotted line) assuming stationary con-\ndensates in separated wells. We notice that the squeezing\nprogresses much more slowly in the dynamical case. In-\ndeed when we separate the clouds, the mean ﬁeld changes\nsuddenly for each component exciting a breathing mode\nwhose amplitude and frequency is diﬀerent for each of the\nFock states involved. In the quasi 2D conﬁguration con-\nsidered here, the breathing of the wave functions is well\ndescribed by a scaling solution in 2D for each condensate\nseparately [18,19] adapted to the case in which the trap\nfrequency is not changed, but the mean-ﬁeld is changed\nsuddenly after separating the two internal states:\nφε(r⊥,t) =e−iηε(t)\nLε(t)eimr2\n⊥˙Lε(t)/2/planckover2pi1Lε(t)φ0/parenleftbiggr⊥\nLε(t)/parenrightbigg\n(89)12 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\n0246810 1214161810−2100102104\nt [ms]ξ2\nFig. 9.SpinsqueezingasthetwoRbcondensatesareseparated\nand put backtogether after an interaction time of about 15 ms .\nRed dashed line: Spin squeezing obtained from the deﬁnition s\n(10)-(12) of the spin operators explicitly including the ov erlap\nbetween the clouds. Blue solid line: extracted spin squeezi ng\nbased of the “instantaneous modes” (42)-(43). Parameters a s\nin Fig.8.\nwith\n˙ηε=˜gεε\n˜gaa¯µ\nL2ε/planckover2pi1(90)\nd2Lε\ndt2=Nε\nN˜gεε\n˜gaaω2\n⊥\nL3ε−ω2\n⊥Lε; (91)\nφ0(r⊥) =/parenleftbigg2\nπR2\n0/parenrightbigg1/2/bracketleftbigg\n1−r2\n⊥\nR2\n0/bracketrightbigg1/2\n.(92)\n¯µis the chemical potential of the stationary condensate\nbeforetheπ/2 pulse, when all the Natoms are in state\na,R0=/radicalbig\n2¯µ/mω2\n⊥is the corresponding Thomas-Fermi\nradius, and ˜ gεεis a reduced coupling constant to describe\nthe interaction between two atoms in the εcondensate in\nquasi 2D system, where we assume that the condensate\nwave functions in the conﬁned direction are Gaussians:\n˜gεε=4π/planckover2pi12aεε\nM/radicalbigg\nMωz\n2π/planckover2pi1(93)\nwithaεεthe 3D scattering length. The initial conditions\nfor (91) are Lε(0) = 1 and ˙Lε(0) = 0.\nWe can use (89) to calculate the squeezing (dotted\ncurve)andwenotethatit reproduceswellthe spinsqueez-\ning curve obtained integrating 5 Gross-Pitaevskii equa-\ntions in 3D (full line). As we studied in detail in Section\n2.4, oscillations of the wave functions cause oscillations\nof the squeezing parameter due to entanglement between\nspatial and spin dynamics. Indeed what we see in the ex-\ntracted spin squeezing curve of Fig.10 (Top) is the begin-\nning of a slow oscillation for the squeezing parameter. In\nFig.10 (Bottom) we show the long time behavior. There\nare indeed times at which the spatial and spin dynamics\ndisentangle, and the dynamical curve and the steady state\ncurve touch (see Sect. 2.4). Unfortunately these times are0    246810 1214161810−210−1100\nt [ms]ξ2Extracted spin squeezing\nScaling solution\nStationary solution\nStationary solution with\n1,2,3−body losses       \n00.050.10.150.20.2510−410−310−210−1100\nt [s]ξ2Scaling solution\nStationary solution\nStationary solution with\n1,2,3−body losses       \nFig. 10. Spin squeezing as a function of time. (Top) compari-\nson between a dynamical calculation and a stationary calcul a-\ntion. Blue solid line: extracted spin squeezing in 3D. Black\ndoted line: 2D scaling solution based on (91). Violet dash-\ndotted line: stationary calculation in 3D without losses. R ed\nsolid line: stationary calculation in 3D with losses. Spins queez-\ning progresses more slowly in the dynamical calculation tha n\nin the stationary calculation. (Bottom) long time behavior .\nBlack doted line: scaling solution. Violet dash-dotted lin e: sta-\ntionary calculation without losses. Red solid line: statio nary\ncalculation with losses. Parameters: χ= 5.003×10−3s−1,\n˜χ= 1.342×10−4s−1,K(a)\n1=K(b)\n1= 0.01s−1,K(a)\n2= 0,\nK(b)\n2= 119×10−21m3s−1[24],K(a)\n3= 6×10−42m6s−1[26].\nThe other parameters are as in Fig.8.\nnot accessible here in presence of losses (in particular the\nhigh two-body lossesin the higher hyperﬁne state). Notice\nthat in the ﬁrst 15 ms of evolution considered in Fig.9 and\nFig.10 (Top) the eﬀect of losses is small and the main lim-\nitation at short times is provided by the spatial dynamics.\nFor a lowernumber of atoms, the sudden changein the\nmean ﬁeld and the consequent oscillations of the squeez-\ning parameter are reduced. In Fig.11 we show the spin\nsqueezingobtainedbysuddenlyseparatingtwoBECofRb\natoms in |F= 1,mF=−1∝an}bracketri}htand|F= 2,mF= 1∝an}bracketri}htstates\nwith 1000 atoms in each component. The dotted line is\na dynamical calculation using the quasi 2D scaling solu-\ntion (91) (and no losses), while the dash-dotted line andPlease give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 13\n0 0.01 0.02 0.03 0.0410−310−210−1100\nt [s]ξ2Scaling solution\nStationary solution\nStationary solution with\n1,2,3−body losses       \nFig. 11. Spin squeezing as a function of time in two small\nRb condensates. Black doted line: scaling solution based on\n(91). Violet dash-dotted line: stationary calculation wit hout\nlosses. Red solid line: stationary calculation with losses . Pa-\nrameters:K(a)\n1=K(b)\n1= 0.01s−1,K(a)\n2= 0,K(b)\n2= 119×\n10−21m3s−1[24],K(a)\n3= 6×10−42m6s−1[26].Theotherparam-\neters:¯Na=¯Nb= 103,ωx,y= 2π×11.82 Hz,ωz= 2π×2 kHz,\nm=87 a.m.u., aaa= 100.44rB,abb= 95.47rB,aab= 98.09rB,\nrBis the Bohr radius. χ= 0.213s−1, ˜χ= 2.763×10−3s−1.\nthe solid line are stationary calculations without and with\nlosses respectively. Note that around t= 0.02s, where the\ndynamical curve and the stationary curve touch, a squeez-\ning of about ξ2∼2×10−2could be reached despite the\nhigh losses in the F= 2 state [31].\n6 Conclusions\nIn conclusion we developed a method to study the en-\ntangled spatial and spin dynamics in binary mixtures of\nBose-Einstein condensates. The method, which is the nat-\nuralextensionofourwork[9] tothe caseofspin squeezing,\nallows a full analytical treatment in some cases and can\nbe used in the general case to study a prioricomplicated\nsituations in 3D without the need of heavy numerics. In-\ncluding the eﬀect of particle losses and spatial dynamics,\nwe have calculated the maximum squeezing obtainable in\na bimodal condensate of Na atoms in |F= 1,mF=±1∝an}bracketri}ht\nstates when the two condensates overlap in space, and we\nhave calculated the squeezing in a bimodal Rb conden-\nsate in which a Feshbach resonance is used to reduce the\ninter-species scattering length as recently realized experi-\nmentally [15]. For Rb we also propose an original scheme\nin which the two components are spatially separated us-\ning state-dependent potentials, recently realized for the\n|F= 1,mF=−1∝an}bracketri}htand|F= 2,mF= 1∝an}bracketri}htstates, and then\nrecombined after a well chosen squeezing time. With this\nmethod we show that ξ2∼2×10−2could be reached\nin condensates of 1000 atoms, despite the high two-body\nlosses in the higher hyperﬁne state.\nYun Li acknowledges support from the ENS-ECNU\nprogram, and A.S. acknowledges stimulating discussionswith M. Oberthaler, J. Est` eve and K. Mølmer. Our group\nis a member of IFRAF.\nA Quantum averages of the ﬁeld operators\nUsing equations (20)-(21), the averages needed to calcu-\nlate squeezing parameter can be written in terms of the\nwave functions φa,φband the phase factor Asolution of\nequation (9):\n∝an}bracketle{tˆψ†\nb(r)ˆψa(r)∝an}bracketri}ht\n=N/summationdisplay\nNa=1N!\n(Na−1)!Nb!|Ca|2(Na−1)|Cb|2NbC∗\nbCa\n×φ∗\nb(Na−1,Nb+1,r)φa(Na,Nb,r)\n×exp{i[A(Na−1,Nb+1)−A(Na,Nb)]//planckover2pi1}\n×[∝an}bracketle{tφa(Na−1,Nb+1)|φa(Na,Nb)∝an}bracketri}ht]Na−1\n×[∝an}bracketle{tφb(Na−1,Nb+1)|φb(Na,Nb)∝an}bracketri}ht]Nb.(94)\n∝an}bracketle{tˆψ†\nb(r)ˆψ†\na(r′)ˆψa(r)ˆψb(r′)∝an}bracketri}ht\n=N−1/summationdisplay\nNa=1N!\n(Na−1)!(Nb−1)!|Ca|2Na|Cb|2Nb\n×φ∗\nb(Na,Nb,r)φ∗\na(Na,Nb,r′)φa(Na,Nb,r)\n×φb(Na,Nb,r′). (95)\n∝an}bracketle{tˆψ†\nb(r)ˆψ†\nb(r′)ˆψa(r)ˆψa(r′)∝an}bracketri}ht\n=N/summationdisplay\nNa=2N!\n(Na−2)!Nb!|Ca|2(Na−2)|Cb|2NbC∗2\nbC2\na\n×φ∗\nb(Na−2,Nb+2,r)φ∗\nb(Na−2,Nb+2,r′)\n×φa(Na,Nb,r)φa(Na,Nb,r′)\n×exp{i[A(Na−2,Nb+2)−A(Na,Nb)]//planckover2pi1}\n×[∝an}bracketle{tφa(Na−2,Nb+2)|φa(Na,Nb)∝an}bracketri}ht]Na−2\n×[∝an}bracketle{tφb(Na−2,Nb+2)|φb(Na,Nb)∝an}bracketri}ht]Nb.(96)\n∝an}bracketle{tˆψ†\nb(r)ˆψ†\nb(r′)ˆψb(r)ˆψa(r′)∝an}bracketri}ht\n=N−1/summationdisplay\nNa=1N!\n(Na−1)!(Nb−1)!|Ca|2(Na−1)|Cb|2NbC∗\nbCa\n×φ∗\nb(Na−1,Nb+1,r)φ∗\nb(Na−1,Nb+1,r′)\n×φb(Na,Nb,r)φa(Na,Nb,r′)\n×exp{i[A(Na−1,Nb+1)−A(Na,Nb)]//planckover2pi1}\n×[∝an}bracketle{tφa(Na−1,Nb+1)|φa(Na,Nb)∝an}bracketri}ht]Na−1\n×[∝an}bracketle{tφb(Na−1,Nb+1)|φb(Na,Nb)∝an}bracketri}ht]Nb−1.(97)14 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\n∝an}bracketle{tˆψ†\na(r)ˆψ†\na(r′)ˆψa(r)ˆψb(r′)∝an}bracketri}ht\n=N−1/summationdisplay\nNa=1N!\n(Na−1)!(Nb−1)!|Ca|2Na|Cb|2(Nb−1)C∗\naCb\n×φ∗\na(Na+1,Nb−1,r)φ∗\na(Na+1,Nb−1,r′)\n×φa(Na,Nb,r)φb(Na,Nb,r′)\n×exp{i[A(Na+1,Nb−1)−A(Na,Nb)]//planckover2pi1}\n×[∝an}bracketle{tφa(Na+1,Nb−1)|φa(Na,Nb)∝an}bracketri}ht]Na−1\n×[∝an}bracketle{tφb(Na+1,Nb−1)|φb(Na,Nb)∝an}bracketri}ht]Nb−1.(98)\nWeusetheseaveragestocalculatethe squeezinginourfull\ndynamical model. In practice we do not sum over all the\nFock states but over a “large enough width” (typically\n>6√\nN) around the average number of atoms ¯Na,¯Nb.\nThe spin squeezing is obtained by equation (15) using the\ndeﬁnitions (10)-(12) for the spin operators.\nB Quantum averages in the modulus-phase\napproach\nWithinthemodulus-phaseapproximation,thescalarprod-\nuct of the wave vectors can be written as\n∝an}bracketle{tφa(Na−β,Nb+β)|φa(Na,Nb)∝an}bracketri}ht\n= exp{iβ/integraltext\nd3r|¯φa(r)|2[χ0(r)+χd(r)]} (99)\n∝an}bracketle{tφb(Na−β,Nb+β)|φb(Na,Nb)∝an}bracketri}ht\n= exp{iβ/integraltext\nd3r|¯φb(r)|2[χ0(r)−χd(r)]} (100)\n∝an}bracketle{tφb(Na−β,Nb+β)|φa(Na,Nb)∝an}bracketri}ht\n=/integraltext\nd3r¯φ∗\nb(r)¯φa(r)exp[i(Na−β)χd(r)−iNbχd(r)]\n×exp[i(N−¯N)χs(r)−iN(|Ca|2−|Cb|2)χd(r)]\n×exp[iβχ0(r)] (101)\n∝an}bracketle{tφa(Na+β,Nb−β)|φb(Na,Nb)∝an}bracketri}ht\n=/integraltext\nd3r¯φ∗\na(r)¯φb(r)exp[−iNaχd(r)+i(Nb−β)χd(r)]\n×exp[−i(N−¯N)χs(r)−iN(|Ca|2−|Cb|2)χd(r)]\n×exp[−iβχ0(r)] (102)\nwhereβ∈Z, and we have used the relation\n/integraltext\nd3r|¯φε|2exp[i(∂Na−∂Nb)θε(¯Na,¯Nb)]\n≃exp[i/integraltext\nd3r|¯φε|2(∂Na−∂Nb)θε(¯Na,¯Nb)].(103)\nByusingtheGross-Pitaevskiiequations(7)for φε(Na,Nb)\nand forφε(¯Na,¯Nb), one obtains\ni/planckover2pi1∂t/bracketleftbigg\n(Na−¯Na)∂θε\n∂Na+(Nb−¯Nb)∂θε\n∂Nb/bracketrightbigg\n¯Na,¯Nb\n= (Nε−¯Nε)gεε|φε|2+(Nε′−¯Nε′)gεε′|φε′|2,(104)whereε∝ne}ationslash=ε′=a,b. Using (104) together with the initial\ncondition (8), we obtain for the phase factor Ain Eq. (9)\n[A(Na−1,Nb+1)−A(Na,Nb)]//planckover2pi1\n=−(Na−1)/integraltext\nd3r|¯φa(r)|2[χ0(r)+χd(r)]\n−Nb/integraltext\nd3r|¯φb(r)|2[χ0(r)−χd(r)] (105)\n[A(Na−2,Nb+2)−A(Na,Nb)]//planckover2pi1\n=−2(Na−2)/integraltext\nd3r|¯φa(r)|2[χ0(r)+χd(r)]\n−2Nb/integraltext\nd3r|¯φb(r)|2[χ0(r)−χd(r)]\n−/integraltext\nd3r{|¯φa(r)|2[χ0(r)+χd(r)]\n+|¯φb(r)|2[χ0(r)−χd(r)]} (106)\nThe averagesand variances of the spin operatorsequa-\ntions (10)-(12) are obtained by equations (94)-(98) after\nspatial integration. We get:\n/integraltextd3r∝an}bracketle{tˆψ†\nb(r)ˆψa(r)∝an}bracketri}ht\n=NC∗\nbCa/integraltext\nd3r¯φ∗\nb(r)¯φa(r)[|Ca|2eiχd(r)+|Cb|2e−iχd(r)]N−1\n×exp[i(N−¯N)χs(r)]exp[−i¯N(|Ca|2−|Cb|2)χd(r)]\n×exp[iχ0(r)]. (107)\n/integraltext\nd3rd3r′∝an}bracketle{tˆψ†\nb(r)ˆψ†\na(r′)ˆψa(r)ˆψb(r′)∝an}bracketri}ht\n=N(N−1)|Ca|2|Cb|2/integraltext\nd3rd3r′¯φ∗\nb(r)¯φa(r)¯φ∗\na(r′)¯φb(r′).\n(108)\n/integraltext\nd3rd3r′∝an}bracketle{tˆψ†\nb(r)ˆψ†\nb(r′)ˆψa(r)ˆψa(r′)∝an}bracketri}ht\n=N(N−1)C∗2\nbC2\na/integraltext\nd3rd3r′¯φ∗\nb(r)¯φa(r)¯φ∗\nb(r′)¯φa(r′)\n×[|Ca|2eiχd(r)+iχd(r′)+|Cb|2e−iχd(r)−iχd(r′)]N−2\n×exp{−i¯N(|Ca|2−|Cb|2)[χd(r)+χd(r′)]}exp{2i[χ0(r)\n+χ0(r′)]}exp{i(N−¯N)[χs(r)+χs(r′)]}\n×exp{−i/integraltext\nd3r′′(|¯φa|2[χ0(r′′)+χd(r′′)]\n+|¯φb|2[χ0(r′′)−χd(r′′)])}. (109)\n/integraltext\nd3rd3r′∝an}bracketle{tˆψ†\nb(r)ˆψ†\nb(r′)ˆψb(r)ˆψa(r′)∝an}bracketri}ht\n=N(N−1)C∗\nbCa|Cb|2/integraltext\nd3r′¯φ∗\nb(r′)¯φa(r′)[|Ca|2eiχd(r′)\n+|Cb|2e−iχd(r′)]N−2exp[−i¯N(|Ca|2−|Cb|2)χd(r′)]\n×exp[iχ0(r′)−iχd(r′)]exp[i(N−¯N)χs(r′)].(110)\n/integraltext\nd3rd3r′∝an}bracketle{tˆψ†\na(r)ˆψ†\na(r′)ˆψa(r)ˆψb(r′)∝an}bracketri}ht\n=N(N−1)C∗\naCb|Ca|2/integraltext\nd3r′¯φ∗\na(r′)¯φb(r′)[|Ca|2e−iχd(r′)\n+|Cb|2eiχd(r′)]N−2exp[i¯N(|Ca|2−|Cb|2)χd(r′)]\n×exp[−iχ0(r′)−iχd(r′)]exp[−i(N−¯N)χs(r′)].(111)\nIn the above expressions χd,χsandχ0are the space and\ntime dependent functions deﬁned in equations (24), (25)\nand (26). In practice it is suﬃcient to evolve ﬁve wave\nfunctionsφa(r,t),φb(r,t) for (¯Na,¯Nb±δNb) and (¯Na±\nδNa,¯Nb) withδNa,b∝ne}ationslash= 0 (to calculate numerically χdχs\nχ0), and with δNa,b= 0 (to calculate the central wave\nfunctions ¯φa,b).Thespinsqueezingisobtainedbyequation\n(15) using the deﬁnitions (10)-(12) for the spin operators.Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 15\nC Equality of χsandχ0in the\nbreathe-together conﬁguration\nEvaluating (104) for ε=a,Na=¯Na;ε=b,Nb=¯Nband\nsubtracting the two relations, on obtains\n∂t/parenleftbigg¯∂θa\n∂Nb−¯∂θb\n∂Na/parenrightbigg\n= 0 (112)\nwherewe usedthe fact that in breathe-togetherconditions\n|¯φa|=|¯φb|. Equation (112) implies that the time deriva-\ntive ofχs−χ0is zero. As for t= 0χs=χ0= 0, we\nconclude that χs=χ0at all times.\nD Extracted spin squeezing quantum\naverages\nBy using the instantaneous modes (42)-(43) and within\nthemodulus-phaseapproach,thequantumaveragesuseful\nto calculate spin squeezing are expressed in terms of the\nfunctions:\nχex\nd(r,r′) =1\n2(∂Na−∂Nb)[θa(r)−θb(r′)](¯Na,¯Nb) (113)\nχex\ns(r,r′) =1\n2(∂Na+∂Nb)[θa(r)−θb(r′)](¯Na,¯Nb) (114)\nχex\n0(r,r′) =1\n2(∂Na−∂Nb)[θa(r)+θb(r′)](¯Na,¯Nb).(115)\nWe obtain:\n∝an}bracketle{t˜b†˜a∝an}bracketri}ht=NC∗\nbCa/integraltext\nd3r1d3r2|¯φb(r1)|2|¯φa(r2)|2\n×[|Ca|2eiχex\nd(r2,r1)+|Cb|2e−iχex\nd(r2,r1)]N−1\n×exp[i(N−¯N)χex\ns(r2,r1)+iχex\n0(r2,r1)]\n×exp[−i¯N(|Ca|2−|Cb|2)χex\nd(r2,r1)] (116)\n∝an}bracketle{t˜b†˜a†˜a˜b∝an}bracketri}ht=N(N−1)|Ca|2|Cb|2(117)\n∝an}bracketle{t˜b†˜b†˜a˜a∝an}bracketri}ht=N(N−1)C∗2\nbC2\na/integraltextd3r1d3r2d3r3d3r4\n×|¯φb(r1)|2|¯φb(r2)|2|¯φa(r3)|2|¯φa(r4)|2\n×{|Ca|2ei[χex\nd(r4,r2)+χex\nd(r3,r1)]\n+|Cb|2e−i[χex\nd(r4,r2)+χex\nd(r3,r1)]}N−2\n×exp{2i[χex\n0(r4,r2)+χex\n0(r3,r1)]}\n×exp{−i¯N(|Ca|2−|Cb|2)[χex\nd(r4,r2)+χex\nd(r3,r1)]}\n×exp{i(N−¯N)[χex\ns(r4,r2)+χex\ns(r3,r1)]}\n×exp{−i/integraltext\nd3r5(|¯φa(r5)|2[χex\n0(r5,r5)+χex\nd(r5,r5)]\n+|¯φb(r5)|2[χex\n0(r5,r5)−χex\nd(r5,r5)])} (118)∝an}bracketle{t˜b†˜b†˜b˜a∝an}bracketri}ht=N(N−1)C∗\nbCa|Cb|2/integraltext\nd3r1d3r2|¯φb(r1)|2|¯φa(r2)|2\n×[|Ca|2eiχex\nd(r2,r1)+|Cb|2e−iχex\nd(r2,r1)]N−2\n×exp[iχex\n0(r2,r1)−iχex\nd(r2,r1)]\n×exp[−i¯N(|Ca|2−|Cb|2)χex\nd(r2,r1)]\n×exp[i(N−¯N)χex\ns(r2,r1)] (119)\n∝an}bracketle{t˜a†˜a†˜a˜b∝an}bracketri}ht=N(N−1)C∗\naCb|Ca|2/integraltext\nd3r1d3r2|¯φa(r1)|2|¯φb(r2)|2\n×[|Cb|2eiχex\nd(r1,r2)+|Ca|2e−iχex\nd(r1,r2)]N−2\n×exp[−iχex\n0(r1,r2)−iχex\nd(r1,r2)]\n×exp[i¯N(|Ca|2−|Cb|2)χex\nd(r1,r2)]\n×exp[−i(N−¯N)χex\ns(r1,r2)] (120)\nIn case the wave functions ¯φa,¯φbare stationary we re-\ncover the stationary two-mode model averages given in\nthe next appendix in the particular case of no losses. The\nspin squeezing is obtained by equation (15) using the def-\ninitions (44)-(46) for the spin operators.\nE Quantum averages with one-body losses:\nExact solution in the non symmetric case\nIn this appendix we give the exact result for quantum\naverages needed to calculate spin squeezing in the case\nof a two-mode model with one-body losses only, in the\ngeneral non-symmetric case.\n∝an}bracketle{ta†a∝an}bracketri}ht=|Ca|2Nexp(−γat) (121)\n∝an}bracketle{ta†a†aa∝an}bracketri}ht=|Ca|4N(N−1)exp(−2γat) (122)\n∝an}bracketle{tb†b†bb∝an}bracketri}ht=|Cb|4N(N−1)exp(−2γbt) (123)\n∝an}bracketle{tb†a†ab∝an}bracketri}ht=|Cb|2|Ca|2N(N−1)exp[−(γa+γb)t] (124)\n∝an}bracketle{tb†a∝an}bracketri}ht=C∗\nbCbe−2ivtNexp/bracketleftbigg\n−1\n2(γa+γb)t/bracketrightbigg\nLN−1\n1(125)\n∝an}bracketle{tb†b†ba∝an}bracketri}ht=|Cb|2C∗\nbCae−2ivtN(N−1)eiχt\n×exp/bracketleftbigg\n−1\n2(γa+3γb)t/bracketrightbigg\nLN−2\n1 (126)\n∝an}bracketle{ta†a†ab∝an}bracketri}ht=|Ca|2C∗\naCbe2ivtN(N−1)eiχt\n×exp/bracketleftbigg\n−1\n2(3γa+γb)t/bracketrightbigg\nLN−2\n−1 (127)\n∝an}bracketle{tb†b†aa∝an}bracketri}ht=C∗2\nbC2\nae−4ivtN(N−1)\n×exp[−(γa+γb)t]LN−2\n2 (128)\nwhere we introduced the function Lβwithβ=−1,1,2\nLβ=|Ca|2\nγa+iβ(χ+ ˜χ)/bracketleftBig\nγaeiβ˜χt+iβ(χ+ ˜χ)e−(γa+iβχ)t/bracketrightBig\n+|Cb|2\nγb−iβ(χ−˜χ)/bracketleftBig\nγbeiβ˜χt−iβ(χ−˜χ)e−(γb−iβχ)t/bracketrightBig\n(129)16 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\nandvgiven by (69).\nF Quantum averages with one, two,\nthree-body losses in the non-symmetric case\nIn this appendix we give the quantum averages useful to\ncalculate spin squeezing for the two-mode model in the\ngeneral non-symmetric case, in presence of one, two and\nthree-body losses.\n∝an}bracketle{ta†a∝an}bracketri}ht=|Ca|2e−λt[N−(∂σ1+∂σ2)]F0(σ1,σ2) (130)\n∝an}bracketle{ta†a†aa∝an}bracketri}ht=|Ca|4e−λt[N−(∂σ1+∂σ2−1)]\n×[N−(∂σ1+∂σ2)]F0(σ1,σ2) (131)\n∝an}bracketle{tb†b†bb∝an}bracketri}ht=|Cb|4e−λt[N−(∂σ1+∂σ2−1)]\n×[N−(∂σ1+∂σ2)]F0(σ1,σ2) (132)\n∝an}bracketle{tb†a†ab∝an}bracketri}ht=|Cb|2|Ca|2e−λt[N−(∂σ1+∂σ2−1)]\n×[N−(∂σ1+∂σ2)]F0(σ1,σ2) (133)\n∝an}bracketle{tb†a∝an}bracketri}ht=C∗\nbCbe−(2iv+λ)t/parenleftbig\n|Ca|2e−iχt+|Cb|2eiχt/parenrightbigN−1\n×[N−(∂σ1+∂σ2)]F1(σ1,σ2) (134)\n∝an}bracketle{tb†b†ba∝an}bracketri}ht=|Cb|2C∗\nbCae−(2iv+λ)t/parenleftbig\n|Ca|2e−iχt+|Cb|2eiχt/parenrightbigN−2\n×eiχt[N−(∂σ1+∂σ2−1)][N−(∂σ1+∂σ2)]\n×F1(σ1,σ2) (135)\n∝an}bracketle{ta†a†ab∝an}bracketri}ht=|Ca|2C∗\naCbe(2iv−λ)t/parenleftbig\n|Cb|2e−iχt+|Ca|2eiχt/parenrightbigN−2\n×eiχt[N−(∂σ1+∂σ2−1)][N−(∂σ1+∂σ2)]\n×G1(σ1,σ2) (136)\n∝an}bracketle{tb†b†aa∝an}bracketri}ht=C∗2\nbC2\nae−(4iv+λ)t/parenleftbig\n|Ca|2e−2iχt+|Cb|2e2iχt/parenrightbigN−2\n×[N−(∂σ1+∂σ2−1)][N−(∂σ1+∂σ2)]\n×F2(σ1,σ2) (137)\nwhereweintroducedthefunctions Fβ(σ1,σ2)andGβ(σ1,σ2)\nFβ(σ1,σ2) =\nexp/braceleftBigg3/summationdisplay\nm=1emσ1γ(m)\na[1−e−imβ(χ+˜χ)t]\nimβ(χ+ ˜χ)[|Ca|2e−iβ(χ+˜χ)t+|Cb|2eiβ(χ−˜χ)t]m\n+emσ2γ(m)\nb[eimβ(χ−˜χ)t−1]\nimβ(χ−˜χ)[|Ca|2e−iβ(χ+˜χ)t+|Cb|2eiβ(χ−˜χ)t]m\n+eσ1+σ2γab[1−e−i2β˜χt]\ni2β˜χ[|Ca|2e−iβ(χ+˜χ)t+|Cb|2eiβ(χ−˜χ)t]2/bracerightbigg\n(138)Gβ(σ1,σ2) =\nexp/braceleftBigg3/summationdisplay\nm=1emσ2γ(m)\nb[1−e−imβ(χ−˜χ)t]\nimβ(χ−˜χ)[|Cb|2e−iβ(χ−˜χ)t+|Ca|2eiβ(χ+˜χ)t]m\n+emσ1γ(m)\na[eimβ(χ+˜χ)t−1]\nimβ(χ+ ˜χ)[|Cb|2e−iβ(χ−˜χ)t+|Ca|2eiβ(χ+˜χ)t]m\n−eσ1+σ2γab[1−ei2β˜χt]\ni2β˜χ[|Cb|2e−iβ(χ−˜χ)t+|Ca|2eiβ(χ+˜χ)t]2/bracerightbigg\n(139)\nwithβ= 0,1,2, and all the expressions should be eval-\nuated inσ1= ln¯Na,σ2= ln¯Nb. The expression of vis\ngiven in (69). The spin squeezing is obtained by equation\n(15) using the deﬁnitions (44)-(46) for the spin operators\n(with ˜a=aand˜b=b).\nG Test of the constant loss rate\napproximation for high asymmetric losses\nThe constant loss rate approximation (61) is in general\nvalid when a small fraction of particles is lost. In the case\nof symmetric condensates, from equation (77) one sees\nthat the best squeezing in presence of losses is of the order\nof the lost fraction. So that ξ(tbest)≪1 guarantees that\nthelostfractionissmallandthe constantlossrateapprox-\nimationisaccurate.In thecaseofasymmetriccondensates\nand asymmetric losses there might be other eﬀects to con-\nsider as the population ratio between the two spin compo-\nnents might change in reality while it remains constant in\nthe constant loss rate approximation. Indeed with the ap-\nproximation (61), the initial phase state remains a phase\nstate through out the whole evolution. As a consequence,\nwhen a quantum jump occurs, only the relative phase and\nthe total number of particle changes (see equation (59)).\nIn Fig.12 and Fig.13 we compare the constant loss rate\napproximation to the exact numerical result in the case\nof overlapping Rb condensates with large asymmetric two\nbody losses considered in Section 4. In Fig.12 we address\nthe case of evenly split condensates ¯Na=¯Nb=N/2 while\nin Fig.13 we address the case of breathe-together param-\neters.\nThe constant loss rate approximation neglects two ef-\nfects: The decrease of the loss rate in time as less and less\nparticles are in the system, and the change of the ratio\n∝an}bracketle{tNa∝an}bracketri}ht/∝an}bracketle{tNb∝an}bracketri}htas particles from the bcomponent are lost. In\nthe case of Fig.12 where we consider initially ∝an}bracketle{tNa∝an}bracketri}ht=∝an}bracketle{tNb∝an}bracketri}ht,\nwhich is the most favorable for squeezing, these two ef-\nfects partially compensates: one tending to degrade and\nthe other to improve the squeezing with respect to reality.\nIn the case of Fig.13 instead, the two eﬀects sum-up, both\nof them tending to degrade the squeezing with respect to\nreality. Note however that even for such large and com-\npletely non-symmetric losses, the constant loss rate ap-\nproximation proves to be rather accurate up to the best\nsqueezing time.Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle 17\n0 0.05 0.1 0.15 0.210−210−1100\nt [s]ξ2Approximate analytical solution\nFull numerical simulation\n0 0.05 0.1 0.15 0.21.61.82x 104Ntot\n0 0.05 0.1 0.15 0.2−101x 104\nt [s](Na− Nb)/2\nFig. 12. (Top) Spin squeezing with two-body losses in a bi-\nmodal Rb condensate as a function of time for symmetrically\nsplit condensates. Blue solid line: exact numerical simula tion\nwith 4000 realizations. Red dash-dotted line: analytical s olu-\ntion with constant loss rate approximation. (Bottom) Corre -\nsponding total number of particles and /angbracketleftSz/angbracketrightas a function of\ntime. Parameters: ¯Na=¯Nb= 104,ω= 2π×42.6 Hz,m=87\na.m.u.,aaa= 100.44rB,abb= 95.47rB,aab= 88.28rB,rBis\ntheBohr radius, χ= 5.367×10−3s−1, ˜χ= 5.412×10−4s−1,v=\n13.758s−1. Particle losses: K(a)\n2= 0,K(b)\n2= 119×10−21m3s−1.\nReferences\n1. M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).\n2. J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, Phys.\nRev. Lett. 83, 1319 (1999); A. Kuzmich, L. Mandel, and\nN. P. Bigelow, Phys. Rev. Lett. 85, 1594 (2000).\n3. D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J.\nHeinzen, Phys. Rev. A 50, 67 (1994).\n4. D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chi -\naverini, W. M. Itano, J. D. Jost, C. Langer, D. J. Wineland\nScience304, 1476 (2004).\n5. G. Santarelli, Ph. Laurent, P. Lemonde, A. Clairon, A. G.\nMann, S. Chang, and A. N. Luiten, and C. Salomon Phys.\nRev. Lett. 82, 4619 (1999).\n6. A. Sørensen, L. M. Duan, I. Cirac, and P. Zoller, Nature\n409, 63 (2001).\n7. B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986).\n8. D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman,\nand E. A. Cornell, Phys. Rev. Lett. 81, 1539 (1998).0 0.05 0.1 0.15 0.210−210−1100\nt [s]ξ2Approximate analytical solution\nFull numerical simulation\n0 0.05 0.1 0.15 0.21.41.72x 104Ntot\n0 0.05 0.1 0.15 0.2−101x 104\nt [s](Na−Nb)/2\nFig. 13. (Top) Spin squeezing with two-body losses in a bi-\nmodal Rb condensate as a function of time in breathe-togethe r\nconﬁguration. Blue solid line: exact numerical simulation with\n4000 realizations. Red dash-dotted line: analytical solut ion\nwith constant loss rate approximation. (Bottom) Correspon d-\ning total number of particles and /angbracketleftSz/angbracketrightas a function of time.\nParameters: ¯Na= 7432, ¯Nb= 12568,χ= 5.392×10−3s−1,\n˜χ= 1.386×10−3s−1,v= 13.850s−1. The other parameters\nare the same as in Fig.12.\n9. A. Sinatra and Y. Castin, Eur. Phys. J. D 8, 319 (2000).\n10. A. Sørensen, Phys. Rev. A 65, 043610 (2002).\n11. S. Thanvanthri, and Z. Dutton, Phys. Rev. A 75, 023618\n(2007).\n12. Y. Li, Y. Castin, and A. Sinatra, Phys. Rev. Lett. 100,\n210401 (2008).\n13. P. Treutlein, P. Hommelhoﬀ, T. Steinmetz, T. W. H¨ asch,\nand J. Reichel, Phys. Rev. Lett. 92, 203005 (2004).\n14. M. Erhard, H. Schmaljohann, J. Kronj¨ ager, K. Bongs, and\nK. Sengstock, Phys. Rev. A 69, 032705 (2004).\n15. A. Widera, S. Trotzky, P. Cheinet, S. F¨ olling, F. Gerbie r,\nI. Bloch, V. Gritsev, M. D. Lukin, and E. Demler, Phys.\nRev. Lett. 100, 140401 (2008).\n16. We can write the ﬁeld operators as ˆψa(r) =a/angbracketleftr|,ˆψb(r) =\nb/angbracketleftr|and use the commutation relations:\n[a/angbracketleftr|,a†\n|φa(Na,Nb)/angbracketright] =/angbracketleftr|φa(Na,Nb)/angbracketright=φa(Na,Nb,r)\n[b/angbracketleftr|,b†\n|φb(Na,Nb)/angbracketright] =/angbracketleftr|φb(Na,Nb)/angbracketright=φb(Na,Nb,r).18 Please give a shorter version with: \\authorrunning and \\titlerunning prior to \\maketitle\n17. A. Sinatra and Y. Castin, Eur. Phys. J. D 4, 247 (1998)\n18. Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys.\nRev. A54, R1753 (1996).\n19. Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996).\n20. S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).\n21. K. Mølmer, Y. Castin, J. Dalibard, J. Opt. Soc. Am. B 10,\n524 (1993); H. J. Carmichael, An Open Systems Approach\nto Quantum Optics Springer, (1993).\n22. A. Sinatra and Y. Castin, Eur. Phys. J. D 4, 247 (1998).\n23. In the expression (65) of |αk|2we replaced Nwith¯Ncon-\nsistently with the constant loss rate approximation.\n24. K. M. Mertes, J. W. Merrill, R. Carretero-Gonzalez, D. J.\nFrantzeskakis, P. G. Kevrekidis, and D. S. Hall, Phys. Rev.\nLett.99190402 (2007).\n25. As we are pretty far from the Feshbach resonance, we as-\nsume for the crossed abtwo-body loss rate the same value\nmeasured in [24] for the |F= 1,mF=−1/angbracketright,|F= 2,mF=\n1/angbracketrightstates.\n26. E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E.\nA. Cornell, and C. E. Wieman, Phys. Rev. Lett. 79, 337\n(1997).\n27. D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S.\nInouye, H.-J. Miesner, J. Stenger, and W. Ketterle, Phys.\nRev. Lett. 802027 (1998).\n28. P. Treutlein, T. W. H¨ asch, J. Reichel, A. Negretti, M. A.\nCirone, and T. Calarco Phys. Rev. A 74022312 (2006).\n29. Uﬀe V. Poulsen and Klaus Mølmer, Phys. Rev. A 65\n033613 (2002).\n30. A. Couvert, T. Kawalec, G. Reinaudi, and D. Gu´ ery-\nOdelin, arXiv: 0708.4197v1.\n31. We checked that similar result can be obtained with diﬀer -\nent geometry where we prepare the condensate in a cigar\nshape and separate them along the longitudinal compo-\nnent."    },    {        "title": "1107.4688v1.Spin__Isospin_and_Strong_Interaction_Dynamics.pdf",        "content": "arXiv:1107.4688v1  [physics.gen-ph]  23 Jul 2011Spin, Isospin and\nStrong Interaction Dynamics\nE. Comay∗\nCharactell Ltd.\nP.O. Box 39019, Tel Aviv 61390\nIsrael\nPACS No: 03.65.-w, 21.10.Hw, 14.20.-c\nAbstract:\nThe structure of spin and isospin is analyzed. Although both spin and isospin\nare related to the same SU(2) group, they represent diﬀerent dy namical eﬀects. The\nWigner-Racah algebra is used for providing a description of bound st ates of several\nDirac particles in general and of the proton state in particular. Iso spin states of the\nfour ∆(1232) baryons are discussed. The work explains the small c ontribution of\nquarks spin to the overall proton spin (the proton spin crisis). It is also proved that\nthe addition of QCD’s color is not required for a construction of an an tisymmetric\nstate of the ∆++(1232) baryon.1. Introduction\nThe isospin notion has been conceived by W. Heisenberg in 1932 (see [1 ], p. 106).\nIt aims to construct a mathematical basis that represents the pr oton-neutron simi-\nlarity with respect to the strong nuclear force. Both spin and isosp in have the same\nSU(2) group structure. Thus, like spin multiplets of a quantum stat e, one combines\ncorresponding states of nuclear isobars in an isospin multiplet. For e xample, the\nground state of the14C,14Oand theJπ= 0+excited state of14Nare members of\nan isospin triplet. Obviously, one must remember that isospin is a usef ulapproxima-\ntionthat neglects proton-neutron diﬀerences that are related to th eir mass and their\nelectromagnetic interactions.\nLaterdevelopments haveshownthattheproton-neutronsimilarit ystemsfromthe\nsimilarity between the u, dquarks. It follows that the usefulness of isospin symmetry\nextends to particle physics. For example, the three pions are memb ers of an isospin\ntriplet. Dueto historical development, isospin notationtakes diﬀer ent formin nuclear\nand particle physics. Here TandIdenote isospin in nuclear and particle physics,\nrespectively. In this work the symbol Tis used, mainly because of the following\nreason. In the case of spin, the symbols Jandjdenote total and single particle\nangular momentum operators, respectively. Similarly, the symbols Tandtdenote\nthe corresponding isospin operators. Thus, due to the same unde rlying SU(2) group,\nisospin relations can be readily borrowed from their corresponding s pin counterparts.\nThe operators Tandtare used in the discussion presented in this work.\nThis work examines states of electrons and quarks. These particle s have spin-\n1/2 and experimental data are consistent with their elementary po intlike property.\nEvidently, a theoretical analysis of an elementary pointlike particle is a much simpler\ntask than that of a composite particle. The discussion begins with an examination of\nrelevant properties of electronic states of atoms. The mathemat ical structure of the\n2SU(2) group is used later for a corresponding analysis of isospin sta tes.\nTwo important conclusions are derived from this analysis. First, it is w ell known\nthat quarks’ spin carry only a small fraction of the entire proton’s spin [2]. This\nexperimental evidence, which is called the second EMC eﬀect and also the proton\nspin crisis, is shown here to be an obvious result of the multi-conﬁgur ation structure\nof states of more than one Dirac particle. Another result is that th e anti-symmetric\nstate of the ∆++(1232) baryon is well understood and there is no need to introduce a\nnew degree of freedom for its explanation. It means that the histo rical starting point\nof the QCD construction has no theoretical basis. (Below, the sym bol ∆ refers to\nthis isospin quartet of baryons.)\nGenerally, in order to simplify notation, the speciﬁc value of normaliza tion factor\nis omitted from the expressions. The second and the third sections analyze spin\nand isospin, respectively. The fourth section provides an explanat ion for the proton\nspin crisis. The ﬁfth section explains the antisymmetric structure o f the ∆++baryon\n(without using color). The last section contains concluding remarks .\n2. Spin States\nA comprehensive discussion of angular momentum can be found in tex tbooks [3].\nIn this short work some elements of this theory are mentioned toge ther with a brief\nexplanation. This is done for the purpose of arriving rapidly at the ma in conclusions.\nA relativistic notation is used and for this reason the jjcoupling [3] takes place.\nLet us begin with a discussion of spin and spatial angular momentum. T hese\nquantities are dimensionless and this property indicates that they maybe coupled.\nNow, the magnetic ﬁeld depends on space and time. Moreover, the t heory must be\nconsistent with the experimental fact where both spatial angular momentum and spin\n3of an electron have the same kind of magnetic ﬁeld. Thus, it is requiredto construct a\nrelativistically consistent coupling of these quantities. This is the the oretical basis for\nthe well known usage of spin and spatial angular momentum coupling in the analysis\nof electronic states of atoms.\nA motionless free electron is the simplest case and the spin-up electr on state is\n(see [4], p. 10)\nψ(xµ) =Ce−imt\n1\n0\n0\n0\n, (1)\nwheremdenotes the electron’s mass.\nA second example is the state of an electron bound to a hypothetica l pointlike\nvery massive positive charge. Here the electron is bound to a spher ically symmetric\nchargeZe. The general form of a jπhydrogen atom wave function is (see [5], pp.\n926, 927)\nψ(rθφ) =/parenleftigg\nFYjlm\nGYjl′m/parenrightigg\n, (2)\nwhereYjlmdenotes the ordinary Ylmcoupled with a spin-1/2 to j,j=l±1/2,\nl′=l±1,F,Gare radial functions and the parity is ( −1)l.\nBy the general laws of electrodynamics, the state must be an eigen function of\nangular momentum and parity. Furthermore, here we have a proble m ofoneelectron\n(the source at the origin is treated as an inert object) and indeed, its wave function\n(2) is an eigenfunction of both angular momentum and parity (see [5], p. 927).\nThe next problem is a set of n-electrons bound to an attractive positive charge\nat the origin. (This is a kind of an ideal atom where the source’s volume and spin\nare ignored.) Obviously, the general laws of electrodynamics hold an d the system\nis represented by an eigenfunction of the total angular momentum and parity Jπ.\nHere a single electron is aﬀected by a spherically symmetric attractiv e ﬁeldandby\nthe repulsive ﬁelds of the other electrons. Hence, a single electron does not move in\n4a spherically symmetric ﬁeld and it cannotbe represented by a well deﬁned single\nparticle angular momentum and parity.\nThe general procedure used for solving this problem is to expand th e overall\nstate as a sum of conﬁgurations. In every conﬁguration, the elec trons’ single particle\nangular momentum and parity are well deﬁned. These angular momen ta are coupled\nto the overall angular momentum Jand the product of the single particle parity is the\nparity of the entire system. The role of conﬁgurations has already been recognized\nin the early decades of quantum physics [6]. An application of the ﬁrst generation of\nelectronic computers has provided a numerical proof of the vital r ole of ﬁnding the\ncorrect conﬁguration interaction required for a description of ev en the simplest case of\nthe ground state of the two electron He atom [7]. The result has pro ved that several\nconﬁgurations are required for a good description of this state an d no conﬁguration\ndominates the others. This issue plays a very important role in the int erpretation of\nthe state of the proton and of the ∆++.\nFor example, let us write down the 0+ground state He gof the Helium atom as a\nsum of conﬁgurations:\nψ(Heg) =f0(r1)f0(r2)1\n2+1\n2++f1(r1)f1(r2)1\n2−1\n2−+f2(r1)f2(r2)3\n2−3\n2−+\nf3(r1)f3(r2)3\n2+3\n2++f4(r1)f4(r2)5\n2+5\n2++... (3)\nHere and below, fi(r), gi(r) andhi(r) denote the two-component Dirac radial wave\nfunction (multiplied be the corresponding coeﬃcients). In order to couple toJ= 0\nthe two single particle jstates must be equal and in order to make an even total\nparity both must have the same parity. These requirements make a severe restriction\non acceptable conﬁgurations needed for a description of the grou nd state of the He\natom.\nHighertwo-electrontotalangularmomentumallowsalargernumber ofacceptable\n5conﬁgurations. For example, the Jπ= 1−state of the He atom can be written as\nfollows:\nψ(He1−) =g0(r1)h0(r2)1\n2+1\n2−+g1(r1)h1(r2)1\n2+3\n2−+g2(r1)h2(r2)1\n2−3\n2++\ng3(r1)h3(r2)3\n2−3\n2++g4(r1)h4(r2)3\n2−5\n2++g5(r1)h5(r2)3\n2+5\n2−+\ng6(r1)h6(r2)5\n2+5\n2−... (4)\nUsing the same rules one can apply simple combinatorial calculations an d ﬁnd a\nlarger number of acceptable conﬁgurations for a three or more ele ctron atom. The\nmain conclusion of this section is that, unlike a quite common belief, the re are only\nthreerestrictions on conﬁgurations required for a good description of a Jπstate of\nmore than one Dirac particles:\n1. Each conﬁguration must have the total angular momentum J.\n2. Each conﬁguration must have the total parity π.\n3. Following the Pauli exclusion principle, each conﬁguration should no t contain\ntwo or more identical single particle quantum states of the same Dira c particle.\nTheserestrictionsindicatethatastatecanbewrittenasasumofm anyconﬁgurations,\neach of which has a well deﬁned single particles angular momentum and parity of its\nDirac particles.\nThe mathematical basis of this procedure is as follows. Take the Hilbe rt sub-\nspace made of conﬁgurations that satisfy the three requirement s mentioned above\nand calculate the Hamiltonian matrix. A diagonalization of this Hamiltonia n yields\neigenvalues and eigenstates. These eigenvalues and eigenstates a re related to a set\nof physical states that have the given Jπ. As pointed out above, calculations show\nthat for a quite good approximation to a quantum state one needs a not very small\n6number of conﬁgurations and that no conﬁguration has a dominant weight. These\nconclusions will be used later in this work.\n3. Isopin States\nSpin and isospin are based on the same mathematical group called SU( 2). Its\nthree generators are denoted jx,jy,jz. An equivalent basis is (see [1], pp. 357-363)\nj+=jx+ijy, j−=jx−ijy, jz. (5)\nAll thejoperators mentioned above commute with the total j2operator. For\nthis reason, if one of them operates on a member of a (2 J+ 1) multiplet of an\nSU(2) irreducible representation then the result belongs to this mu ltiplet. The two\nj±operators are of a particular importance. Thus, let ψJ,Mdenote a member of such\na multiplet and one ﬁnds\nJzJ−ψJ,M= (M−1)J−ψJ,M. (6)\nThis relation means that J−castsψJ,MintoψJ,M−1\nJ−ψJ,M=/radicalig\nJ(J+1)−M(M−1)ψJ,M−1, (7)\nwhere the appropriate coeﬃcient is written explicitly. Analogous rela tions hold for\ntheJ+operator.\nLet us turn to isospin. The required operators are simply obtained b y taking the\nmathematical structure of spin and replacing the total spin opera torJand the single\nparticle spin operator jby the corresponding isospin operators T, t. (Here, like in\nthe spin case, M, mdenote the eigenvalue of Tz, tz, respectively.) The issue to be\nexamined is the structure of the isospin multiplet of the four baryon s:\n∆−,∆0,∆+,∆++. (8)\n7These ∆(1232) baryons have the lowest energy of the family of the ∆ baryons [8].\nThe ∆++baryon has three uquarks and ψ∆(uuu) denotes its state. Therefore, its\nisospin state is T= 3/2, M= 3/2 and the isospin component of the wave function is\nsymmetric with respect to an exchange of any pair of quark.\nLet us examine the operation of T−on ∆++.\nT−ψ∆(uuu) = (t1−+t2−+t3−)ψ∆(uuu) =ψ∆(duu)+ψ∆(udu)+ψ∆(uud),(9)\nwhereti−operates on the ith quark. This is the way how one obtains a yet unno r-\nmalized expression for the ∆+baryon from that of ∆++. A successive application of\nT−yields expressions for every member of the isospin quartet (8).\nNow, the ∆++state is symmetric with respect to its quark constituents and the\nsame property holds for the operator T−=t1−+t2−+t3−. Hence, also the ∆+is\nsymmetric with respect to its uudquark states. This argument proves that isospin\nspace ofeverymember of the baryonic quartet (8) is symmetric. The same result c an\nbe obtained from a diﬀerent argument. Quarks are fermions and th eir overall state\nmust be antisymmetric with respect to an interchange of any pair of quarks. Now,\nthe isospin operators used above do not aﬀect other coordinates of quarks. It means\nthat for every members of the isospin quartet (8), the entire sym metry of the other\ncoordinates remain antisymmetric and the isospin coordinate is symm etric.\nThe data conﬁrms the similarity between members of an isospin multiple t. Thus,\nfor example, the mass diﬀerence between the ∆0and ∆++baryons is less than 3\nMeV [8], whereas the mass diﬀerence between the ∆ multiplet and the n ucleons is\nabout 300 MeV. This evidence shows the goodness of the isospin not ion, where strong\ninteractions dominate the state of members of an isospin multiplet an d the eﬀect of\n8all other interactions can be regarded as a small perturbation.\n4. The Proton Spin Crisis\nThe proton’s Jπ= 1/2+state is determined by three valence uudquarks. The\nnon-negligible probability of the existence of an additional quark-an tiquark pair (see\n[1], p. 282) indicates that it is a highly relativistic system. The discussio n of section\n2 holds for the spin-1/2 point-like quarks and the expansion in conﬁg urations is a\nuseful approach. Here the three single particle jπrepresent the uudquarks, in that\norder. Evidently, each conﬁguration must satisfy the three requ irement written few\nlines below (4). However, the Pauli exclusion principle of restriction 3 does not hold\nfor thedquark. Thus, in analogy to (3) and (4) one expands the proton’s wa ve\nfunction as a sum of terms of speciﬁc conﬁgurations. A truncated expression for this\nexpansion is shown below:\nψ(uud) =f0(r1)f0(r2)h0(r3)1\n2+1\n2+(0)1\n2++f1(r1)f1(r2)h1(r3)1\n2−1\n2−(0)1\n2++\nf2(r1)g2(r2)h2(r3)1\n2+1\n2+(1)1\n2++f3(r1)g3(r2)h3(r3)1\n2−1\n2−(1)1\n2++\nf4(r1)g4(r2)h4(r3)1\n2+1\n2−(0)1\n2−+f5(r1)g5(r2)h5(r3)1\n2+1\n2−(1)1\n2−+\nf6(r1)g6(r2)h6(r3)1\n2+3\n2+(1)1\n2++f7(r1)g7(r2)h7(r3)1\n2−3\n2+(1)1\n2−+\nf8(r1)g8(r2)h8(r3)1\n2+1\n2+(1)3\n2++f9(r1)g9(r2)h9(r3)1\n2−1\n2−(1)3\n2++\nfa(r1)ga(r2)ha(r3)1\n2−3\n2−(1)1\n2++fb(r1)gb(r2)hb(r3)1\n2+3\n2−(1)1\n2−+\nfc(r1)gc(r2)hc(r3)1\n2+1\n2−(1)3\n2−+... (10)\nThe symbols 0...9,a,b,care used for enumerating the terms. Here, like in (3) and (4),\nfi(r), gi(r) andhi(r) denote the Dirac two-component radial wave function of the\nuudquarks, respectively (multiplied be the corresponding coeﬃcients) . In each term,\nthe number in parentheses indicates how the two angular momenta o f theuuquarks\n9are coupled. Below, Juudenotes the value of this quantity.\nThe following remarks explain the form of these terms. An important issue is the\ncoupling of the two uuquark that abide by the Pauli exclusion principle. For this\nreason,Juuis given explicitly in each term. Another restriction stems from the ru le of\nangular momentum addition. Thus, for every term, the following rela tion must hold\nin order to yield a total spin-1/2 for the proton: Juu=jd±1/2. These rules explain\nthe speciﬁc structure of each term of (10) which is described below .\nIn terms 0,1the two spin-1/2 are coupled antisymmetrically to Juu= 0 and the\ntwo radial function are the same. In terms 2,3these spins are coupled symmetrically\ntoJuu= 1 and antisymmetry is obtained from the two orthogonal radial fu nctions.\nIn terms 4,5the diﬀerent orbitals of the uuquarks enable antisymmetrization. Thus,\nthe two spin-1/2 functions are coupled to Juu= 0 andJuu= 1, respectively. The\nradial functions are not the same because of the diﬀerent orbitals . In terms 6,7the\nspins are coupled to Juu= 1. In terms 8,9we have a symmetric angular momentum\ncouplingJuu= 1 and the antisymmetry is obtained from the orthogonality of the\nradial function fi(r), gi(r). Terms a,bare analogous to terms 6,7, respectively. In term\ncthe diﬀerent uuorbitals enable antisymmetrization and they are coupled to Juu= 1.\nA comparison of the expansion of the He atom ground state (3) and that of the\nproton (10) shows the following points:\n1. If the expansion is truncated after the same value of a single par ticle angular\nmomentum then the number of terms in the proton’s expansion is sign iﬁcantly\nlarger.\n2. This conclusion is strengthened by the fact that the proton has a non-negligible\nprobability of an additional quark-antiquark pair. An inclusion of this pair\nincreases the number of acceptable conﬁgurations.\n3. Calculations show that the number of conﬁgurations required fo r the ground\n10state spin-0 of the two electron He atom is not very small and that t here is\nno single conﬁguration that dominates the state [7]. Now the proton is a spin-\n1/2 relativistic particle made of three valence quarks. Therefore, it is very\nreasonable to assume that its wave function takes a multiconﬁgura tion form.\nUsing angular momentum algebra, one realizes that in most cases an in dividual\nquark does not take the proton’s spin direction. This is seen on two le vels. First,\nthe upper and the lower parts of the quark single particle function h avel=j±1/2.\nFurthermore, the relativistic quark state indicates that the coeﬃ cients of the upper\nand the lower part of the Dirac four component function take a simila r size. Hence,\nfor the case where j=l−1/2, the Clebsch-Gordan coeﬃcients [3] used for coupling\nthe spatial angular momentum and the spin indicate that the spin of e ither the upper\nor the lower Dirac spinor has no deﬁnite direction and that the coeﬃc ient of the spin\ndown is not smaller than that of the spin up (see [3], p. 519).\nLet us turn to the coupling of the quark spins. The 3-quark terms c an be divided\ninto two sets having juu= 0 andjuu>0, respectively. For juu= 0 one ﬁnds that the\nsingle particle jd= 1/2 and this spin is partially parallel to the proton’s spin. For\ncases where juu>0, the proton’s quark spins are coupled in a form where they take\nbothup anddown direction so that they practically cancel each oth er. The additional\nquark-antiquark pair increases spin direction mixture. It can be co ncluded that the\nquark spin contribute a not very large portion of the proton spin an d the rest comes\nfrom the quark spatial motion. This conclusion is supported by expe riment [9].\n5. The State of the ∆++Baryon\nIn textbooks it is argued that without QCD, the state of the ∆++baryon demon-\nstrates a ﬁasco of the Fermi-Dirac statistics (see [10], p. 5). The a rgument is based\n11on the claim that the ∆++takes the lowest energy state of the ∆ baryons [11] and\ntherefore, its spatial wave function consists of three single part icle symmetric s-waves\nof each of its three uuuquarks. Now the Jπ= 3/2+state of the ∆ baryons shows that\nalso their spin is symmetric. It means that the ∆++is regarded to have space, spin\nand isospin symmetric components of its wave function. As stated a bove, textbooks\nclaim that this outcome contradicts the Fermi-Dirac statistics. How ever, using the\nphysical issues discussed in this work and the following energy level d iagram of the\nnucleon and the ∆ baryons, it is proved that this textbook argumen t is incorrect.\nn p938∆−∆0∆+∆++1232\nFig. 1: Energy levels of the nucleon and the ∆isospin\nmultiplets (MeV).\n•As explained in section 3, all members of an isospin multiplet have the sa me\nsymmetry. Hence, if there is a problem with the Fermi-Dirac statistic s of the\n∆++then the same problem exists with ∆+and ∆0. It follows that if the above\nmentioned textbook argument is correct then it is certainly incomple te.\n•The data described in ﬁg. 1 shows that ∆+is an excited state of the proton.\nHence, its larger mass is completely understood. Thus, there is no p roblem\nwith the Fermi-Dirac statistics of the ∆+baryon. Analogous relations hold for\nthe neutron and the ∆0baryons. Using the identical statistical state of the\nfour ∆ baryons (8), one realizes that there is no problem with the Fe rmi-Dirac\nstatistics of the ∆++and the ∆−baryons.\n12•The multi-conﬁguration structure of a bound system of Dirac part icles is known\nfor about 50 years [7]. In particular, the multi-conﬁgurations stru cture of all\nbaryons (like in (10)) proves that, contrary to the above mention ed textbook\nargument (see [10], p. 5), the single particle spatial wave functions of the three\nuquarks of the ∆++baryonare not a pure s-wave .\n6. Conclusions\nThis work uses the Wigner-Racah mathematical structure and pro ves two very\nimportant points. It explains the small contribution of quark’s spin t o the overall\nproton spin. Therefore, it eliminates the basis for the proton spin c risis. It also\nproves that everything is OK with the Fermi-Dirac statistics of the ∆++baryon. It\nfollows that there is no need to introduce the QCD’s color degree of f reedom in order\nto build an antisymmetric wave function for this baryon.\n13References:\n* Email: elicomay@post.tau.ac.il\nInternet site: http://www.tau.ac.il/ ∼elicomay\n[1] D. H. Perkins Introductions to High Energy Physics (Addison-Wesley, Menlo\nPark, 1987) 3rd edn.\n[2] J Ashman et al. (EMC) Phys. Lett. B206, 364 (1988).\n[3] A. de-Shalit and I. Talmi, Nuclear Shell Theory (Academic, New York, 1963).\n[4] J. D. Bjorken and S. D. Drell Relativistic Quantum Mechanics (McGraw, New\nYork, 1964).\n[5] A. Messiah, Quantum Mechanics (Dover, Mineola, 1999).\n[6] H. A. Bethe, Intermediate Quantum Mechanics (Benjamin, New York, 1964).\n(see p. 109).\n[7] A. W. Weiss, Phys. Rev. 122, 1826 (1961)\n[8] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 1 (2008).\n[9] S.E. Kuhn, J.-P. Chen, E. Leader, Prog. Part. Nucl. Phys. 631 (2009).\n[10] F. Halzen and A. D. Martin, Quarks and Leptons (Wiley, New York, 1984).\n[11] Today more than 10 diﬀerent ∆ baryonic multiplets are identiﬁed [8 ].\n14"    },    {        "title": "1301.0421v1.Coherent_spin_dynamics_of_nanomolecules_and_magnetic_nanoclusters.pdf",        "content": "arXiv:1301.0421v1  [cond-mat.mes-hall]  3 Jan 2013Coherent spin dynamics of nanomolecules and\nmagnetic nanoclusters\nV.I. Yukalov1and E.P. Yukalova2\n1Bogolubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\n2Laboratory of Information Technologies,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nAbstract\nSpin dynamics of nanomolecules and nanoclusters are analyz ed. The nanosizes of\nthese objects make it possible to consider them as single-do main magnets with a large\ntotal spin, where the motion of the spins of all atoms, compos ing a nanocluster, occurs\nin a coherent way. Another meaning of coherence in spin dynam ics is the coherent spin\nmotion of several nanomolecules or nanoclusters. Diﬀerent a pproaches for treating\nspin dynamics are compared and the main mechanisms inﬂuenci ng the spin motion\nare studied. Spin dynamics of separate magnetic nanomolecu les and nanoclusters are\ninvestigated, as well as the spin dynamics of the ensembles o f these nano-objects.\n1 Introduction\nMagnetic nanomolecules and nanoclusters enjoy many similar proper ties because of which\nthe dynamics of their magnetization can be described by the same ty pe of equations. This\nis why, we consider both these nano-objects together. Of cours e, there is diﬀerence in their\nstructure and parameters which we shall take into account and ch aracterize them by the\nappropriate models. The detailed description of general physical p roperties and applications\nof diﬀerent magnetic nanoparticles can be found in review articles [1- 9]. Here we brieﬂy\nmention those of the properties and parameters that will be neces sary for the following\nconsideration.\nIt is worth stressing that there exist two types of magnetic nanop articles. One large\nclass consists of nanoclusters and nanomolecules, whose magnetic moments are formed by\nelectron spins. Another type includes nanomolecules that possess magnetic moments solely\ndue to polarized proton spins. Examples are propanediol C 3H8O2, butanol C 4H9OH, and\nammonia NH 3. In such nanomolecules, there is no any other magnetic moment exc ept that\ncaused by polarized protons. So, here the proton magnetic momen t is not a contribution,\nbut the main object.\nThe magnetic moment of an atom is composed of electron and proton moments, with the\nelectron magnetic moment µe=−geµBS=/planckover2pi1γeS=−µBand the proton magnetic moment\n1µp=gpµNS=/planckover2pi1γpS, wherege= 2 and gp= 5.586 are the electron and proton Land´ e\nfactors,µB=|e|/planckover2pi1/2meandµN=|e|/planckover2pi1/2mpare the Bohr and nuclear magnetons, γeandγp\nare the electron and proton gyromagnetic ratios. Since the proto n mass is larger than that\nof an electron, mp/me∼103, the proton magnetic moment is essentially smaller, µp/µe∼\n10−3. The electron and proton radii are re∼10−15cm andrp∼10−13cm, respectively. An\natom is called magnetic, when its total magnetic moment is nonzero. T he total spin of a\nmagnetic atom can be between 1/2 and S∼10, hence, its magnetic moment can be of order\n1µB−10µB. Atom radii are of order rA∼10−9−10−8cm. Examples of magnetic atoms are\nFe (Iron), Co (Cobalt), Ni (Nickel), Gd (Gadolinium), and Cr (Chrom ium).\nMagnetic nanomolecules are composed of many magnetic atoms an, a s is clear from\ntheir name, are of the nanometer size. An important property of a magnetic nanomolecule\nis that its total magnetic moment can be treated as being due to an e ﬀective total spin.\nGenerally, the molecule spin can be directed either up or down, with an energy barrier\nbetween these directions of order EA∼10−100 K. At high temperatures, above the blocking\ntemperature TB∼1−10 K, a magnetic molecule behaves as a superparamagnetic particle,\nwhosespinrandomlyoscillatesbetweentheupanddownpositions. Wh ilebelowtheblocking\ntemperature the spin is frozen in one of the directions.\nMagnetic nanoclusters are also made of magnetic atoms that are as sembled together in\na random way. This distinguishes them from magnetic molecules, wher e atoms are strictly\nconnected by chemical bonds. The sizes of nanoclusters can be in t he range between 1 nm\nand100nm, containingabout100 −105atoms. Thesevaluesdeﬁnethe coherence radius Rcoh,\nbelow which a nanocluster is in a single-domain state and can be treate d as a large particle\nwith an eﬀective spin. A cluster, with a size larger than Rcoh, separates into domains with\nopposite magnetizations. Similarly to magnetic molecules at low temper ature, the magnetic\nmoment of a nanocluster, below the blocking temperature TB∼10−100 K, is frozen in one\nof two possible directions. The eﬀective spin of a nanocluster is form ed by electron spins\nand can be as large as S∼100−105.\nThe often considered nanoclusters are made of the magnetic atom s of Fe, Ni, and Co.\nThey can be made of oxides, such as NiO, Fe 2O3, NiFe 2O4or alloys, such as Nd 2Fe14B,\nPr2Fe14B,Tb2Fe14B,DyFe 14B,Pr2Co14B,Sm 1Fe11Ti1,Sm1Fe10V2,Sm2Fe17N23,Sm2Fe17C22,\nSm2Co17, Sm2Co5. To protect nanoclusters from oxidation, one coat them with grap hene or\nnoble metals, forming the double-component nanoclusters, such a s Fe-Au, Co-Au , Co-Ag,\nCo-Cu, Co-Pt, Co-Pd, Ni-Au, Ni-Ag, Ni-Pd, and Mn−-Au. The coating is done be means of\nchemical reactions or laser ablationtechniques. The nanoclusters are produced by employing\nthermal decomposition, microemulsion reactions, and thermal spr aying.\nMagnetic nanoclusters and nanomolecules ﬁnd numerous application s, among which we\ncan mention magnetic chemistry, biomedical imaging, medical treatm ent, genetic engineer-\ning, waste cleaning, information storage, quantum computing, and creation of radiation\ndevices. Since both nanomolecules and nanoclusters possess many common properties and\ncan be considered as single particles with a large spin, we shall often t alk on nanoclusters,\nimplying that similar eﬀects can be realized with both of them, molecules as well as clusters.\nThe use of these nano-objects requires the existence of two pro perties that contradict\neach other. From one side, to be able to keep memory, a cluster has to enjoy a stable state\nwith itsspin frozen inonedirection. But fromanother side, inorder t o beable to manipulate\nthe cluster magnetization, there should exist a way of suppressing the anisotropy. And it\nis necessary that the spin manipulation could be done suﬃciently fast , so that the cluster\n2magnetization could be quickly reversed. Recall that thermal reve rsal is characterized by\nthe Arrhenius law giving the longitudinal relaxation time T1∼exp{EA/kBT}, whereEA\nis the anisotropy energy, so that, at temperatures below the bloc king temperature, the\nmagnetization is frozen.\nMagnetization reversal can be realized by diﬀerent methods, by ap plying transverse con-\nstant or alternating magnetic ﬁelds and short magnetic ﬁeld pulses [1 0]. To achieve fast\nreversal, one needs to ﬁnd optimal values for the amplitude, frequ ency, and duration of such\nﬁeld pulses.\nA very eﬃcient method of achieving ultrafast magnetization revers al of magnetic nan-\noclusters has been suggested [11] by employing the acceleration eﬀ ect caused by a resonator\nfeedback ﬁeld. The eﬃciency of this method is due to self-optimizatio n of the spin motion\nproducing the resonator ﬁeld acting back on the spins. Historically, this eﬀect was described\nby Purcell [12] and considered by Bloembergen and Pound [13] using c lassical phenomeno-\nlogical equations. Such equations are not suﬃcient for describing d iﬀerent regimes of spin\nmotion. Microscopic theory of spin dynamics has been developed bein g applied to polarized\nproton spins of such molecules as propanediol C 3H8O2, butanol C 4H9OH, and ammonia NH 3\n(see review articles [4,14]) and to magnetic molecules [15-19].\nThe aim of the present paper is threefold. First, we concentrate o n the spin dynamics of\nnanoclusters, comparing the peculiarity of their spin motion with tha t of proton and molec-\nular spins. Second, we analyze the role of other eﬀects, such as th e Nyquist-noise triggering\nand Dicke correlation, studying their inﬂuence on the spin dynamics o f nanoclusters. We\nshow that these eﬀects are negligible as compared to the Purcell eﬀ ect. And, third, we\ncompare diﬀerent approaches to describing spin dynamics, demons trating the advantage of\nusing a microscopic approach based on quantum equations of motion .\n2 Phenomenological classical equations\nDynamics of the magnetic moment Mof a magnetic particle is usually described by the\nclassical equation\ndM\ndt=−|γS|M×Heff+R, (1)\nin which γSis the giromagnetic ratio of the particle with spin SandRis a relaxation term.\nThe eﬀective magnetic ﬁeld is given by the variational derivative Heff=−δE/δMof the\nparticle energy E. The length of the magnetic moment is conserved, when the right-h and\nside of the equation dM2/dt= 2M·Ris zero.\nChoosing the relaxation term in the form\nR=−α|γS|\nMM×(M×Heff), (2)\none gets the Landau-Lifshitz equation, where αis a dissipation parameter and M≡ |M|.\nUnder form (2), |M|is conserved. The equation was initially derived [20] for describing\nenergy dissipation in the process of magnetic domain wall motion inside bulk ferromagnetic\nmatter. Though it is often applied for treating the dynamics of ferr omagnetic particles [21].\nTaking the relaxation term as\nR=α\nMM×dM\ndt, (3)\n3one comes to the Gilbert equation [22]. This equation, up to a renotat ion of parameters, is\nequivalent to the Landau-Lifshitz equation. Hence, it has the same region of applicability,\nthough it is also used for describing the magnetization rotation of ma gnetic particles [10].\nAnother form of the relaxation term has been advanced by Bloch [23 ] as\nR=−Mx−M∗\nx\nT2ex−My−M∗\ny\nT2ey−Mz−M∗\nz\nT1ez, (4)\nwhereM∗isanequilibrium magnetization, eαareunit coordinatevectors, andtherelaxation\nparametersarecharacterized bythelongitudinalrelaxationtime T1andtransverse relaxation\ntimeT2. Thelatterisalsocalledthedephasingtime. Foranensembleof Nmagneticparticles\nwith a large average spin polarization\ns≡1\nSNN/summationdisplay\nj=1/an}bracketle{tSz\nj/an}bracketri}ht, (5)\nthe transverse term has to be renormalized [16,24] as 1 //tildewideT2= (1−s2)/T2.\nTheLandau-Lifshitsequationhasasingledissipationparameter αandpreserves spherical\nsymmetry, thus, describing isotropic magnetization rotation. Bec ause of these properties, it\nis appropriate for bulk macroscopic ferromagnetic matter with sph erical magnetic symmetry.\nIt may also be used for magnetic clusters, possessing this symmetr y, which, however, is a\nrather rare case.\nThe Bloch equation has two relaxation parameters, T1andT2. Therefore it can describe\nmoregeneral situation ofanisotropicrelaxation, which ismorerealis tic fortreating nanoclus-\nters in a medium or below the blocking temperature, when T2≪T1. The Bloch equations\nhave been employed for considering the electron and nuclear spin mo tion in a strongly coher-\nent regime [25-29] and for spin-polarized129Xe gas [30]. But these equations cannot describe\nthe whole process of spin relaxation starting from an incoherent qu antum stage, for which\na microscopic approach is necessary [29,31,32]. The initial stage of sp in relaxation is trig-\ngered by quantum spin ﬂuctuations that can be identiﬁed with noneq uilibrium spin waves\n[4,16,31-33].\n3 Microscopic quantum approach\nIn a self-consistent quantum approach, we start with a microscop ic spin Hamiltonian ˆHthat\nis a functional of spin operators S. The evolution equations are given by the Heisenberg\nequations of motion\ni/planckover2pi1dS\ndt= [S,ˆH]. (6)\nThe advantage of using the quantum approach is in the following. Firs t, it takes into\naccount quantum eﬀects that can be important for small clusters . Hence, it is more general.\nSecond, at the initial stage of free spin relaxation, quantum spin ﬂu ctuations are of principal\nimportance, being the triggering mechanism for starting the spin mo tion. Third, being based\non an explicit spin Hamiltonian makes it possible to control the used app roximations and to\nhave well deﬁned system parameters.\n4We assume that a magnetic cluster isinserted into anmagnetic coil, of nturns andlength\nl, of a resonant electric circuit characterized by resistance R, inductance L, and capacity C.\nThe coil axis is taken along the axis x. Moving magnetic moments induce in the coil the\nelectric current jdescribed by the Kirchhoﬀ equation\nLdj\ndt+Rj+1\nC/integraldisplayt\n0j dt=−dΦ\ndt+Ef, (7)\nin which the magnetic ﬂux Φ = 4 πnMx/clis formed by the mean transverse magnetization\nMx=µ0/summationtextN\nj=1/an}bracketle{tSx\nj/an}bracketri}ht, whereµ0≡/planckover2pi1γS. HereEfis an additional electromotive force, if any.\nThe resonator natural frequency and circuit damping, respectiv ely, are\nω=1√\nLC, γ=R\n2L. (8)\nThe coil current creates the magnetic ﬁeld\nH=4πn\nclj (9)\nthat is the solution to the equation\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηdmx\ndt, (10)\nwhereη≡V/Vcoilis the ﬁlling factor and\nmx≡Mx\nV=µ0\nVN/summationdisplay\nj=1/an}bracketle{tSx\nj/an}bracketri}ht\nis the transverse magnetization density. The external electromo tive force is omitted. The\nﬁeldHis the feedback ﬁeld, created by moving spins and acting back on the m.\n4 Dynamics of a single nanocluster\nThe typical Hamiltonian of a nanocluster is\nˆH=−µ0B·S−D(Sz)2+D2(Sx)2+D4/bracketleftbig\n(Sx)2(Sy)2+(Sy)2(Sz)2+(Sz)2(Sx)2/bracketrightbig\n,(11)\nwhere the total magnetic ﬁeld\nB=B0ez+B1ex+Hex (12)\nconsists of an external constant ﬁeld B0, weak transverse anisotropy ﬁeld B1, and the feed-\nback resonator ﬁeld H. The anisotropy parameters D,D2,D4are deﬁned by the particular\ntype of considered nanoclusters.\nThe main attention will be payed to the investigation of spin dynamics s tarting from a\nstrongly nonequilibrium initial state, where the magnetization is direc ted opposite to the\nconstant external magnetic ﬁeld B0.\n5First, we study the inﬂuence of the thermal Nyquist noise of the co il in order to un-\nderstand whether it can trigger the spin motion in a nanocluster. Fo r the thermal-noise\nrelaxation time, we ﬁnd\ntT=4γVcoil\n/planckover2pi1γ2\nSωtanh/parenleftbiggω\n2ωT/parenrightbigg\n, (13)\nwhereωT≡kBT//planckover2pi1isthethermal frequency deﬁned bytemperature T. At lowtemperatures,\nbelow the blocking temperature, say at T= 1 K, we have ωT∼1012s−1. Then the thermal-\nnoise relaxation time is\ntT≃2γVcoil\n/planckover2pi1γ2\nSωT/parenleftbiggω\nωT≪1/parenrightbigg\n. (14)\nOn the other side, for the reversal time, caused by the resonato r feedback ﬁeld, we have\ntrev≃Vcoil\nπ/planckover2pi1γ2\nSS. (15)\nThe ratio of the latter to the thermal time (13) is trev/tT∼ωT/2πγS. For the typical values\nT= 1 K,γ∼1010s−1, andS∼103, this ratio is small: trev/tT∼10−2. Therefore the\nthermal Nyquist noise does not play any role in the spin dynamics of a n anocluster.\nWe have accomplished numerical solution of the evolution equations f or nanocluster\nparameters typical of Fe, Ni, and Co nanoclusters. The Zeeman fr equency is taken as\nω0≡2µBB0//planckover2pi1∼1011s−1. For the feedback rate, we have γ0≡πη/planckover2pi1γ2\nSS/Vcoil∼1010\ns−1. The typical anisotropy parameters satisfy the relations D/(/planckover2pi1γ0)∼10−3,D2/(/planckover2pi1γ0)∼\n10−3,D4/(/planckover2pi1γ0)∼10−10. At the initial time, the spin is assumed to be directed along the axis\nz. The resonator natural frequency is taken to be in resonance wit h the Zeeman frequency\ndeﬁned by the ﬁeld B0. The behavior of the spin polarization (5) is shown in Fig. 1, where\nwe compare the spin motion in the presence of the resonator ( h/ne}ationslash= 0) and in the absence of\nthe latter ( h= 0). Clearly, without the resonator feedback ﬁeld, the spin is block ed, while\nin the presence of the resonator, it reverses in short time trev∼10−10s.\n012345678−1−0.8−0.6−0.4−0.200.20.40.60.81s(t)\nth ≠ 0h = 0\nFigure 1: Spin reversal of a single nanocluster,\nwith parameters typical of nanoclusters made of\nFe, Ni, and Co.\n5 Dynamics of nanocluster assemblies\nThe ensemble of nanoclusters is described by the Hamiltonian\nˆH=/summationdisplay\niˆHi+1\n2/summationdisplay\ni/negationslash=jˆHij, (16)\n6where the indices i,j= 1,2,...,Nenumerate nanoclusters. The single nanocluster Hamil-\ntonians are\nˆHi=−µ0B·Si−D(Sz\ni)2+D2(Sx\ni)2+D4/bracketleftbig\n(Sx\ni)2(Sy\ni)2+(Sy\ni)2(Sz\ni)2+(Sz\ni)2(Sx\ni)2/bracketrightbig\n,(17)\nwith the total magnetic ﬁeld\nB=B0ez+Hex. (18)\nThe interaction term takes into account the dipolar spin interaction s\nˆHij=/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (19)\nthrough the dipolar tensor Dαβ\nij=µ2\n0/parenleftBig\nδαβ−3nα\nijnβ\nij/parenrightBig\n/r3\nij, in which rij≡ |rij|,nij≡rij/rij,\nandrij≡ri−rj.\nOne sometimes says that spin systems are similar to atomic systems, where transition\ndipoles are correlated by means of the photon exchange through t he common radiation ﬁeld.\nThis correlation leads to coherent atomic radiation called the Dicke su perradiance [34]. One\nsays that moving spins also radiate electromagnetic ﬁeld that could y ield the correlated spin\nmotion, in the same way as in the Dicke eﬀect. To check whether this is so, we need to\ncompare the time trad, required for inducing spin correlations through the common radiat ion\nﬁeld with the spin dephasing time T2. As the radiation time [35,36] for nanoclusters, we\nhave\ntrad=3c3\n2/planckover2pi1γ2\nSω3S, (20)\nwhile the spin dephasing time is\nT2=1\n/planckover2pi1ργ2\nSS. (21)\nFor the typical nanocluster density ρ∼1020cm−3andS∼103, the spin dephasing time is\nT2∼10−10s. While for the radiation time (20), with ω∼1011s−1, we have trad∼108s = 10\nyears. The ratio of times (20) and (21) is extremely large: trad/T2= 3c3ρ/(2ω3)∼1018. This\ntells us that the spin motion in no way can be correlated through elect romagnetic radiation.\nThat is, the Dicke eﬀect has no relation to the coherent spin motion. But spins can be\ncorrelated only through the Purcell eﬀect requiring the presence of a feedback ﬁeld caused\nby a resonator.\nThe feedback rate due to the resonator is\nγ0=πηρ/planckover2pi1γ2\nSS . (22)\nThe reversal time for Ncorrelated nanoclusters becomes\ntrev=1\nγ0=Vcoil\nπ/planckover2pi1γ2\nSSN=t1\nrev\nN, (23)\nwheret1\nrevis the relaxation time (15) for a single nanocluster inside the same coil.\nWe solved the evolution equations for the nanocluster assemblies inv olving the scale\nseparation approach [4,14] that is a generalization of the Krylov-Bo golubov [37] averaging\nmethod. Four classes of spin objects have been investigated.\n7(i) Polarized nuclear materials, such as propanediol C 3H8O2, butanol C 4H9OH, and am-\nmonia NH 3, with the parameters: S= 1/2,ρ= 1022cm−3,T= 0.1K,B0∼104G,ω0∼\n108s−1,λ∼102cm,T1∼105s,T2∼10−5s,τ≡1/γ∼10−6s . Recall that in these\nnanomolecules the magnetization is due to polarized proton spins.\nThe following characteristic times are found: thermal-noise time tT∼1016s∼109years,\nradiation time trad∼1015s∼108years, and reversal time trev∼10−6s.\nTherefore, neither the Nyquist thermal noise nor the photon exc hange through the radi-\nated ﬁeld play any role in the relaxation process. Spin dynamics, resu lting in the magnetiza-\ntion reversal, is completely due to the action of the resonator feed back ﬁeld. As is explained\nabove, the same concerns nanomolecules and nanoclusters\n(ii)Nuclearpolarizedferromagnets, whereprotonspinsarepolariz edandinteractthrough\nhyperﬁne forces with electrons participating in forming ferromagn etic order. In such ma-\nterials, the electron subsystem plays the role of an additional reso nator enhancing eﬀective\nnuclear correlations. Being interested in the motion of nuclear spins , under a ﬁxed mean\nelectron magnetization, we ﬁnd the reversal time trev∼10−9s.\n(iii) Molecular magnets, such as Mn 12and Fe 8, with the typical parameters: S= 10,ρ=\n1020−1021cm−3,TB= 1K,B0∼105G,ω0∼1013s−1,λ∼10−2cm,ωA≡EA//planckover2pi1∼1010−\n1012s−1,T1∼105−107s,T2∼10−10s. The reversal time is trev∼10−11s.\n(iv) Magnetic nanoclusters composed of Fe, Ni, and Co, at T= 1 K, with the typical\nparameters: S= 103,ρ= 1020cm−3,TB= 10−40K,B0∼104G = 1T,ω0∼1011s−1,λ∼\n1cm,T1∼1034s∼1027years,T2∼10−10s. The reversal time can be very small reaching the\nvaluetrev∼10−12s.\nIn the case of magnetic molecules and, especially, nanoclusters, be cause of their high\nspins, thesystemofmanyclusterscanproducequitestrongcohe rentradiationofthemaximal\nintensity\nImax∼2µ2\n0\n3c3S2ω4N2\ncoh, (24)\nwhereNcoh∼ρλ3is the number of clusters in a coherent packet. The intensity of rad iation\nof magnetic molecules, with Ncoh∼1014, is of order Imax∼105W. And for magnetic\nnanoclusters, with Ncoh∼1020, the radiation intensity can reach Imax∼1012W.\nThere can happen several regimes of spin dynamics depending on th e initial spin polar-\nization, the strength of a triggering pulse, and the eﬀective couplin g parameter\ng≡γγ0ω0\nγ2(γ2\n2+∆2), (25)\nwhereγ2≡1/T2and ∆≡ω−ω0is the detuning from resonance. These regimes for\nnanoclusters can be classiﬁed analogously to those occurring for n uclear magnets [4,16]:\nincoherent free relaxation, weakly coherent free induction, weak ly coherent superradiance,\nstrongly coherent pure superradiance, strongly coherent trigg ered superradiance, pulsing\nsuperradiance, and punctuated superradiance [38].\nIt is important to stress that the existence of magnetic anisotrop y in magnetic nanoclus-\nters does not preclude the realization of fast spin reversal, provid ed the external magnetic\nﬁeld is suﬃciently strong. The inﬂuence of the anisotropy energy EAon the spin reversal\nof a nanocluster system is shown in Fig. 2, where A≡EA/ω0. This regime corresponds to\npure spin superradiance.\n80 0.05 0.1 0.15 0.2−1−0.500.51\nts(t)\nA = 0\nA = 0.5A = 1\nFigure 2: Inﬂuence of magnetic anisotropy on the\nspinreversal ofanensemble ofmanynanoclusters,\nwith the parameters typical of Fe, Ni, and Co.\nCoherent dynamics in the spin assemblies, formed by magnetic nanom olecules, have an\nimportant diﬀerence from the spin dynamics in an ensemble of magnet ic nanoclusters. Mag-\nnetic molecules are identical and form the systems with well organize d crystalline lattices.\nWhile magnetic nanoclusters vary in their shapes, sizes, and total s pins, which results in an\nessential nonuniform broadening. Computer simulations, accomplis hed together with V.K.\nHenner and P.V. Kharebov, demonstrate that this nonuniformity d oes not destroy coherent\nspin motion. A detailed analysis of the computer simulations, with nonu niform nanocluster\ndistributions, will be presented in a separate publication.\nIn conclusion, we have considered spin dynamics in magnetic nanomole cules and nan-\noclusters, starting from a strongly nonequilibrium state, with the m agnetization directed\nopposite to the applied external magnetic ﬁeld. We have compared s everal methods of de-\nscribing the spin dynamics, showing that a microscopic approach, ba sed on the quantum\nequations of motion, is the most accurate. We also have analyzed th e inﬂuence of diﬀerent\neﬀects on spin dynamics. The eﬀects of the Nyquist-noise triggerin g and of Dicke correla-\ntions are found to be negligible for spin systems. This principally disting uishes spin systems\nfrom atomic systems or quantum dot systems [39], where correlatio ns, leading to coherent\nradiance, are caused by the Dicke eﬀect of interactions through t he common radiation ﬁeld.\nThe feedback ﬁeld, developing in the resonator, reaches rather h igh values, of the order\nof the applied constant magnetic ﬁeld. Such a strong feedback ﬁeld suppresses the inﬂuence\nof mutual cluster interactions. Generally, in an ensemble of nanoclu sters of suﬃciently high\ndensity, in addition to dipole interactions, there can appear exchan ge interactions [40] that\ncan inﬂuence equilibrium properties of nanoclusters. But in the cons idered case of strongly\nnonequilibrium spin dynamics, the exchange interactions are also sup pressed by the self-\norganized resonator feedback ﬁeld.\nThis important conclusion can be formulated as follows: Coherent spin dynamics are\ncompletely governed by the Purcell eﬀect that is caused by th e action of the resonator feedback\nﬁeld.\nReferences\n[1] BarbaraB, ThomasL,Lionti F,Chiorescu I,andSulpiceA1999 J. Magn. Magn. Mater.\n200167\n[2] Wernsdorfer W 2001 Adv. Chem. Phys. 11899\n9[3] Ferre J 2002 Topics Appl. Phys. 83127\n[4] Yukalov V I and Yukalova E P 2004 Phys. Part. Nucl. 35348\n[5] Wang J and Zeng X C 2009 in: Nanoscale Magnetic Materials and Applications\n(Springer, Berlin), p. 35\n[6] Bedanta S and Kleemann W 2009 J. Phys. D 42013001\n[7] Berry C C 2009 J. Phys. D 42224003\n[8] Beveridge J S, Stephens J R, and Williams M E 2011 Annu. Rev. Anal. Chem. 4251\n[9] Hoang V V and Ganguli D 2012 Phys. Rep. 51881\n[10] Bauer M, Fassbender J, Hillebrands B, and Stamp R L 2000 Phys. Rev. B 613410\n[11] Yukalov V I and Yukalova E P 2012 J. Appl. Phys. 111023911\n[12] Purcell E M 1946 Phys. Rev. 69681\n[13] Bloembergen N and Pound R V 1954 Phys. Rev. 958\n[14] Yukalov V I and Yukalova E P 2000 Phys. Part. Nucl. 31561\n[15] Yukalov V I 2002 Laser Phys. 121089\n[16] Yukalov V I 2005 Phys. Rev. B 71184432\n[17] Yukalov V I, Henner V K, Kharebov P V and Yukalova E P 2008 Laser Phys. Lett. 5\n887\n[18] Yukalov V I, Henner V K and Kharebov P V 2008 Phys. Rev. B 77134427\n[19] Yukalov V I and Yukalova E P 2011 Laser Phys. Lett. 8804\n[20] Landau L D and Lifshitz E M 1935 Phys. Zeits. Sow. 8153\n[21] Henner V, Raikher Y and Kharebov P 2011 Phys. Rev. B 84144412\n[22] Gilbert T L 1956 PhD Thesis (Illinois Institute of Technology, Chica go)\n[23] Bloch F 1946 Phys Rev. 70460\n[24] Abragam A and Goldman M 1982 Nuclear Magnetism: Order and Disorder (Clarendon,\nOxford)\n[25] Yukalov V I 1992 Laser Phys. 2559\n[26] Fokina N P, Khutsishvili K O and Chkhaidze S G 1992 J. Exp. Theor. Phys. 1021013\n[27] Khutsishvili K O and Chkhaidze S G 1992 Physica B 17654\n[28] Fokina N P, Khutsishvili K O and Chkhaidze S G 1992 Physica B 179171\n10[29] Yukalov V I 1996 Nucl. Instrum. Methods Phys. Res. A 37345\n[30] Yoshimi A, Asahi K, Sakai K, Tsuda M, Yogo K, Ogawa H, Suzuki T and Nakagura M\n2002Phys. Lett. A 30413\n[31] Yukalov V I 1995 Phys. Rev. Lett. 753000\n[32] Yukalov V I 1996 Phys. Rev. B 539232\n[33] Rueckriegel A, Kreisel A and Kopietz P 2012 Phys. Rev. B 85054422\n[34] Dicke R H 1954 Phys. Rev. 9399\n[35] Yukalov V I and Yukalova E P 2005 Laser Phys. Lett. 2302\n[36] Yukalov V I 2005 Laser Phys. Lett. 2356\n[37] Bogolubov N N and Mitropolsky Y A 1961 Asymptotic Methods in the Theory of Non-\nlinear Oscillations (Gordon and Breach, New York)\n[38] Yukalov V I and Yukalova E P 2002 Phys. Rev. Lett. 88257601\n[39] Yukalov V I and Yukalova E P 2010 Phys. Rev. B 81075308\n[40] Kondratyev V N and Lutz H O 1998 Phys. Rev. Lett. 814508\n11"    },    {        "title": "0905.0482v1.Collective_dynamics_of_interacting_Ising_spins__Exact_results_for_the_Bethe_lattice.pdf",        "content": "arXiv:0905.0482v1  [cond-mat.dis-nn]  4 May 2009Collectivedynam icsofinteracting Ising spins:Exactresu ltsfortheBethelattice\nA.L.Buri n\nDepartmentofChemistry,Tul ane University,New Orl eans LA ,70118\nN.V.Prokofev\nDepartmentofPhysics,University ofM assachusetts,Amher st,M A 01003\nI.S.Tupi tsyn\nPaci\fcInstituteofTheoreticalPhysics,University ofBri tish Col umbia,Vancouver,BC V6T1Z1,Canada\n(Dated:November19,2021)\nW e study the l ow temperature dynami csi n \fl msmade ofmol ecul armagnets,i . e. crystal s\ncomposed ofmol ecul eshavi ng l arge el ectroni c spi n Si n thei rground state. Theel ectroni c spi n\ndynami csi smedi ated by coupl i ng to a nucl earspi n bath;thi s coupl i ng al l owstransi ti onsfora\nsmal lfracti on ofel ectroni c spi ns between thei rtwo energy mi ni ma, Sz=\u0006S,underresonant\ncondi ti onswhenthechangeoftheZeemanenergyi nmagneti cd i pol ar\fel dofotherel ectroni cspi ns\ni scompensatedbyi nteracti onwi thnucl earspi ns.Transi ti onsofresonantspi nscanresul ti nopeni ng\norcl osi ngresonancesi nthei rnei ghborsl eadi ngtothecol l ecti vedynami csatsu\u000eci entl yl argedensi ty\nP0ofresonantspi ns.W eformul ateand sol vetheequi val entdyn ami cpercol ati on probl em forthe\nBethel atti ce(BL)ofspi nsi nteracti ngwi th znei ghborsand \fnd thatdependi ngon thedensi ty of\nresonantspi ns P0and thenumberofnei ghbors zthesystem hasei therone(2 < z <6)ortwo\n(z\u00156)ki neti ctransi ti onsat P0=Pc1\u0019e\u00001=3=(3z)andP0=Pc2\u0019e\u00001=z.Theformertransi ti on\ni sconti nuousand associ ated wi th the formati on ofan i n\fni t e cl usterofcoupl ed resonantspi ns\nsi mi l arl y tothestati cpercol ati on transi ti on occurri nga tP0\u00191=z.Thel attertransi ti on, z>5,i s\ndi sconti nuousand associ ated wi th thei nstantaneousi ncre asei n thedensi tyofresonantspi nsfrom\nthesmal lval ue \u00181=ztonearuni ty.Experi mentali mpl i cati onsofourresul tsare di scussed. \nPACS numbers:7080.Le,72.20.Ee,72.25.-b,87.14.Gg\nI. IN TROD U CTION \nPercol ati ontheorydescri besthe\row i nheterogeneous\nmedi a. Iti ssuccessful l y appl i ed to a vari ety ofphysi -\ncal ,chemi cal ,bi ol ogi calandevensoci alprocessesrangi n g\nfrom thehoppi ngconducti vi tyi ndopedsemi conductors1\nto the evol uti on of l arge geneti c networks and sel f-\norgani zed cri ti cal i ty2,3. Fori nstance,thel ow tempera-\ntureconducti vi ty ofhoppi ngi nsul atorscan bemodel ed\nby the equi val entnetwork ofresi stancesrepl aci ng el e-\nmentaryel ectron hoppi ngevents1.Thepercol ati onthe-\nory\fndstheopti mum setofi nterconnected open chan-\nnel scharacteri zedbyresi stancesnotexceedi ngacertai n\nmaxi mum and formi ngan i n\fni tecl ustertogui deel ec-\ntronsthroughthesampl e. \nOpen percol ati on channel sareusual l y treated assta-\nti onaryi n ti me.Iti srel ati vel yeasytostudy thestati c\ncasenumeri cal l y. M oreoverthereexi stanal yti calsol u-\nti onsi ncl udi ngpercol ati onontheBethel atti ce(BL)and\ni nl ow-di mensi onalsystems4,5,6,7,8.Inal lcasestheperco-\nl ati on ki neti ctransi ti on i sfound tobeconti nuous.Sys-\ntem parametersi ncl udi ng,fori nstance,di \u000busi on coe\u000e-\nci ent,correl ati on radi usand averagesi zeofpercol ati ng\ncl ustersal lshow scal i ngbehavi ornearthecri ti calpoi nt\nsi mi l arl ytotheoneforsecondorderphasetransi ti ons. \nThe modelofstati c percol ati on i soften an approxi -\nmati on. Fl owi ng parti cl esi nteractwi th each otherand\nthei rmoti oncanopenorcl osepercol ati onchannel sl ead-\ni ngto,fori nstance,conducti vi tynoi se9.Thesi gni \fcanceof\ructuati onsforthecooperati vedynami csofquantum\ndefects,spi nsi nspi ngl asses,protonsi ni oni cconductors\nand el ectronsi n hoppi ng i nsul atorswaspoi nted outi n\nRefs. 10,11,12,13,14,15.In amorere\fned anal ysi sofperco-\nl ati on oneshoul d i ncl udethepossi bi l i ty thatthe\rows\nthroughopenchannel scanopenorcl oseotherchannel s. \nThesedynami ccorrel ati onscana\u000bectki neti ctransi ti ons\ni n a fundamentalway and l ead to the cooperati vedy-\nnami cs,e.g.percol ati onmayoccuratsmal l erdensi tyof\nopen channel sand becomedi sconti nuous.In thi spaper\nwe suggesta si mpl e modelofdynami c percol ati on for\nthel ow-temperatureki neti csofthesystem ofi nteract-\ni ngmagneti cmol ecul esand obtai n an exactdescri pti on\noftheki neti ctransi ti onson BL.Accordi ngtotheargu-\nmentsofRefs. 16,17theuseofBL i smorejusti \fedi nour\ncasethan forthestati cpercol ati on becauseofthesi mi -\nl ari tyi n thephasespacestructureforBethel atti ceand\nmany-bodysystem ofi nteracti ngspi ns.Inbothcasesthe\nsi zeofthephasespacegrowsexponenti al l ywi ththesys-\ntem si ze(2N forNspi nsandz(z\u0000 1)N \u00001forthenumber\nofnodesi n BL wi th thecoordi nati on number zandN\nshel l softhetree). \nInthepresentworkwestudythedynami cpercol ati on\nmodelon theBethel atti ceand then projectourresul ts\nontheel ectroni cspi ndynami csi n\fl msmadeofmagneti c\nmol ecul essuch as M n12, Fe8,etc.Thesemol ecul esare\nusual l ycomposedoftransi ti onmetal s,formi ngthemag-\nneti ccore,coupl ed to vari ousorgani cl i gands18. They\npossessl argeel ectroni cspi n Si nthegroundstate(both2\nM n12andFe8mol ecul esarecharacteri zedbytheground\nstate\\centralspi n\" S= 10).Atl ow temperaturesonl y\nthetwol oweststatesofeach mol ecul e,characteri zedby\nthe spi n projecti ons Sz\ni=\u0006 Sto the easy axi s z,are\noccupi ed and each mol ecul ecan be model ed asa two-\nl evelsystem wi th two statescorrespondi ngto theIsi ng\npseudo-spi n1 =2.Thetunnel i nggap2\u0001 obetweenthetwo\nl owestspi nstatesi nzeroexternaltransverse\fel di sti ny,\n\u0018 10\u000011\u0000 10\u00007K.Iti smuchsmal l erthanotherrel evant\nparametersi ncl udi ngthestrengthofthedi pol ari nterac-\nti on between thenearest-nei ghborspi ns, UD \u0018 0: 1K, \nandthehal f-wi dthofthedi stri buti onsofthenucl earbi -\nases, Eo\u0018 10\u00002\u0000 10\u00003K.Underthesecondi ti onsonl y\nspi nsexposed toatotall ongi tudi nalbi as(external\fel d\npl usi nternaldi pol ardemagneti zati on\fel d)smal l erthan\nEocane\u000eci entl ytransferbetweenthei rtwostatesdueto\nenergyexchangewi ththenucl earspi nbath20,21,22.These\nspi nsare the resonant,or\\open\",ones. Atthe same\nti me,si nce Uo\u0015 Eo,each spi n \ri p a\u000bectsal li tsnei gh-\nborsandmaycreatenew resonances(ordestroyexi sti ng\nones,seeSec.II).Theassoci atedcol l ecti vedynami csof\nspi nsi sstudi edwi thi nthedynami cpercol ati onmodeli n\nSecs.III,IV (seeFi g.1). \nTobemorespeci \fc,onemaybei nterestedi nthecor-\nrel ati onfuncti on C(t)=N\u00001P \nihSz\ni(t)Sz\ni(0)iwhereNi s\nthetotalnumberofspi ns,and theaveragei stakenover\ni ni ti alcondi ti ons,andevol uti onhi stori es.Ani nteresti ng\nquanti ty to i nvesti gatei n thei n\fni teti mel i mi twoul d\nbethefracti on ofspi n whi ch neverchangethei rmagne-\nti zati onW \u0003. Thi sfracti on can beeasi l y seen to resul t\ni nthe\fni teval ueof C(t! 1 )si ncedynami cal l yfrozen\nspi nscontri buteuni tytothesum above.Theotherques-\nti onconcernsspi ndi \u000busi onandthepossi bi l i tytotransfer\nenergy and magneti zati on overl argedi stancesi n whi ch\ncaseoneshoul d bel ooki ng fortheonsetofpercol ati ng\ncl ustersofmobi l espi ns. \nThee\u000bectofdynami cand percol ati on transi ti onson\nspi n rel axati on i n \fl ms ofmagneti c mol ecul es,based\non exampl esof M n12and Fe8,i sdi scussed i n Sec. V. \nOurgoalhere i sto i nvesti gate the case ofcompl etel y\ndemagneti zed sampl es at rel ati vel y hi gh temperatures\nkBT > UD toavoi dthe\\di pol arorderi ng\"e\u000bectsandre-\nl atedsampl e-geometrydependentdemagneti zati on\fel ds. \nIn thi s l i mi t the resonant spi ns are nearl y uni forml y\nand randoml y di stri buted i n the system bul k contrary\ntothecaseofstrongl ypol ari zedsampl eswhereresonant\nspi ns form spati al l y ordered structures,the \\resonant\nsurfaces\",and i ni ti alcondi ti onspl ay an i mportantrol e\ni nthesystem ki neti cs22. \nW e predi ct the reducti on ofthe rel axati on rate by\nmany ordersofmagni tude i n the vi ci ni ty ofthe tran-\nsi ti onpoi nt.Atthesameti metheabruptchangeofthe\nrel axati on ratei ssmeared outby i tsconti nuousdepen-\ndenceonthetotall ongi tudi nalbi as,whi chmayobscure\ntheexperi mentalobservati onofthetransi ti onpoi nt. II. COLLECTIVE DYN AM ICS OF\nIN TERACTIN G SPIN S\nSi ncethemodelofspi ndynami csi sal readyformul ated\ni n greatdetai li n Refs. 20,21,22,herewebri e\ryoutl i nei ts\nmai n featuresand then i ntroducei tsBetheLatti cever-\nsi on.Asi twasmenti onedi ntheIntroducti on,theweak-\nnessofspi n tunnel i ng i n mol ecul armagnetsal l owsan\naccuraterepresentati on ofthesystem i n termsofIsi ng\npseudospi ns1 =2coupl edtoeachotherbythel ong-range\nmagneti cdi pol ari nteracti on UijSz\niSz\njwi th\nUij=UD a31\u0000 3n2\nij\nR3\nij; nij=Rij\nRij; (1)\nwhereUD i sthestrengthofthedi pol e-di pol ei nteracti on\nofspi ns; ai sthel atti ceconstant;and Riji sthevector\nconnecti ng mol ecul es iandj. Forthesakeofsi mpl i c-\ni ty wel i mi tourconsi derati on to thecaseof2 Dsquare\nl atti ce. In thi smodeleach spi n Sii ssubjected to the\n\\mol ecul ar\"\fel dofal lotherspi ns\nUi=X \njUijSz\nj (2)\nandtheexternalmagneti c\fel d \u0016h,where\u0016i stherel ated\nmagneti cmoment. \nThe spi n dynami csi sassoci ated wi th the spi n tun-\nnel i ngbetweenstates Sz=\u0006 1=2.Thetransi ti onofeach\nspi nbetweenthesetwostatesrequi resthem toacqui reor\nrel easethel ongi tudi nalenergyEq.(1)tosomethermal \nbathbecauseoftheenergyconservati on.Thecoupl i ngof\nspi nstophononsusual l yresponsi bl eforl ow-temperature\ndynami csi ndi el ectri csi sextremel yweakandcanbene-\ngl ected. Therel axati on takespl acedueto thei nterac-\nti onofthecentralel ectroni cspi nswi th thenucl earspi n\nbath20,21,22. Si nce i n the l i mi t\u0001 o<< Eoonl y spi ns\ni nsi dethe\\resonancewi ndow\"2 Eoareal l owed to tun-\nnel ,theapproxi mateconstrai nt j \u0016h+Uij < Eodeter-\nmi nesthesubspaceof\\resonantspi ns\".Inthi sform the\nconstrai nti smostsui tabl eforthedynami cpercol ati on\nmodelstudi ed bel ow i n Secti onsIIIand IV. M oreac-\ncurate dependence ofthe rel axati on ti me on energy i s\nconsi deredi nSecti onV. \nTheonl yspi nsthatareal l owedtotunneli nanygi ven\ncon\fgurati onaretheresonantones.Transi ti onsofreso-\nnantspi nscanchangethestatusofthei rnei ghborsfrom\nnon-resonanttoresonantandvi ceversabychangi ngthei r\nl ocalbi asfrom bei ng l argerthan Eoto bei ng smal l er\n(see Fi g. 1. Atthe same ti me i n the new con\fgura-\nti on thetransi ti onsofnei ghbors,di \u000bused i ntothereso-\nnancewi ndow,canpushtheformerresonantspi nsoutof\nresonance. Dependi ng on thedensi ty ofresonantspi ns\nP0\u0018 Eo=UD thereexi sttwopossi bi l i ti es. If P0i svery\nsmal l(i . e. ,the di stance between the resonantspi nsi s\nl arge),thetransi ti onsofresonantspi nscanessenti al l ya f-\nfectonl ythei rl ocalenvi ronmentandspi nsi nthesampl e\ncanbeseparatedi ntoasmal l\\mobi l e\"andl arge\\i mmo-\nbi l e\" groups. Themobi l egroup consi stsofpercol ati ng3 \n \n \n \n \n \n \n \n \n \n             \n(1) \n(2) \n(3) \n FIG.1: Evol uti onof\\percol ati ng\"(or\\open\")spi nsa\u000becti ng\nthei rnei ghborsi n 2 Dl atti ce.W hen someoftheopen spi ns\n(shown by dashed red arrows) make transi ti ons (shadowed\narrows for steps (2) and (3)) they swi tch the state ofthe\nnei ghbori ngspi nsbetween \\open\"and \\cl ose\". \nspi ns,capabl eofmaki ngtransi ti onsi nthecourseofsys-\ntem evol uti on,whi l ethei mmobi l egroupconsi stsofnon-\npercol ati ng spi nswhi ch cannot\ri p i n spi teofthe\fel d\n\ructuati onscausedbymobi l espi ns.Ifthedensi ty P0i s\nl arge,theresonantspi ntransi ti onscreateordestroyres-\nonancesaroundthem l eadi ngtothecol l ecti vedynami cs\ni nvol vi ngpracti cal l yal lspi nsaftersu\u000eci entl yl ongti me . \nIn whatfol l owswe i nvesti gatethe ki neti c transi ti on\nbetween these two regi mes. The study ofreal i sti c2 D\nl atti ceofspi ns,coupl ed by thedi pol ari nteracti on,can\nnotbedoneexactl yanal yti cal l yand requi resnumeri cal \nsi mul ati ons. However,we can sol vea si mi l arprobl em\non theBethel atti cewi th random i nteracti onsbetween\nthenei ghbors.W eassumethatspi ns1 =2occupyal lsi tes\noftheBethel atti ceand each spi n i nteractswi th al li ts\nznei ghbors. Thei nteracti on constant Uijbetween two\nnei ghbori ngspi nsi sassumedtobetheGaussi anrandom\nvari abl ewi th thezeroaverageand di spersi on UD .Thi s\nassumpti oni sapproxi mate-di pol ari nteracti onsi nareal \nsystem i nvol venotonl ynearestnei ghborsbutal sospi ns\nseparated from thegi ven oneby morethan onel atti ce\nconstant.Thel ongrangenatureofdi pol ari nteracti ons\ncannotbenegl ectedi nthree-di mensi ons,buti nthetwo-\ndi mensi onalsystem thel atti cesum of1 =r3i s\fni teand\nthenearestnei ghborapproxi mati oni sjusti \fed.Iti scl ear\nthatthemodelwestudyandtherealprobl em di \u000berfrom\neachother;howevertheycanbel ongtothesameuni ver-\nsal i ty cl assbecausethephase-spacestructurefori nter-\nacti ngIsi ngspi nsi ssi mi l artothatfortheBethel atti ce. \nThephasespacegrowsexponenti al l ywi ththenumber N\nofspi nsasi nmany-bodyprobl em16,17,whi chpermi tsus\nto negl ect\\cl osel oops\" fordi \u000berentevol uti on pathsof\nthesystem i n i tsphasespacethusmaki ngi tsi mi l arto\ntheBethel atti ce. \nAs menti oned above,we restri ct our consi derati ons\nto the l i mi tofl ow temperatures,where each mol ecul e\ncan sti l lbe model ed asa two-l evelsystem,butweas-\nsumekBT > U D to avoi d the di pol ar orderi ng and, \nconsequentl y,any l ong-rangestati sti calcorrel ati onsbe -tween spi ns.Indeed,such correl ati onsshoul d a\u000bectthe\nki neti c transi ti on. Thi s assumpti on i s not restri cti ve\nsi nce,fori nstance,i n Fe8andM n12theregi meofpure\nground-statetunnel i ng can bereached attemperatures\nT.0: 4KandT.0: 8Krespecti vel y whi ch i sl arger\nthanUD i n both systems19.W eal soconsi deronl ycom-\npl etel ydepol ari zedsystem here. \nIII. DYN AM IC PERCOLATION PROBLEM ON \nBETH E LATTICE\nIn theBethel atti cethespi n i sresonant(open)i fi ts\nl ocalbi as(orZeemanspl i tti ng) Ui=P \njUijSj,Eq.(2), \ni ssmal l erthan Eo.Si ncethedi pol ar\fel dsdi stri buti on\ni n thedepol ari zed l i mi ti ssi mi l arto theGaussi an one\n(see Fi g. 4 bel ow),the fracti on ofresonantspi ns (i . e. , \ntheprobabi l i ty thatthespi n i sopen)i sdetermi ned by\nP0\u0019 2Eo=(UD p\n2\u0019z)\u001c 1. W e expect(and thi swi l l \nbecon\frmed by the\fnalresul t)thatthecol l ecti vedy-\nnami cscantakepl acei feachspi nhasapproxi matel yone\nresonantnei ghbor(cf. Ref. 7),whi ch l eadsto theesti -\nmateforthetransi ti on poi ntas P0c\u0018 1=z. Thi shap-\npensatEo\u0018 UD p \n\u0019=(2z). Nearthe transi ti on poi nt\nonecan approxi matel y i gnorecorrel ati onsbetween di f-\nferentcon\fgurati onsofnei ghborscorrespondi ngtoopen\nstatesofthegi venspi n.Indeed,i fthespi ni sopeni nthe\ngi ven con\fgurati on theprobabi l i ty thati twi l lbeopen\nagai nafter k< zturnsofnei ghborscanbeesti matedas\nPk\u0018 2Eo=(UD p\n2\u0019k). Thenumberofnei ghbortransi -\nti onsZneededtobri ngthegi venspi nbacktotheopen\nstatecanbeesti matedsetti ngP Z\nk=1Pk\u0019 1.Onecansee\nthatatEo\u0018 UD p \n\u0019=(2z)thenextopeni ngofthegi ven\nspi n can beexpected,on average,afterturnsofaround\nz=2nei ghborssothenew resonantenvi ronmenti sful l y\ndi \u000berentfrom theprevi ousone. \nUnderaboveassumpti onsonecan formul atethefol -\nl owi ngmodelofdynami cpercol ati onontheBL,seeFi g. \n1.Consi deral atti ceoccupi ed byspi ns1 =2havi ngran-\ndom projecti on Sz=\u0006 1=2. Assumethati n any gi ven\ncon\fgurati on ofspi nsthe rul esfordeci di ng aboutthe\nspi n statesarethe same: the probabi l i ty thata gi ven\nspi n i sopen (i . e. ,\\dynami c\")i n each ofthe2zcon\fgu-\nrati onsofnei ghborsi s P0.Inotherwords,thedynami cs\nofthemodeli s\fxedbyhavi ngatabl eforeachspi nwi th\n2zopen and cl osestates(thefracti on ofcl osed statesi s\nW 0= 1\u0000 P0).Correl ati onsbetweenopenstatesofnei gh-\nborsi n di \u000berentcon\fgurati onsarei gnored. Onl y open\nspi nsareal l owedtochangethei rstates.Asti meevol ves, \nal lopen spi nshaveequalchancestomakeatransi ti on. \nThespi noverturna\u000bectsal li ts znei ghborsi nawaythat\nthey haveto change\\status\" accordi ngto thei rtabl es. \nOurmai n goali sto study thecooperati vedynami csof\nspi ns,characteri zedbythefracti on P\u0003= 1\u0000 W \u0003,whi ch\nwerei nvol ved i nto dynami csatsomestage(i fthespi n\never\ri psi n thecourseofevol uti on wecal li t\\percol at-\ni ng\")andto\fndoutwhenani n\fni tepercol ati ngcl uster\ni sformedbythosepercol ati ngspi ns. 4\nThe probl em under consi derati on has common fea-\ntures and di \u000berences wi th the bootstrap percol ati on\nprobl em25,26used asa mathemati cal l y i deal i zed model \nforsuch phenomena asnucl eati on and growth appl i ed\ni n the study ofcrack formati on,magneti c al l oys,hy-\ndrogen mi xtures,and computerstoragearrays. Ithas\nbeenextensi vel ystudi edbothforthe2 Dl atti ceandBL. \nThebootstrappercol ati onprobl em canbeformul atedi n\ntermsofspi ns1 =2pl acedi n al ll atti cesi tes.Thespi n i s\nopenandal l owedto\ri pi fthenumberofnei ghborswi th\nSz= +1=2 exceedstheprede\fned number0 < n\u0014 z. \nThe cooperati vedynami csi sdetermi ned by the i ni ti al \ndensi tyP0ofspi nswi th posi ti veprojecti onsand i n the\nmosti nvesti gated caseof n= 2thi sdynami csvani shes\ni n a di sconti nuousmanneratsu\u000eci entl y smal l P0;for\nthe2Dl atti cethethreshol ddensi ty P0capproacheszero\nwi th i ncreasi ngthesystem si ze.In spi teofsi mi l ari ti es, \nourmodeldi \u000bersqual i tati vel ybecausethebootstrapper-\ncol ati on resul tsi n thei rreversi bl eevol uti on ofspi n con-\n\fgurati on towardsthe i ncrease i n the densi ty ofspi ns\nwi thposi ti veprojecti on,whi l ei nourcasethedensi tyof\nopen spi nspracti cal l y doesnot\ructuateand westudy\na reversi bl eequi l i bri um dynami cs. Oneconsequenceof\nthi sdi \u000berencei svani shi ng P0ci n thebootstrap perco-\nl ati on probl em i n the2 Dl atti cewhi l ei n ourcasethere\ni sno cooperati vedynami csatsu\u000eci entl y smal ldensi ty\nofopen spi ns P0<2\u0000z=z(z= 4 i n 2D)becausenon-\npercol ati ng spi nsform an i n\fni tecl usterbl ocki ng such\ndynami cs,cf.Ref. 1. \nTheexactdescri pti on ofthedynami cpercol ati on for\nBL can beobtai ned si mi l arl y to Refs. 5,25(seeFi g. 2). \nFi rst,wecal cul atethedensi ty P\u0003ofpercol ati ng spi ns. \nNotethatpercol ati ngspi nscanbei n thecl osestatefor\nsome ti me. Thuspercol ati ng spi nsi ncl ude open spi ns\nandal lotherspi nswhi chcanenteranopenstateatsome\npoi nti n ti me.In whatfol l owsweshow thatthedensi ty\nP\u0003canundergoasharprai se(seeFi g.3)wi thi ncreasi ng\nthedensi ty ofopen spi ns P0abovesomecri ti calval ue\nPc2.Thi sdi sconti nuoustransi ti onhappensafterthefor-\nmati on ofthe i n\fni te percol ati ng cl usterat P0=Pc1\nwhi chpromotesthecooperati vedynami cs. \nIV. K IN ETICS TRAN SITION S IN BETH E\nLATTICE\nConsi dertheprobabi l i ty W \u0003= 1\u0000 P\u0003thatagi venspi n\ni snon-percol ati ng,i . e.i ti sneveri nvol vedi ntodynami cs\ndespi tesomeofi ts znei ghborsmaki ngtransi ti onsonBL, \nseeFi g.2.Thi sprobabi l i tydependson thestatesofal l \ni tsnei ghbors,whi chcanbetreatedasun-correl ateddue\ntothespeci \fcproperti esofBL.Eachnei ghborofthese-\nl ectednon-percol ati ngspi ni scharacteri zedbythecondi -\nti onalprobabi l i ty W e> W \u0003thati ti sal sonotpercol ati ng\n(seeFi g.2).Al lnei ghborsarei n i denti calsi tuati on by\nconstructi on. \nThe probabi l i ty W \u0003thatthe spi n i snotpercol ati ng\ncanbedetermi nedbyconsi deri ngdi \u000berentl ocalenvi ron-mentsdi sti ngui shedbythenumberofpercol ati ngnei gh-\nborsk= 0; 1;: : : z,seeFi g.2.Thereare z! =(k! (z\u0000 k)! )i n-\ndependentwaystobesurroundedby kpercol ati ngspi ns. \nIn each ofthem thesel ected spi n wi l lexperi enceal l2k\npossi bl estatesi nthecourseofevol uti on(assumi ngthat\npercol ati ngspi ns\ri panunl i mi tednumberofti mes;thi s\ni sde\fni tel ytruebeforethei n\fni tecl usteri fformed)and\ni neachstatei thastoremai ncl osed.Thecorrespondi ng\nprobabi l i tyi s( W 0)2k(recal lthat W 0= 1\u0000 P0).Summi ng\nupal ltheprobabi l i ti esfordi \u000berent kweget\nW \u0003=zX \nk=0z! (1\u0000 W e)kW z\u0000k\ne\nk! (z\u0000 k)! W 2k\n0: (3)\nThe probabi l i ty W efor nei ghbors of the gi ven non-\npercol ati ngspi n tobenon-percol ati ngaswel li sde\fned\nsi mi l arl y to Eq. (3)(seeFi g. 2). Oneshoul d consi der\nonl ytheremai ni ng z\u0000 1nei ghborscharacteri zedbythe\nsameprobabi l i ty W etobenon-percol ati ng.Accordi ngl y\nweobtai nthesel f-consi stentequati onfortheprobabi l i ty\nW ei ntheform\nW e=z\u00001X \nk=0(z\u0000 1)! (1\u0000 W e)kW z\u0000k\u00001\ne\nk! (z\u0000 k\u0000 1)! W 2k\n0: (4)\nIn Fi g.3thesol uti on ofEqs.(3),(4)forthedensi ty\nofpercol ati ngspi ns P\u0003= 1\u0000 W \u0003i sshown forBL wi th\nz= 4,5,6and 7nei ghbors.Onecan seethatfor z= 6\nandz= 7 thereexi ststhe di sconti nuoustransi ti on at\nP0=Pc2(l abel Pc1i sreserved fortheformati on ofthe\ni n\fni tepercol ati ngcl uster,seebel ow)wherethedensi ty\nofpercol ati ngspi ns P\u0003jumpsfrom thesmal lval ue P\u0003\u0018 \n1=zal mosttouni ty P\u0003\u0018 1\u0000 (W 0)2z.AtP0> Pc2the\nvastmajori tyofspi nsbel ongtothei n\fni tepercol ati ng\ncl uster. \nz-k k\nz-1-k’ k’ z-1-k’’ k’’ k\nek z\ne ) W(1 W −×−\nk' \nek' 1 z\ne ) W(1 W −×− −W*\nWe\nFIG.2: Con\fgurati ons ofnei ghbori ng spi nsi n the Bethe\nl atti ceand thei rprobabi l i ti es.Red dotted arrowsshow per -\ncol ati ng spi nsand bl ack sol i d arrowsshow non-percol ati ng\nspi ns.Bl uel i nesconnectnei ghbori ngspi ns. \nThephysi csbehi nd thejump ofthepercol ati ng spi n\ndensi ty atthetransi ti on poi nt Pc2can bei l l ustrated i n5\nthel i mi tofl argenumberofnei ghbors z\u001d 1when Eq. \n(4)can besi mpl i \fed bytaki ngadvantageof Pe;P0\u001c 1\nand expandi ng(1 \u0000 P0)2k\u0019 1\u0000 2kP0(recal lthat Pe=\n1\u0000 W e)\nPe\u0019 P0(1+Pe)z\u00001\u0019 P0exp(zPe): (5)\nThi sequati onhasasol uti ononl yfor P0< Pc2\u0019 e\u00001=z, \nwhi ch agrees wi th the numeri calsol uti ons for Pc2at\nz\u0015 7. In ouropi ni on the di sconti nuoustransi ti on i s\ncaused by an aval anche-typegrowth ofthepercol ati ng\nsi tenumberi nthevi ci ni tyofotherpercol ati ngsi tes.For\ni nstance,i fthe densi ty ofpercol ati ng si tes P\u0003reaches\ntheval ue1 =zthenapproxi matel yeachsi tehasoneopen\nnei ghbor.Itsprobabi l i tytobecomepercol ati ngi ncreases\nby thefactorof2 whi ch l eadsto theformati on ofnew\nopensi tes.Thi sprocessl eadstothejumpi nthedensi ty\nofpercol ati ngspi nstonearuni ty. \n0 0.1 0.2 0.3 \nP00.0 0.2 0.4 0.6 0.8 1.0 P*n=7n=6n=5 n=4\nPc1(6)Pc2(6)\nFIG.3:Dependenceofthepercol ati ngsi tesdensi ty P\u0003onthe\ndensi tyofopenspi ns P0forz= 4,5,6and7nei ghbors.There\ni sadi sconti nuouschangei n P\u0003forz>5wherei tsdensi tyi n-\nstantaneousl y jumpsto \u00191.Transi ti on poi nts Pc1\u00190: 0546, \nPc2\u00190: 1085areshown for z= 6. \nThedi sconti nuoustransi ti on wasfound i n the boot-\nstrap percol ati on probl em on Bethel atti ce25,butatef-\nfecti veparameter P0= 1=(z\u0000 1)2.Iti si nteresti ngthati n\nspi teoftheabsenceofki neti ctransi ti oni nthebootstrap\npercol ati on probl em i n thei n\fni te2 Dl atti ce,thetran-\nsi ti on i n the\fni tesystem i sessenti al l ydi sconti nuous26. \nThi sgi vesussomehopethatthedi sconti nuousdynami c\npercol ati ontransi ti onmaytakepl acei nourprobl em ap-\npl i edtoreal i sti cl atti cesof\fni tedi mensi ons( d>1)si m-\ni l arl ytoaBethel atti ce. \nThe cooperati ve dynami csi n the ensembl e ofi nter-\nacti ng spi nsari sesi n the presence ofthe i n\fni te cl us-\nterofpercol ati ngsi tes.Such cl usterobvi ousl yexi stsat\nP0> Pc2,buti tmaybeformedatsmal l erdensi ty P0of\nopensi tes.Indeed,theprobabi l i tytohaveapercol ati ng\nsi tenearanotherpercol ati ngsi tei saboutafactorof2\nl argerthan i n theabsenceofpercol ati ng nei ghborsbe-\ncauseofthedoubl i ngi nthenumberofexpl oredcon\fgu-rati ons.Thecorrespondi ngcri ti calpoi nt P0=Pc1where\nthei n\fni tecl usterofpercol ati ngspi nsi sformedcanal so\nbefound exactl y.Itcan beshown (seeAppendi x)that\ntheformati onofthei n\fni tecl usteri sdetermi nedbythe\nequati on\n(z\u0000 1)(1+F0\u0000 2F1)=\n1+ (z\u0000 1)2[ F2(1\u0000 F0)+F2\n1+F0\u0000 2F1] ; (6)\nwhereFm ,seeEq.(13)i n theAppendi x,i stheproba-\nbi l i ty of\fndi nganon-percol ati ngspi n surrounded by z\nnei ghborswhere m= 0; 1; 2ofthem are de\fnitel yperco-\nl ati ng,2\u0000 mnei ghborsare de\fnitel ynon-percol ati ngand\ntheremai ni nggroupof z\u0000 2nei ghborscancontai nboth\npercol ati ngand non-percol ati ngspi ns.Forsmal l P0we\nhaveFm \u0019 1\u0000 2m P0\u0001 (m(1\u0000 W e)+W e)z\u00002.Thenumer-\ni calanal ysi softhetransi ti onpoi nt Pc1correspondi ngto\ntheformati on ofan i n\fni tepercol ati ngcl ustercon\frms\nthatPc1< Pc2. In thez\u001d 1 l i mi tone\fndsthatthe\ni n\fni tecl usteri sformedat P0> Pc1\u0019 e\u00001=3=(3z)< Pc2\ni nagreementwi ththenumeri calsol uti onofEq.(6). \nV. D ISCU SSION .H OW CAN TH E K IN ETIC\nTRAN SITION BE OBSERVED IN M OLECU LAR\nM AG N ETS?\nBel ow weconsi dertheki neti ctransi ti oni na2D square\nl atti ceofspi nsrepresenti ngmagneti cmol ecul esusi ngthe\nresul tsobtai ned forthe Bethe l atti ce. Thi sconsi dera-\nti onshoul dbeappl i cabl etotherecentl ysynthesi zedtwo-\ndi mensi onalcrystal sofM n 12mol ecul es27.Assumethat\ntheeasyaxi si sperpendi cul artothesampl epl ane.Si nce\nonl yspi nsi nsi detheresonancewi ndow canchangethei r\nstate,thedensi ty ofresonantspi nsi n the l ongi tudi nal \nmagneti c\fel d hcanbeesti matedas\nP0\u0019 2g(\u0016h)Eo; (7)\nwhereg(E)i sthe probabi l i ty densi ty forthe Zeeman\nspl i tti ngE(l ongi tudi nalbi as)ori gi natedfrom thespi n-\nspi n i nteracti on (seeFi g.4)and \u0016i sthemagneti cmo-\nmentofthemol ecul e. \nTostudy theki neti ctransi ti on usi ngtheprevi ousre-\nsul tsfortheBethel atti ceonehastoi ntroducethenum-\nberofnei ghborsparameter z.W eassumethatthenum-\nberofnei ghbori ngspi nscanbeesti matedasthenumber\nofspi nswhoseresonancecan bea\u000bected by thetransi -\nti on ofthegi ven spi n.Forthecrudeesti matewecount\nnei ghbori ngspi nsasthosecoupl ed tothegi ven spi n by\nthei nteracti onexceedi ngthewi dthoftheresonantwi n-\ndow 2Eo. In thel i mi t Eo\u001c UD onecan esti matethi s\nnumberi naquasi -conti nuum approachas\nz\u0019 \u0019(UD =2Eo)2=3: (8)\nAccordi ngtothesol uti on fortheBethel atti ce(seeFi g. \n3)thesharpchangeoftransi ti onrateforal mostal lspi ns\nshoul d takepl acenearthepoi nt P0=Pc2. Assumi ng6\n-5 0 5\nµh/UD00.05 0.1 0.15 0.2 g(µh)UD\nµhc/UD\nFIG.4:Rescal edprobabi l i tydensi tyforthespi nenergyspl i t-\nti ng on a modelsquarel atti ce ofmagneti c mol ecul es. The\npredi cted transi ti on poi nti sshown bysol i d l i nes. \nz\u001d 1 thecrudeesti mateforthe transi ti on poi ntcan\nbemadeusi ngtheapproxi materel ati onshi p Pc2\u0019 e\u00001=z\n(seeSec.IV).Accordi ngl yweget\nEc\u0019 UD \n2(\u0019eg(\u0016h)UD )3: (9)\nThedi stri buti on ofthedi pol arbi asenergi es g(E)i n a\nsampl ecan beeasi l y cal cul ated,seeFi g. 4. Themi ni -\nmum threshol d i sreal i zed atzero\fel d (maxi mum den-\nsi tyofresonances)where g(0)\u0019 0: 19=UD sothatweget\nEc2(0)\u0019 0: 12UD . \nUsi ngtheaboveanal ysi softheki neti ctransi ti on one\ncan approxi matel y characteri ze the spi n rel axati on for\ndi \u000berentmol ecul armagnets.In Fe8onehasUD \u0019 130\nmK andEo\u0019 6 mK23so even i n the absenceofthe\nexternal\fel d the system i si n the l ocal i zati on regi me. \nThesi tuati on i sdi \u000berenti n M n12,whereUD \u0019 70 mK\nandEo\u0019 80 mK24andthecol l ecti vespi ndynami csex-\ni stsatzero magneti c\fel d accordi ngto thetheory,Eq. \n(9). However,appl i cati on ofthe externall ongi tudi nal \nmagneti c\fel d can reducetheresonanceprobabi l i tyand\nresul ti n thel ocal i zati on. Usi ng Eq. (9)onecan esti -\nmatetheval ueoftheexternal\fel dcorrespondi ngtothe\ntransi ti onpoi ntfrom\ng(\u0016hc)UD \u0019 1\n\u0019e\u0012UD \n2Eo\u00131=3\n\u0019 0: 09 (10)\nandthetransi ti oni n M n12takespl aceatthe\fel d hc\u0019 \n2: 8UD =\u0016\u0018 0: 2T(seeFi g. 4).Onecan al soexpectthat\nthe transi ti on rate shoul d decrease nearthe transi ti on\npoi nt. \nAl though theory predi ctsthe absenceofcooperati ve\ndynami csat h > hcthereal i tyi smorecompl i cated be-\ncausespi nswhi chdonotbel ongtothetransi ti onwi ndow\nyethavesmal lbut\fni tetransi ti on rate. Indeed,spi nswi th theZeeman energy E > Eocan maketransi ti ons; \nhoweverthei rtransi ti on ratei sbecomi ng exponenti al l y\nsmal l 20,21,22\n\u001c\u00001/exp(\u0000 j Ej =Eo); (11)\nbecausesuchtransi ti onrequi resi mul taneous\ri psofl arge\nnumberofnucl earspi ns.Then,wepredi cttheexponen-\nti al l y smal ltransi ti on ratewhi ch can beesti mated fol -\nl owi ngthestandardpercol ati ontheoryapproach1asthe\nrateEq. (11)atthe Zeeman spl i tti ng Eequalto the\nthreshol dval ue Ec,Eq.(9)\n\u001c\u00001/exp(\u0000 Ec=Eo)\u0019 exp(\u0000 (g(\u0016hc)=g(\u0016h))3): (12)\nThi sdependencei si l l ustrated i n Fi g.5toshow theex-\npected strong change i n the rate ofspi n rel axati on at\nl arge\fel ds. Unfortunatel y,thedynami cssl owi ngdown\ntakespl aceat\fel ds h\u0019 2hc,wherethedensi ty ofres-\nonantspi nsi sal ready very smal l(seeFi g. 4)and the\npredi cted reducti on oftherel axati on rateby many or-\ndersofmagni tude i smoredi \u000ecul tto study. The dy-\nnami ctransi ti on i tsel fbecomesa crossoverbecausethe\ntransi ti onratei saconti nuousnon-vani shi ngfuncti onof\nenergy,Eq.(11). \n0 1 2 3 4\nµh/UD1x10 -7 1x10 -5 1x10 -3 1x10 -1 τo/τMn12\nFe8\nFIG.5:Dependenceoftherel axati on rate \u001c\u00001on themag-\nneti c\fel dnearthetransi ti onpoi ntcal cul atedusi ngtheex po-\nnenti alapproach Eq.(12). \u001coi sthecharacteri sti crel axati on\nti me. \nInaddi ti ontotheappl i cati onofthel ongi tudi nalmag-\nneti c\fel d onecan al soa\u000bectthetransi ti on by di l uti ng\nFe8magnets.Thereducti on oftheconcentrati onofFe 8\nmol ecul esto x\u001c 1 wi l lamountforchangi ng UD =E0\nrati o,andthusprovi deaknobfordetermi ni ngthetran-\nsi ti on poi ntexperi mental l y.Assumi ngthatthee\u000becti ve\nconstantformagneti cdi pol ari nteracti on scal espropor-\nti onal l y to the x3=2onecan esti matethatat x\u0014 0: 25\nthecol l ecti vedynami csi n Fe8\fl mswi l ltakepl ace. 7\nVI. CON CLU SION \nW e consi dered the dynami c percol ati on probl em on\nthe Bethe l atti ce ofspi ns 1 =2. Dynami cs was i ntro-\nduced through the open spi nscapabl e to changethei r\nstatesand a\u000bectthestatusofnei ghbori ngspi nsswi tch-\ni ng them between open and cl osed states. The prob-\nl em wassol vedexactl y.W efoundtwoki neti ctransi ti ons\ni ncl udi ng theconti nuoustransi ti on associ ated wi th the\nformati onofthei n\fni tepercol ati ngcl usterand thedi s-\nconti nuoustransi ti onassoci atedwi ththeaval anche-type\ngrowthi nthenumberofpercol ati ngsi tes.Thi smodelap-\nproxi matel y descri besthel ow-temperaturedynami csof\nmol ecul armagnetssti mul ated by thei ri nteracti on wi th\nthenucl earspi nbath.Inthi smodelopenspi nsarethose\nhavi ngthesmal lZeemanspl i tti ngcomparedtothei rhy-\nper\fnei nteracti on.Thesharp ki neti ctransi ti on i spre-\ndi ctedforthe2 Dl atti ceofmagneti cmol ecul es,however\ntheabruptchangeoftherel axati on ratei ssmeared out\nby i tsconti nuousdependenceon thetotall ongi tudi nal \nbi asEq.(11),whi ch may obscuretheexperi mentalob-\nservati onofthetransi ti onpoi nt. \nW e acknowl edge val uabl e di scussi ons wi th Phi l i p\nStamp and thesupportby thePaci \fcInsti tuteofThe-\noreti calPhysi cs. Thework ofAB i ssupported by the\nTul aneResearchandEnhancementFund.IT thanksthe\nTul aneResearchand EnhancementFund forsupporti ng\nhi svi si ttotheTul aneUni versi ty. \nVII. APPEN D IX\nConsi derthe probabi l i ty P(N)thattwo spi nssepa-\nratedbyNsi tesareconnectedthroughpercol ati ngsi tes. \nIti scl earthati fthi sprobabi l i tydecreaseswi th Nsl ower\nthanthei nversenumberofpathsofl ength Nstarti ngat\nthegi vensi te,( z\u0000 1)\u0000N ,thenthei n\fni tecl usterofperco-\nl ati ngsi tesi sformed.Thi scri teri oni sal soappl i cabl eto\nthestati cpercol ati onprobl em onBL wheretheconnec-\nti onprobabi l i tyi sgi venby PN \n0.Accordi ngl y,thei n\fni te\ncl usteri sformedat P0\u0015 1=(z\u0000 1)i nful lagreementwi th\ntheexactsol uti on7. \nEach si te ofthe l i ne connecti ng spi ns0 and N+ 1\nhasz\u0000 2nei ghborswhi ch donotbel ongtothel i ne.It\ni sconveni entfortherestofthedi scussi on to i ntroduce\na shortnotati on for the probabi l i ty of\fndi ng a non-\npercol ati ng spi n (or n-spi n)when m= 0; 1; 2 ofthem\narede\fnitel ypercol ati ng,2 \u0000 mnei ghborsare de\fnitel y\nnon-percol ati ngandtheremai ni nggroupof z\u0000 2nei gh-\nborscan contai n both percol ati ng and non-percol ati ng\nspi ns. \nFm =z\u00002X \nk=0(z\u0000 2)! (1\u0000 W e)kW z\u0000k\u00002\ne\nk! (z\u0000 k\u0000 2)! W 2k+ m \n0; (13)\ncf. Eqs.(3)and (4). Indeed, Fm are al lwe need to\nknow tocal cul atetheprobabi l i tythatagi venl i nespi ni s\nnon-percol ati ngwhenthestatesofi tsl i nenei ghborsareknown.Si nce W ei sknown asafuncti on of W 0,wecan\nconcentrateon properti esofthel i nespi nsal one.Note, \nthatthi sdecomposi ti on ofprobabi l i ti esEq.(13)i spos-\nsi bl eduetotheBethel atti cefactori zati on around non-\npercol ati ng spi ns. In otherwordsdi \u000berentreal i zati ons\nforspi nsbel ongi ng to branchesofBL separated by n-\nspi nsarei ndependentofeach other.Si ncethestatesof\ntheendspi ns(0and N+ 1)cannotchangescal i ngprop-\nerti esofthe l ong l i ne connecti ng them wesel ectthem\nto be n-spi nsto si mpl i fy the deri vati on of P(N). W e\nthenconsi der P(N)ascompl ementarytotheprobabi l i ty\nofhavi ngan arbi traryl i nedecomposi ti on i n termsof p-\nandn-spi nswi thatl eastone n-spi namong N >0,i . e. \n1=X \nfng\u001an;p(fng); (14)\nwherei n each sequence fngal l N+ 2spi nsaredi vi ded\ni nto2s+1segmentsofal ternati ng n-andp-spi nsasfng=\n(n1;p1;n2:::ps;ns+1)wi thni;pi\u0015 1(nii sused forn-\nspi nsandpiforp-spi ns).Theprobabi l i tyofaparti cul ar\nsequencei sdenotedas \u001an;p[ n1;p1;:::;ps;ns+1] .Cl earl y, \nn1+p1+\u0001 \u0001 \u0001 +ps+ns+1=N+ 2. \nDuetofactori zati onprovi dedbyn-spi ns(noi nforma-\nti on can be exchanged between the nei ghborsofnon-\npercol ati ng spi ns and thus al lbranchesofBL around\nthem arestati sti cal l yi ndependent)wehave\n\u001an;p=FN \n0; (fors= 0orn1=N+ 2); (15)\n\u001an;p=\u001ae(n1)\u001ae(ns+1)s\u00001Y \nj=2\u001an(nj)sY \nj=1P(pj); (fors>0)\nHere\u001an(m)i stheprobabi l i ty ofhavi ng a cl usterof m\nn-spi nsi narow,and, \u001ae(m)i sasi mi l arquanti tyforthe\n\frstand l astgroupsofn-spi ns.Theseprobabi l i ti escan\nbei mmedi atel ycomputedbycounti nghow manyn-spi ns\nhavemp-spi nsasthei rnei ghbors(recal lthatthe end\nspi ns(0and N+ 1)areassumedtobenon-percol ati ng)\n\u001an(1) =F2; \u001an(m >1)=F2\n1Fm \u00002\n0; \n\u001ae(1) = 1; \u001ae(m >1)=F1Fm \u00002\n0: (16)\nThesel f-consi stentequati on (14)for P(N)i ssol ved us-\ni ng generati ng functi onal P(x)=P 1 \nN =1xN P(N)(and\nsi mi l arl y de\fned \u001ae(x)=P 1 \nN =1xN \u001ae(N)and\u001an(x)=P 1 \nN =1xN \u001an(N))whi chtransforms(14)i nto\nx\n1\u0000 x=F0x\n1\u0000 F0x+\u001a2\ne(x)P(x)\nx21 X \nk=0[ \u001an(x)P(x)] k; (17)\nwi th\u001ae(x)=x+F1x2=(1\u0000 F0x)and\u001an(x)=F2x+\nF2\n1x2=(1\u0000 F0x).Afterel ementaryal gebrawe\fnd\nP(x)=x3(1\u0000 F0)\n(1\u0000 x)(1\u0000 F0x)\u001a2e(x)+x3(1\u0000 F0)\u001an(x); \n(18)\nwhi chcanbefurthersi mpl i \fedto\nP(x)=x(1\u0000 F0)\n(1\u0000 x)(1\u0000 F0x+ 2F1x))+x2[ F2\n1+F2(1\u0000 F0)] : 8\nThei n\fni tecl usteri sformedi f P(N)decreaseswi th N\nsl owerthan1 =(z\u0000 1)N ).Thi smeansthatthethreshol di s\ndetermi nedbythedi vergenceof P(x)atx= (z\u0000 1)whi ch\ni sonl y possi bl ei fthedenomi natori n Eq.(18)i szero. \nThustheformati on ofthei n\fni tecl usteri sdetermi ned\nbytheequati on\n(z\u0000 1)(1+F0\u0000 2F1)=\n1+ (z\u0000 1)2[ F2(1\u0000 F0)+F2\n1+F0\u0000 2F1] ; (19)\n1B.I.Shkl ovski iand A.L.Efros,El ectroni cProperti esof\nDoped Semi conductors,Spri nger,Berl i n,1984. \n2S.Bornhol dt,Sci ence 310,449(2005). \n3R.Al bertand A.L.Barabasi ,Rev.M od.Phys. 74,47\n(2002)\n4PaczuskiM ,M asl ov S,Bak P. ,Phys.Rev.E 53,414-443\n(1996). \n5M .E.Fi sher,J.W .Essam,J.M ath.Phys.(N.Y. ) 2,609\n(1961). \n6S.Ki rkpatri ck,Phys.Rev.Lett. 36,69(1976). \n7K.Bi nder, W .Kob, Gl assy M ateri al s and Di sordered\nSol i ds: An Introducti on to Thei r Stati sti calM echani cs, \nW orl dSci enti \fcPubl i shi ngCompany,2005;K.Bi nder,A. \nP.Young,Rev.M od.Phys. 58,801(1986). \n8J.Cardy,J.Phys.A 25,L201(1992). \n9A.L.Buri n,B.I.Shkl ovski i ,V.I.Kozub,Y.M .Gal peri n, \nand V.Vi nokur,Phys.Rev.B 74,075205,2006. \n10Yu.Kagan,L.A.M aksi mov,Zh.Eksp.Teor.Fi z. 87,348\n(1984). \n11A.L.Buri n,K.N.Kontor,L.A.M aksi mov,Local i za-\nti on and Del ocal i zati on i n theParamagneti cPhaseofthe\nTransverse Isi ng M odel , Theoreti cal and M athemati cal \nPhysi cs, 85,pp.1223-1230,1990. \n12G.H.Fredri ckson,H.C.Andersen,Phys.Rev.Lett. 53, \n1244(1984). \n13V.I.Kozub,S.D.Baranovski i ,I.Shl i mak,Sol i d State\nCommuni cati ons 113,587(2000). \n14S.D.Drugger,M .A.Rather,A.Ni tzan,Phys. 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S.Tupi tsyn,Phys.Rev.B 69,014401\n(2004). \n24W .W ernsdorfer,R.Sessol i ,and D.Gatteschi ,Europhys. \nLett.47 (2),254 (1999);cond-mat/9904450.Experi men-\ntal l y observed val ue Eo\u001981m Ki si n agreementwi th\ntheoreti calcal cul ati onsgi vi ng Eo\u001984m K(I. S.Tupi tsyn\nand P. C. E.Stamp,unpubl i shed). \n25Ai zenman M .and Lebowi tzJ. ,J.Phys.A 21,3801-3813, \n1988. \n26Hol royd,A.Prob.Th.and Rel ated Fi el ds125,195-224, \n2003. \n27Z.Sal man,K.H.Chow,R.I.M i l l er,A.M orel l o,T.J. \nParol i n,M .D.Hossai n,T.A.Keel er,C.D.P.Levy,W . \nA.M acFarl ane,G.D.M orri s,H.Saadaoui ,D.W ang,R. \nSessol i ,G.G.Condorel l i ,R.F.Ki e\r,Nano Lett. 7(6), \n1551-1555,2007. "    },    {        "title": "1208.5681v2.Non_Markovian_Dynamics_of_Spin_Squeezing.pdf",        "content": "arXiv:1208.5681v2  [quant-ph]  16 Apr 2013Non-Markovian Dynamics of Spin Squeezing\nPeng Xue\nDepartment of Physics, Southeast University, Nanjing 2111 89, China\n(Dated: July 18, 2018)\nWe evaluate the spin squeezing dynamics of Nindependent spin-1 /2 particles with exchange\nsymmetry. Each particle couples to an individual and identi cal reservoir. We study the time\nevolution of spin squeezing underthe inﬂuence of diﬀerent d ecoherence. The spin squeezing property\nvanishes with evolution time under Markovian decoherence, while it collapses quickly and revives\nunder non-Markovian decoherence. As spin squeezing can be r egarded as a witness of multipartite\nentanglement, our scheme shows the collapses and revivals o f multipartite entanglement under the\ninﬂuence of non-Markovian decoherence.\nKey words: spin squeezing, non-Markovian decoherence\nPACS numbers: 42.50.St, 03.65.Yz, 03.67.Pp, 06.20.Dk\nI. INTRODUCTION\nQuantum correlation has been playing a central role in quantum infor mation science and has also found many\npromising applications such as achieving interferometric [1–4] and e nhancing the signal-to-noise ratio in spectroscopy\n[5, 6] beyond the standard quantum noise limit. The spin squeezed st ate is one kind of quantum correlated states\n[7, 8] with reduced ﬂuctuations in one of the collective spin componen ts, which can be used to improve the precision\nof atomic interferometers and atomic clocks. As an important quan tum correlation, entanglement is based on the\nsuperposition principle combined with the Hilbert space structure, w hile spin squeezing is originated from another\nfundamental principle of quantum mechanics—the uncertainty prin ciple. It has been proved that the spin squeezing\nis closely related to and implies quantum entanglement [3, 5, 9–11]. As a measure of multipartite entanglement spin\nsqueezing is relative easy to be operated and measured.\nTo evaluate the potential application of quantum correlations such as spin squeezing and entanglement, it is there-\nfore essential to include a realistic description of noise in experiment s of interests [12]. The dynamics of entanglement\nin open systems has been broadly studied [13]. A peculiar aspect of th e entanglement dynamics is the well known “en-\ntanglement sudden death” phenomenon [14–16] and recently the “ sudden death” of spin squeezingduring a Markovian\nprocess has been investigated [17, 18]. The unidirectional ﬂow of inf ormation in which the decoherence and noise act2\nconsistently, characterizes a Markovian process. However, the re are some systems such as condensed-matter systems\nwhich are strongly coupled to the environment and the coupling leads to a diﬀerent regime where information also\nﬂows back into the system from the surroundings, which characte rizes a non-Markovian process. Memory eﬀects\ncaused by the information ﬂowing back to the system during a non-M arkovian process can temporarily interrupt the\nmonotonic increases or decreases of distinguishability such as spin s queezing parameter. In this paper we study the\nspin squeezing dynamics of Nindependent spin-1 /2particles with exchange symmetry which are coupled to individual\nand identical non-Markovian decoherence channels and show the c ollapses and revivals of spin squeezing.\nII. SPIN SQUEEZING DEFINITIONS\nWe consider an ensemble of Ntwo-level particles with lower (upper) state |↓∝an}bracketri}ht(|↑∝an}bracketri}ht). Adopting the nomenclature of\nspin-1/2 particles, we introduce the total angular momentum\n/vectorJ=N/summationdisplay\nj=1/vectorSj, (2.1)\nwhere/vectorSj\nz=1\n2ˆσj\nz=1\n2/parenleftig\n|↑∝an}bracketri}htj∝an}bracketle{t↑|−|↓∝an}bracketri}htj∝an}bracketle{t↓|/parenrightig\n. At this point, it is convenient to introduce the following deﬁnition of s pin\nsqueezing parameter [5, 19]\nξ2=N/parenleftbig\n∆J/vector n⊥/parenrightbig2\nmin\n∝an}bracketle{t/vectorJ∝an}bracketri}ht2. (2.2)\nHere the minimization is over all directions denoted by /vector n⊥, perpendicular to the mean spin direction /vector n=∝an}bracketle{t/vectorJ∝an}bracketri}ht/|∝an}bracketle{t/vectorJ∝an}bracketri}ht|.\nIfξ2<1 is satisﬁed, the spin squeezing occurs and the N-qubit state is entangled.\nThere are also other deﬁnitions for spin squeezing parameters whic h might show diﬀerent sensitivities to the deco-\nherence channels. We introduce another parameter deﬁned by T´ oth et al. [11]\nξ′2=λmin\n∝an}bracketle{t/vectorJ2∝an}bracketri}ht−N/2, (2.3)\nwhereλminis minimum eigenvalue of the matrix Γ = ( N−1)Υ+Cwith Υ kl=Ckl−∝an}bracketle{tJk∝an}bracketri}ht∝an}bracketle{tJl∝an}bracketri}htfork,l∈ {x,y,z}the\ncovariance matrix and C= [Ckl] withCkl=∝an}bracketle{tJlJk+JkJl∝an}bracketri}ht/2 is the global correlation matrix.\nIII. ONE-AXIS TWISTED SPIN SQUEEZED STATES\nNow we introduce one kind of spin squeezed states—one-axis twiste d spin squeezed states. Consider an ensemble of\nNspin-1/2particleswithexchangesymmetryandassumethatthedynamical propertiesofthesystemcanbedescribed3\nby collective operators Jα,α=x,y,z. The one-axis twisting Hamiltonian [20–22] is an Ising-type Hamiltonian\nˆH=/summationdisplay\nj/negationslash=k1\n4f(j,k)/parenleftbig\nI−ˆσj\nz/parenrightbig\n⊗/parenleftbig\nI−ˆσk\nz/parenrightbig\n, (3.1)\nwhich involves all pairwise interactions with coupling constant f(j,k).\nThe one-axis twisted spin squeezed state [23–25] can be prepared by the evolution of the above Hamiltonian\n|ψt∝an}bracketri}ht= exp/parenleftig\n−iˆHt/parenrightig\n|+∝an}bracketri}ht⊗N=/productdisplay\nj/negationslash=kexp/bracketleftbigg\n−i\n4f(j,k)tˆσj\nzˆσk\nz/bracketrightbigg\n|+∝an}bracketri}ht⊗N, (3.2)\nwhere|+∝an}bracketri}ht= (|↑∝an}bracketri}ht+|↓∝an}bracketri}ht)/√\n2. If we choose the evolution time to satisfy f(j,k)t=mπwithman integer, the state |ψt∝an}bracketri}ht\nis a product state. If f(j,k)t= (2m+1)π/2,|ψt∝an}bracketri}htbecomes a graph state. For 0 <f(j,k)t<π/2,|ψt∝an}bracketri}htis a one-axis\ntwisted spin squeezed state characterized by spin squeezing para meterξ(ξ′).\nThe spin squeezing parameter ξof the one-axis twisted spin squeezed state with all coupling coeﬃcie nts satisfying\nf(j,k)t=αtakes this form\nξ2=1−(N−1)/bracketleftbig√\nA2+B2−A/bracketrightbig\n/4\ncos2N−2α, (3.3)\nwhere\nA= 1−cosN−2(2α),B= 4sinαcosN−2α. (3.4)\nThe mean spin direction for the one-axis twisted spin squeezed stat e is\n/vector n= (cos(Nα),sin(−Nα),0), (3.5)\nand the orthogonal direction is\n/vector n⊥= (−cosφsin(−Nα),cosφcos(Nα),sinφ). (3.6)\nThe minimum spin squeezing parameter with respect to αis obtained ξ∝1/N1/3shown in Fig. 1.\nThe spin squeezing parameter with another deﬁnition ξ′of the one-axis twisted spin squeezed state above takes the\nform\nξ′2=min(a,b)\n(1−1/N)(1+cosN−22α)/2+1/N, (3.7)\nwhere\na= 1−(N−1)(/radicalbig\nA2+B2−A)/4, (3.8)\nb= 1+(N−1)/bracketleftbig\n(1+cosN−22α)/2−cos2N−2α/bracketrightbig\n.4\nIV. EVOLUTION OF SPIN SQUEEZING IN THE PRESENCE OF DECOHEREN CE\nFor a single qubit coupled to a noisy channel which is described by a the rmal reservoir, the evolution of this qubit\nis governed by a general master equation of a Lindblad form\nd\ndtχ=i/bracketleftig\nˆHr,χ/bracketrightig\n+Lχ, (4.1)\nwhere the reference system is\nˆHr=∆\n2N/summationdisplay\nj=1ˆσj\nz (4.2)\nwith ∆ is the strength of the external ﬁeld. The optimal spin squeez ed states are eigenstates of the Hamiltonian [26].\nWhereas, the incoherent processes are described by the supero peratorL:\nLχ=−b\n2(1−s)[ˆσ+ˆσ−χ+χˆσ+ˆσ−−2ˆσ−χˆσ+] (4.3)\n−b\n2s[ˆσ−ˆσ+χ+χˆσ−ˆσ+−2ˆσ+χˆσ−]\n−2c−b\n8[2χ−2ˆσzχˆσz],\nwith ˆσ±= (ˆσx±iˆσy)/2. Forb= 0,c=γand an arbitrary s, the generator Eq. (4.3) describes the coupling between\nthe qubit and a dephasing channel. For s= 1/2 andb=c=γ, the qubit is coupled to a depolarizing channel.\nWhereas, for s= 1 andb= 2c=γ, that is coupled to a decay channel (pure damping channel).\nEquivalently, onecanuse the resulting completelypositive map Ewithχ′=Eχto describethe decoherencechannels\nEχ=3/summationdisplay\nj=0pjˆσjχˆσj. (4.4)\nwithχa density matrix for a single-qubit state and/summationtext3\nj=0pj= 1. These decoherence channels are of practical\ninterests in quantum information science. This class contains for ex ample: (i) for p0=/parenleftbig\n1+3κ2/parenrightbig\n/4 andp1=p2=\np3=/parenleftbig\n1−κ2/parenrightbig\n/4 withκ=e−γtEdescribing a depolarizing channel; (ii) for p0=/parenleftbig\n1+κ2/parenrightbig\n/2,p1=p2= 0 and\np3=/parenleftbig\n1−κ2/parenrightbig\n/2 a dephasing channel. Finally, the decay channel is described\nEχ=E0χE†\n0+E1χE†\n2, (4.5)\nwith the Kraus operators E0=\n1 0\n0κ\nandE1=\n0√\n1−κ2\n0 0\n[27].\nThe quantum master equations with the time-local structures are also very useful for the description of non-\nMarkovian processes. Suppose we have a time-local master equat ion of the form\nd\ndtχ=i/bracketleftig\nˆHr,χ/bracketrightig\n+K(t)χ, (4.6)5\nwhereK(t) is a time-dependent generator and takes the similar form in Eq. (4.3 ) with time-dependent parameter γ(t).\nIf the relaxation rate γ(t) is positive, the generator K(t) takes the Lindblad form for each ﬁxed t≥0. In the\nMarkovian regime, information encoded in the qubit state leaks to its surroundings and |κ(t)|is a monotonically\ndecreasing function of times. In the non-Markovian environment, in contrast, information also ﬂows back into the\nsystem of qubit state and a revival of distinguishability (here the sp in squeezing parameter) can be observed in the\ntime evolution. With the deﬁnition of non-Markovianity [28], we see th at an increase of |κ(t)|leads to a negative rate\nγ(t) in the generator K(t). For example, we consider the case of a Lorentzian reservoir spe ctral density which is on\nthe resonance with the spin qubit transition frequency and leads to an exponential two point correlation function\nf(t) =1\n2η0γe−γt. (4.7)\nThe rateκ(t) is deﬁned as the solution of the integrodiﬀerential equation\nd\ndtκ(t) =−/integraldisplayt\n0dt′f(t−t′)κ(t′) (4.8)\ncorresponding to an initial condition κ(0) = 1. The parameter γ, deﬁning the spectral width of the coupling between\nthe qubit and reservoir, is connected to the reservoir correlation timeτ≈γ−1. For a weak coupling regime η0<γ/2,\nthe relaxation time is greater than the reservoir correlation time an d the behavior of κ(t) is a Markovian exponential\ndecay. In the strong coupling regime η0>γ/2, the reservoir correlation time is greater than the relaxation time and\nnon-Markovian eﬀects become relevant [13, 28–36]. Thus we obtain\nκ(t) =e−γt\n2/bracketleftbigg\ncos/parenleftbiggdt\n2/parenrightbigg\n+γ\ndsin/parenleftbiggdt\n2/parenrightbigg/bracketrightbigg\n, (4.9)\nwhered=/radicalbig\n2η0γ−γ2.\nWe would be interested in the eﬀect of decoherence on the spin sque ezing properties of a system including N\ntwo-level particles. A decoherence channel individually coupling to e ach qubit is considered in this paper, and the\nevolution of the kth qubit is described by the map Ekwith Pauli operators ˆ σj(j= 0,1,2,3) acting on qubit k. We\nare interested in the dynamical evolution of a given one-axis twisted spin squeezed state ψofNqubits in the presence\nof decoherence. The initial state ψevolves to a mixed state ρ(t) given by\nρ(t) =E1E2...EN|ψt∝an}bracketri}ht∝an}bracketle{tψt|. (4.10)\nFor a one-axis twisted spin squeezed state, we consider three kind s of decoherence channels. The modiﬁed mean6\nspin direction and the orthogonal direction are calculated as\n/vector n′= (cos(∆t−Nα),sin(∆t−Nα),0), (4.11)\n/vector n′\n⊥= (−cosφsin(∆t−Nα),cosφcos(∆t−Nα),sinφ).\nUnder the inﬂuence of the individual dephasing channels which are th e main type of decoherence for a spin ensemble,\nthe spin squeezing parameter of the one-axis twisted spin squeeze d state evolves to\nξ2\ndeph(t) =ζ\ncos2N−2α, (4.12)\nwhere\nζ= 1+1\n4κ2(t)(N−1)/parenleftbigg\nA−A2\n√\nA2+B2/parenrightbigg\n−1\n4κ(t)(N−1)B2\n√\nA2+B2(4.13)\nThen the spin squeezing parameter of the one-axis twisted spin squ eezed state evolves to\nξ2\ndepol(t) =ζ\nκ2(t)cos2N−2α. (4.14)\nunder depolarizing. Whereas, the spin squeezing parameter of the one-axis twisted spin squeezed state evolves to\nξ2\ndamp(t) =ζ\n/braceleftig\nκ(t)cosN−1α+[1−κ(t)]/bracerightig2. (4.15)\nunder damping.\nWith the other deﬁnition shown in Eqs. (2.3) and (3.7), the spin squee zing parameter ξ′of the one-axis twisted\nspin squeezed states coupled to individual dephasing, depolarizing a nd damping channels evolves as followings\nξ′2\ndeph(t) =ζ\n(1−1/N){κ(t)2+[1−κ(t)2](1+cosN−22α)/2}+1/N, (4.16)\nξ′2\ndepol(t) =ζ\n(1−1/N)κ(t)2+1/N, (4.17)\nξ′2\ndamp(t) =ζ\n1+(1−1/N)κ(t)(1−κ(t))[1−cosN−1α+(1+cosN−22α)/2]. (4.18)\nThe time evolution of spin squeezing parameters ξandξ′of 10 particles prepared initially in one-axis twisted\nspin squeezed state coupled to individual dephasing, depolarizing an d damping channels are shown in Figs. 2-7,\nrespectively. We compare the evolution of spin squeezing under Mar kovian and non-Markovian decoherence. Here\nwe consider a Lorentzian reservoir and thus the deocherence fun ctionκ(t) can be written as exponential decay e−γt/2\nmodiﬁed by a periodical time-dependent function cos( dt/2)+γ/dsin(dt/2). In short time regime, the spin squeezing\nproperty collapses and revives under the inﬂuence of either of thr ee kinds of non-Markovian decoherence. With time7\nincreasing, the part of the exponential decay becomes more impor tant and the spin squeezing property vanishes ﬁnally\nas that under Markovian decoherence does. The spin squeezing ev olving under Markovian decoherence gives a lower\nbound of the envelope of that evolving under non-Markovian decoh erence shown in Figs 2-7. The two parameters\nshow diﬀerent sensitivities to diﬀerent decoherence channels. Bot h parameters are robust to damping channel in Figs.\n4 and 7 and the squeezing property represented by the paramete rsξandξ′lasts fort= 1000. Coupled to the other\ntwo decoherence channels (dephasing and depolarizing channels), the parameter ξis more sensitive than ξ′. In Figs. 2\nand 5, the disappearance times for spin squeezing representing by the parameters xiandξ′under dephasing are about\nt= 259.99 andt= 318.13, respectively. In Figs. 3 and 6, the disappearance times for spin squeezing representing\nbyξandξ′under depolarizing are about t= 68.76 andt= 106.01, respectively. Both parameters ξandξ′are more\nsensitive to the depolarizing.\nV. CONCLUSION\nIn summary, we study the dynamics processes of the spin squeezin g of a spin ensemble in which each spin is coupled\nto an independent and identical decoherence channel. We analytica lly calculate the dynamics of the spin squeezing\nparameters under three diﬀerent types of decherence. As we kn ow the Heisenberg scaling 1 /Nin the decoherence-free\ncase can be achieved. In the presence of Markovian decoherence the spin squeezing property of one-axis twisted\nstates vanishes with the evolution time. Whereas, in the presence o f non-Markovian decohernce and in the short\ntime limit, the spin squeezing property collapses and revives with the e volution time due to short-time memory eﬀect\nduringnon-Markovianprocessing. With timeincreasing, thespin squ eezingvanishesﬁnallyevenundernon-Markovian\ndecoherence. As spin squeezing can be regarded as a measure/wit ness of multiqubit entanglement, thus our scheme\nfor the ﬁrst time shows the collapses and revivals of multiqubit entan glement under non-Markovian decoherence.\nAcknowledgments\nWe would like to thank Yongsheng Zhang and Xiangfa Zhou for useful conversations. This work has been supported\nby the National Natural Science Foundation of China under Grant N os 11004029 and 11174052, the Natural Science\nFoundation of Jiangsu Province under Grant No BK2010422, the Ph .D. Program of the Ministry of Education of\nChina, the Excellent Young Teachers Program of Southeast Univer sity and the National Basic Research Development8\nProgram of China (973 Program) under Grant No 2011CB921203.\n[1] C.M. Caves, Phys. Rev. D 23 (1981) 1693-1708.\n[2] B. Yurke, Phys. Rev. Lett. 56 (1986) 1515-1517.\n[3] M. Kitagawa, M. Ueda, Phys. Rev. Lett. 67 (1991) 1852-185 4.\n[4] D. Ulam-Orgikh, M. Kitagawa, Phys. Rev. A 64 (2001) 05210 6.\n[5] D.J. Wineland, J.J. Bollinger, W.M. Itano, F.L. Moore, D .J. Heinzen, Phys. Rev. A 46 (1992) R6797-R6800.\n[6] A. Andr´ e, M.D. Lukin, Phys. Rev. A 65 (2002) 053819.\n[7] J. Ma, X.G. Wang, C.P. Sun, F. Nori, Phys. Rep. 509 (2011) 8 9-165.\n[8] P. Xue, Phys. Rev. A 86 (2012) 023812.\n[9] A.Søensen, L.M. Duan, J.I. Cirac, P. Zoller, Nature 409 ( 2001) 63-66.\n[10] X.G. Wang, B.C. Sanders, Phys. Rev. A 68 (2003) 012101.\n[11] G. T´ oth, C. Knapp, O. G¨ uhne, H.J. Briegel, Phys. Rev. L ett. 99 (2007) 250405.\n[12] H.P. Breuer, F. 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Zou, G.C. Guo, H. Breuer , E. Laine, J. Piilo, Europhysics Letters 97 (2012) 10002.10\n02004006008001000\nN0.20.40.60.81Ξopt\nFIG. 1: The plot of the spin squeezing parameter ξfor a one-axis twisted spin squeezed state v.s. the number of the qubits N\noptimized with respect to α.\n02004006008001000\nt0.60.811.21.41.61.8ΞoptwithrespecttoΑ\nFIG. 2: The time evolution of spin squeezing of a one-axis twi sted spin squeezed state coupled to non-Markovian dephasin g\nchannel representing by ξv.s. the time twithN= 10,γ= 0.01 andη0= 10 (in black solid line). For comparison, we plot the\nevolution of the spin squeezing under Markovian dephasing w ithκ(t) =e−0.005t(in red dashed line).11\nFIG. 3: The time evolution of spin squeezing of a one-axis twi sted spin squeezed state coupled to non-Markovian depolari zing\nchannel representing by ξv.s. the time twithN= 10,γ= 0.01 andη0= 10 (in black solid line). For comparison, we plot the\nevolution of the spin squeezing under Markovian depolarizi ng with κ(t) =e−0.005t(in red dashed line).\n02004006008001000\nt0.60.70.80.911.1ΞoptwithrespecttoΑ\nFIG. 4: The time evolution of spin squeezing of a one-axis twi sted spin squeezed state coupled to non-Markovian damping\nchannel representing by ξv.s. the time twithN= 10,γ= 0.01 andη0= 10 (in black solid line). For comparison, we plot the\nevolution of the spin squeezing under Markovian damping wit hκ(t) =e−0.005t(in red dashed line).12\n02004006008001000\nt0.60.811.21.41.6Ξ'optwithrespecttoΑ\nFIG. 5: The time evolution of spin squeezing of a one-axis twi sted spin squeezed state coupled to non-Markovian dephasin g\nchannel representing by ξ′v.s. the time twithN= 10,γ= 0.01 andη0= 10 (in black solid line). For comparison, we plot the\nevolution of the spin squeezing under Markovian dephasing w ithκ(t) =e−0.005t(in red dashed line).\n02004006008001000\nt0.511.522.53Ξ'optwithrespecttoΑ\nFIG. 6: The time evolution of spin squeezing of a one-axis twi sted spin squeezed state coupled to non-Markovian depolari zing\nchannel representing by ξ′v.s. the time twithN= 10,γ= 0.01 andη0= 10 (in black solid line). For comparison, we plot the\nevolution of the spin squeezing under Markovian depolarizi ng with κ(t) =e−0.005t(in red dashed line).13\nFIG. 7: The time evolution of spin squeezing of a one-axis twi sted spin squeezed state coupled to non-Markovian damping\nchannel representing by ξ′v.s. the time twithN= 10,γ= 0.01 andη0= 10 (in black solid line). For comparison, we plot the\nevolution of the spin squeezing under Markovian damping wit hκ(t) =e−0.005t(in red dashed line)."    },    {        "title": "2303.05833v2.Chiral_spin_channels_in_curved_graphene__pn__junctions.pdf",        "content": "Chiral spin channels in curved graphene pnjunctions\nDario Bercioux,1, 2,∗Diego Frustaglia,3,†and Alessandro De Martino4,‡\n1Donostia International Physics Center (DIPC),\nManuel de Lardizbal 4, E-20018 San Sebasti´ an, Spain\n2IKERBASQUE, Basque Foundation for Science, Euskadi Plaza, 5, 48009 Bilbao, Spain\n3Departamento de F´ ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain\n4Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom\n(Dated: September 28, 2023)\nWe show that the chiral modes in circular graphene pnjunctions provide an advantage for spin\nmanipulation via spin-orbit coupling compared to semiconductor platforms. We derive the effective\nHamiltonian for the spin dynamics of the junction’s zero modes and calculate their quantum phases.\nWe find a sweet spot in parameter space where the spin is fully in-plane and radially polarized for\na given junction polarity. This represents a shortcut to singular spin configurations that would\notherwise require spin-orbit coupling strengths beyond experimental reach.\nI. INTRODUCTION\nGraphene has attracted exceptional interest as a quan-\ntum material with Dirac cones at the Fermi energy and\nother unique electronic properties [1–3]. One appealing\nfeature is the possibility of tuning electrostatically the\ncharge carriers’ polarity in pnjunctions of linear [4–9]\nand circular shape [10–18]. The latter have been created\nby different means, such as the tip potential of a scan-\nning tunneling microscope [10, 14, 16, 18] or by placing\nimpurities in the substrate [11, 13]. In both approaches,\nexperiments have shown that it is possible to single out\nand steer individual electronic eigenstates. Importantly,\npnjunctions are essential building blocks for graphene-\nbased electron-optical elements and edge-state interfer-\nometers [14, 19–21] also exploiting the so-called snake\nstates [9, 22, 23].\nThe electronic spin degree of freedom is usually ne-\nglected in the study of graphene pnjunctions because of\nthe weak atomic spin-orbit coupling (SOC) of carbon [24–\n27]. However, theoretical predictions followed by exper-\nimental realizations proved that strong SOCs can be in-\nduced, e.g., by proximity with transition metal dichalco-\ngenide (TMD) substrates [28–37]. These advances open\nthe exciting possibility of including the spin functional-\nity in graphene-based electron optics, with the further\nbenefit that the versatility of pnjunctions allows for the\ndesign of curved waveguides for spin and charge carri-\ners. This is particularly interesting in view of the intense\ncurrent theoretical and experimental research activity on\nthe spin dynamics triggered by SOC in curved geome-\ntries [38–41]. The effects of SOC in graphene have been\nalso investigated in other geometries [42–45].\nIn this article, we investigate circular pnjunctions in\nthe presence of (i) a perpendicular magnetic field, cou-\npled to the electronic charge (developing Landau levels\n∗dario.bercioux@dipc.org\n†frustaglia@us.es\n‡ademarti@city.ac.ukin the quantum Hall regime) and spin (through Zeeman\ncoupling), and (ii) proximity-induced SOCs of different\ntypes. We provide the exact solution of graphene’s Dirac\nequation for this system and formulate an effective one-\ndimensional (1D) model for the spin and angular dynam-\nics of the states localized at the pninterface. This resem-\nbles the model for semiconductor rings subject to Rashba\nSOC (RSOC) [46], with a meaningful difference: the chi-\nral nature of the propagating modes. We identify a re-\nmarkable sweet spot in the parameter space, where the\nspin eigenstates align locally with the effective magnetic\nfield produced by the SOC. This point coincides with the\nRabi condition for electronic spin resonance in a magnetic\nfield and represents a shortcut to adiabatic spin dynamics\nunavailable in its semiconductor equivalent. We confirm\nthis result within the original full model and propose a\nset-up to identify this sweet spot via spin interferometry,\nopening a promising route to spin state manipulation in\ngraphene.\nThe article is organized in the following way: In Sec. II,\nwe introduce the model system. In Sec. III, we present\na low-energy model for the system under investigation,\nwhere we show the presence of the sweet spot in the pa-\nrameter space. In Sec. IV, we provide a proposal for an\ninterferometric experiment to detect the presence of this\nsweet spot. We discuss in Sec. V the interpretation of\nthe experimental proposal and its range of validity. Fi-\nnally, in Sec. VI, we provide our conclusions. All the\ntechnical details are presented in the Supplemental Ma-\nterial (SM) [47].\nII. MODEL\nThe low-energy model for graphene with proximity-\ninduced SOCs reads\nH=H0+Hspin, (1)\nwhere H0is the Dirac Hamiltonian in a perpendicular\nmagnetic field\nH0=vF(τσxΠx+σyΠy) +V, (2)arXiv:2303.05833v2  [cond-mat.mes-hall]  27 Sep 20232\n(c)\n(a)\nB\nx\ny\nz\n(b)Back gateTMD\n-10\n -5\n 0\n-2\n-1\n0\n1\n2\nj\n-10\n -5\n 0\nj\nFigure 1. (a) Sketch of the system, with pandnregions\ndrawn in yellow and blue. (b) Energy spectrum versus angular\nmomentum jforV0= 0.51,ξ0= 5.1,λR= 0.5 and λZ=\nλKM= 0. (c) Same as in (b) but for λZ= 0.1. In (b) and (c),\nthe red dots highlight the zeroth Landau levels.\nwith Fermi velocity vFand kinetic momentum Π=\n−iℏ∇+e\ncA, with A=B\n2(−y, x) in the symmetric gauge.\nHere, τ=±1 denotes the valley index and σ= (σx, σy)\nare Pauli matrices in sublattice space [3]. The potential\nV(r) =V0sign(R−r), (3)\ndefines a circular pnjunction of radius R, with a p-doped\nregion for r < R (the “dot”), and a n-doped region for\nr > R . The system is sketched in Fig. 1(a). The spin-\ndependent part Hspin=HZ+HR+HKM+HVZincludes\nboth Zeeman and SOCs terms [28, 29, 48]:\nHZ=λZsz, (4a)\nHR=λR\n2(τσxsy−σysx), (4b)\nHKM=λKMτσzsz, (4c)\nHVZ=λVZτsz. (4d)\nHere λZ=gsµB\n2B, and s= (sx, sy) denotes the Pauli\nmatrices in spin space. The terms HR,HKM, and HVZ\nare the Rashba, Kane-Mele, and valley-Zeeman SOC, re-\nspectively [24, 26, 49]. Precise estimates for the SOCs\ndepend on the specific heterostructure, e.g., the relative\norientation between graphene and substrate [35, 36]. The\nRSOC and the VZSOC range from few hundredths of\nmeV up to few meV, while the KMSOC is typically much\nsmaller [35, 37]. We are mainly concerned with the effects\nof the Zeeman and RSOC terms. The valley-Zeeman\nterm can be included by means of a valley-dependent\nshift of the Zeeman coupling and will be considered sep-\narately in the discussion section below. For λVZ= 0,the valley degree of freedom just leads to a degeneracy\nfactor, so we can focus on a single valley and set τ= +1.\nThroughout this paper, we measure lengths in units of\nmagnetic length ℓB=p\nℏc/eB = 25.65 nm /p\nB[T] and\nenergies in units of cyclotron energy ℏωc=ℏvF/ℓB≈\n26 meVp\nB[T], and assume a typical field B∼1 T [37].\nIn this model, the wave function is a four-component\nspinor ΨT= (Ψ A↑,ΨB↑,ΨA↓,ΨB↓). The Hamiltonian\nHcommutes with the total angular momentum J=\nLz+1\n2(σz+sz), with Lz=−i∂θthe orbital angular mo-\nmentum, hence its eigenstates Ψ j(r), expressed in terms\nof confluent hypergeometric functions [47, 50–52], can be\nlabelled by an integer j∈Z. The spectrum is illustrated\nin Figs. 1(b) and 1(c). In particular, we find two “zero-\nenergy” Landau levels (LLs), the “top” (T) and “bot-\ntom” (B) zero modes, highlighted in red in the figures.\nIn the absence of SOCs, they have zero energy for V0= 0,\nbut develop a dispersion in jfor finite V0[22, 47]. Their\nenergy at j= 0 and at j≪ − 1 approaches the value\nof the potential V(r) inside and outside the dot, respec-\ntively, see Fig. 1(b). In the presence of RSOC, the two\nmodes acquire a spin splitting, similar to the case of a\ntwo-dimensional electron gas (2DEG) [27, 53]. A finite\nZeeman coupling produces an additional vertical split-\nting—see Fig. 1(c). We present in the SM [47] the exact\nsolution of the model (1), including a detailed analysis of\nthe spin splitting as a function of λR.\nIII. EFFECTIVE 1D MODEL\nIn order to describe the low-energy physics around\nthe Fermi energy (set at the charge neutrality point,\nEF= 0), we introduce an effective 1D Hamiltonian for\nthe zero modes localized at the pninterface. We follow\nan analogous derivation for a semiconductor ring with\nRSOC [54], see the SM [47] for details. We first per-\nform a unitary transformation, H → ˜H=UHU−1, with\nU=eiσz\n2(θ+π\n2)eisz\n2θ. In this rotating frame, we factorize\nthe wave function as ˜Ψ = ˜ψ0(r)˜χ(θ), where ˜ψ0(r) is the\nsublattice spinor for the (spin degenerate) zero mode of\nthe radial part of ˜H0, and ˜ χ(θ) is a spinor in spin space,\ncontaining the angular dependence. The projection of\n˜Honto the zero mode ˜ψ0(r) leads to the effective 1D\nHamiltonian controlling the dynamics of ˜ χ(θ):\n˜Heff=ω0(Lz+ Φ) + ( ωZ−ω0\n2)sz−ωRsx.(5)\nThe frequencies in Eq. (5) are defined by\nω0=Dσx\nrE\n0, (6a)\nωZ=λZ+λKM⟨σz⟩0, (6b)\nωR=λR\n2⟨σx⟩0, (6c)\nwhere ⟨. . .⟩0denotes the (radial) expectation value in the\nstate ˜ψ0(r). (We note that σxis the azimuthal compo-\nnent of the velocity operator in the rotating frame.) The3\n10\n 20\n 30\n 40\n 50\n0.05\n0.06\n0.07\n0.08\n0.09\n0.10\n0.11\n0.12\nξ 0ω 0\n1 0\n 1 2\n 1 4\n 1 6\n 1 8\n-1 .0\n-0 .5\n0 .0\n0 .5\n1 .0\n1 0\n 1 2\n 1 4\n 1 6\n 1 8\n-1 .0\n-0 .5\n0 .0\n0 .5\n1 .0\n28 26\n 30\n 32\n 34\n-1 .0\n-0 .5\n14\n0 .5⟨s  ⟩⟨s  ⟩ ⟨s  ⟩\n1 .0\nξ 0\nξ 0 ξ 0(a) (c)\n(d)\n(b)\nFigure 2. Sweet spot identification in the full model. (a)\nThe angular frequency ω0as a function of ξ0=R2/2ℓ2\nB, at\na fixed magnetic field. The green and red horizontal lines\ndescribe two representative values of 2 ωZ, and the vertical\ndashed lines the corresponding values of ξ0at which the res-\nonance condition 2 ωZ=ω0is realized. (b)-(d) The exact\nexpectation values of the radial and perpendicular spin com-\nponents in the top and bottom modes as a function of ξ0,\nforλZ= 0.047 and λR= 0.2 in (b) and for λZ= 0.033 and\nλR= 0.2,0.3 in (c) and (d) respectively. In (b)-(d), the top\ncurve shows ⟨sr⟩B, the bottom one ⟨sr⟩T, and the two central\nones⟨sz⟩Tand⟨sz⟩B. In all the panels, V0= 0.51.\nparameter Φ ≈ξ0=BπR2/Φ0denotes approximately\nthe magnetic flux through the dot in units of the flux\nquantum Φ 0. Since ˜Hspinis treated perturbatively, this\nprojection is justified as long as ℏωcis much larger than\nthe Zeeman and SOCs. The Hamiltonian (5) describes\na 1D spinful chiral mode propagating along the curved\npninterface, with angular velocity controlled by the gate\nvoltage difference across the junction. Importantly, the\npolarity of the junction determines the signs of ω0and\nωR[55]. For V0>0 both are positive. Inverting the\npolarity, V0→ −V0, reverses the propagation direction,\nchanging both signs. This feature has crucial implica-\ntions for the experimental setup discussed below.\nDiagonalizing ˜Heff, we obtain the eigenvalues\nEm,±=ω0(m+ Φ)±r\n(ωZ−ω0\n2)2+ω2\nR, (7)\nwhere m∈Zunder periodic boundary conditions. This\nformula predicts a linear dependence of the energy on\nm, which we observe in the exact solution close to zero\nenergy, and provides an approximate analytical expres-\nsion for the slope of the dispersion. The correspondingeigenstates are\n˜χm,+=eimθ\n√\n2π\u0012\ncosγ\n2\n−sinγ\n2\u0013\n, (8a)\nχm,−=eimθ\n√\n2π\u0012\nsinγ\n2\ncosγ\n2\u0013\n, (8b)\nwhere\neiγ=ωZ−ω0\n2+iωRq\n(ωZ−ω0\n2)2+ω2\nR. (9)\nWe find a sweet spot for ωZ=ω0\n2(γ=π\n2), where\nthe spin eigenstates (8) point along the radial direction\nin the xyplane for any value of ωR. This situation is\nremarkable. It recalls the Rabi condition for spin reso-\nnance in the rotating wave approximation (RWA), with\nthe difference that there is no Bloch-Siegert shift [56]\nas a function of the driving amplitude (represented by\nωR): here, the RWA is exact. Notice that an inver-\nsion of the junction polarity, changing the chirality of\nthe propagating spin channels ( ω0→ −ω0), would take\nthe system off-resonance. This is in sharp contrast to the\ncase of semiconductor-based Rashba rings [46, 57], where\ncounter-propagating channels coexist, and a full in-plane\nalignment of the spinors is only achieved in the adiabatic\nlimit of very large RSOC ( ωR≫ω0) [46].\nThe resonance condition, exact in the projected\nmodel (5), holds with excellent accuracy also in the full\nmodel (1). This is shown in Fig. 2, where for simplic-\nity we set λKM= 0. Here, we define the angular fre-\nquency ω0as the expectation value ⟨σx/r⟩λR=0on the\nj-state closest to zero energy. From Fig. 2(a), we can\nsee that ω0decreases as a function of the radius Rand\npresents a staircase behavior due to the discreteness of j.\nIn Figs. 2(b)-(d), we show the expectation values of the\nperpendicular and radial components of the spin, szand\nsr, in the top and bottom j-states closest to zero energy\nfor different sets of parameters. We observe that at the\nvalue of ξ0where the resonance condition ωZ=ω0\n2is re-\nalized, ⟨sz⟩is almost zero, whereas ⟨sr⟩is close to 1. The\nresults in Fig. 2 show an excellent agreement between the\nprediction of the projected model and the full solution.\nIn particular, they confirm that the resonance condition\nis independent of the RSOC. The small discrepancies are\ndue to the coupling of the zero modes to the higher LLs\nvia the RSOC, neglected in the projected model. We\npresent additional results, including the effect of λKM, in\nthe SM [47].\nIV. EXPERIMENTAL PROPOSAL\nWe propose two setups based on linear and circular\npnjunctions to implement interferometric circuits for\nspin carriers. Thanks to the chiral nature of the propa-\ngating channels, we find that, depending on the junction\npolarity, the interferometers respond differently to the4\nZeeman coupling ωZ(assuming λKM= 0 for simplicity),\nmaking possible a unique geometric characterization of\nthe propagating spin states.\nFigure 3 depicts the circuits’ architecture built upon n\n[Fig. 3(a)] and p[Fig. 3(b)] dots. Contact 1 at voltage\nVis the carrier source, while the grounded contacts 2\nand 3 act as drains. The grounded contact 4 contributes\nwith an empty channel. Importantly, either setup can be\nturned into the other by simply inverting the pnpolarity,\nrelabeling the contacts, and swapping voltages, meaning\nthat a single sample could realize both interferometer in\nthe laboratory.\nCarriers injected from contact 1 propagate along a lin-\nearpnjunction. Traveling toward contact 2, they can en-\nter the circular pnjunction with probability 0 < τ1<1,\nfrom which they can escape at the opposite end towards\ncontact 3 with probability 0 < τ2<1. The tunnel bar-\nriers τ1andτ2operate as beam splitters (BSs) for the\nchiral modes. Their spin-dependent probability ampli-\ntudes are determined by projecting the propagating spin\nmodes on the local basis [47].\nWe calculate the quantum conductance G21from con-\ntact 1 to contact 2 for the zero modes following the\nLandauer-B¨ uttiker approach [58, 59]. (By unitarity,\nG21+G31= 2e2/h, since we are considering a single\nvalley.) Obtaining the quantum transmission requires\nthe combination of the BS scattering matrices [59], tak-\ning into account the spin-dependent phases mπgathered\nby the carriers propagating between the tunnel barriers\nalong the circular junction [47]. These phases are ob-\ntained by setting Em,s= 0 in Eq. (7), where mis not\nnecessarily an integer for open pnjunctions, since peri-\nodic boundary conditions do not apply in the presence\nof contact leads. Figures 3(c)-(e) summarize our main\nresults. We plot the conductance G21for the two op-\nposite junction polarities, as a function of dimension-\nless Rashba QR=ωR/ω0and Zeeman QZ=ωZ/ω0\ncoupling strengths. Without loss of generality, we set\nτ1=τ2= 1/2 (50% BSs) and Φ ∈N. Other settings can\nmodify the relative amplitudes and phases of the pat-\nterns, but their general composition remains the same.\nWe observe that the patterns in Figs. 3(c) and 3(d) differ\nby a relative ∆ QZ= 1 shift along the Zeeman axis. This\nshift reveals significant information on the spin-state ge-\nometry of propagating channels, as explained below.\nIn Fig. 3(e) we plot G21forQZ= 0 (solid line) and\nQZ= 1/2 (dashed line). For QZ= 0, the result holds\nfor both nandppolarities. Here we find quasi-periodic\noscillations as a function of QR, which tend to be peri-\nodic for QR≫1. This limit corresponds to the regime\nof adiabatic spin dynamics, where the local spin quanti-\nzation axis is expected to point along the radial Rashba\nfield with γ→π/2 in Eq. (8). Moreover, after a round\ntrip around the dot, the spin carriers collect a geometric\nphase φg=−Ω/2, with Ω = 2 π(1−cosγ) the solid angle\nsubtended by the spin states on the Bloch sphere. In the\nadiabatic limit, one finds φg→ −π. Similar results have\nbeen reported for semiconductor Rashba rings [46, 57].\n(a)\n (b)\n0\n0.5\n1\n1.5\nQZ\nτ1\nτ2\nempty channel\nnn\nnp p\nV\nτ1τ2\nempty channel\np\np\npn n\n22 4\n4\nV11 3\n3\nQR\nQR(c) (d)\n(e)QRFigure 3. (a) The circuit’s architecture with a n-doped dot.\n(b)The same as in (a), but with opposite junctions’ polarity.\n(c)-(d) Differential conductance G21for the circuits in (a) and\n(b), respectively, as a function of the dimensionless Rashba\nand Zeeman coupling strengths. (e) Cut of the differential\nconductance for the cases in (c) and (d) with QZ= 0 (blue\nline) and for the case in (d) with QZ= 1/2 (red dashed line).\nThe two polarities respond very differently to QZ. For\nthendot [see Eq. (9) and Fig. 3(c)], we find that QZacts\nto the detriment of in-plane spinor polarization, which\nstill requires large RSOC intensities QR. On the con-\ntrary, for the pdot [see Eq. (9) and Figs. 3(d) and 3(e)],\nat the sweet spot QZ= 1/2 we find perfectly periodic\noscillations corresponding to fully in-plane spin states\n(γ=π/2) regardless of the RSOC intensity, picking up\na geometric phase φg=−π.\nV. DISCUSSION\nAll relevant features of Fig. 3(d) are captured by a\nlow-order semiclassical expansion of the conductance in\nterms of Feynman paths corresponding to single windings\naround the pdot [47]. In this approximation, we find\nG21≈1 + cos ϕABcosϕS, (10)5\nwith\nϕAB= 2πΦ, (11a)\nϕS= 2πs\u0012\nQZ−1\n2\u00132\n+Q2\nR, (11b)\nwhere ϕABandϕSare independent phase contributions\noriginating in the orbital and spin degrees of freedom, re-\nspectively. Equation (10) reproduces well the pattern of\nFig. 3(d) showing circular wavefronts centered at QR= 0,\nQZ= 1/2. For QZ= 0, we find from Eq. (10) that\nϕS= 2πQRsinγ−πcosγ= 2πQRsinγ−(π+φg).\nThis phase reduces to ϕS≈2πQRin the adiabatic limit\nQR≫1, leading to periodic oscillations of G21as a func-\ntion of QR. Thus, a strong RSOC drives the spin eigen-\nstates to be in-plane, such that γ→π/2 and φg→ −π.\nThe physical realization of this formal limit is difficult in\nthe laboratory due to the required field intensities. Alter-\nnatively, we find here a shortcut by setting QZ= 1/2. In\nthis sweet spot, the spin phase contribution reduces ex-\nactly to ϕS= 2πQReven for weak RSOC fields, which as-\nsures in-plane spin eigenstates that introduce a πphase-\nshift of purely geometric origin.\nWe emphasize that this precise characterization of\nthe propagating spin channels boils down to their chi-\nral nature, in contrast to the case of semiconductor\nRashba rings, where counter-propagating modes coex-\nist [46, 57, 60]. The chirality also protects the sweet spot\nfrom the effect of random impurities. Moreover, we ex-\npect that small deviations from a perfectly circular shape,\nbreaking the rotational symmetry might induce small os-\ncillations of the out-of-plane component of the spin and\nthus blur the sweet spot, but will not qualitatively alter\nthe physics discussed here [61].\nFinally, we briefly address the effect of the VZSOC. In\nthe effective model (5), it leads to a valley-dependent\nshift ωZ→ωZ+τλVZ. Hence, at ωZ=ω0/2, the\nspin states (8) will have a residual out-of-plane compo-\nnent, opposite at the two valleys. The valley-resolved\nconductances will be periodic functions of QRonly forλR≫λVZ[47], see Eqs. (10) and (11b). The selection of\nsubstrates inducing the weakest possible VZSOC [35, 36]\nis thus essential to observing the effects described in this\nwork.\nVI. CONCLUSIONS\nWe have shown that the chiral spin channels in curved\ngraphene pnjunctions with proximitized SOCs can be\nprecisely characterized and controlled. We uncovered a\nsweet spot in the parameter space enabling an efficient\nmanipulation of spin-state configurations without requir-\ning a strong RSOC, which is difficult to achieve exper-\nimentally. This opens up new possibilities for explor-\ning quantum-state geometry and advancing spintronics\nin graphene. Curved pnjunctions thus offer a versatile\nplatform for investigating spin dynamics phenomena in-\nduced by SOCs, providing an alternative to traditional\nsemiconductor systems.\nACKNOWLEDGMENTS\nWe thank R. Egger and K. Richter for helpful com-\nments on the manuscript. D.B. acknowledges the sup-\nport from the Spanish MICINN-AEI through Project\nNo. PID2020-120614GB-I00 (ENACT), the Transna-\ntional Common Laboratory Quantum −ChemPhys , the\nfinancial support received from the IKUR Strategy under\nthe collaboration agreement between Ikerbasque Founda-\ntion and DIPC on behalf of the Department of Educa-\ntion of the Basque Government and the Gipuzkoa Provin-\ncial Council within the QUAN-000021-01 project. D.F.\nacknowledges support from the Spanish MICINN-AEI\nthrough Project No. PID2021-127250NB-I00 (e-QSG)\nand from the Andalusian Government through PAIDI\n2020 Project No. P20-00548 and FEDER Project No. US-\n1380932.\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. 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THE HAMILTONIAN\nIn this section, we illustrate the complete low-energy Hamiltonian for a graphene monolayer with proximitized\nspin-orbit couplings (SOCs). Following [29] (see also [48]), the full Hamiltonian reads:\nH=H0+H∆+HZ+HR+HKM+HVZ, (SE1)\nwhere\nH0=vF(τzσxΠx+σyΠy) +V, (SE2a)\nH∆= ∆σz, (SE2b)\nHZ=λZsz, (SE2c)\nHR=λR\n2(τzσxsy−σysx), (SE2d)\nHKM=λKMτzσzsz, (SE2e)\nHVZ=λVZτzsz. (SE2f)\nHere, vF≈106m/s is the graphene’s Fermi velocity, Π=−iℏ∇+e\ncAthe kinetic momentum, A=B\n2(−y, x,0) the\nvector potential in the symmetric gauge (we assume B >0), and V(r) the potential defining the circular pnjunction.\nThe symbols τ/σ/sdenote the valley/sublattice/spin Pauli matrices. The Hamiltonian (SE1) is diagonal in valley\nspace. It includes the sublattice-symmetry breaking term H∆, the Zeeman term HZ(with λZ=gsµB\n2B), the Kane\nand Mele (or intrinsic) SOC HKM, the Rashba SOC HR, and the valley-Zeeman SOC HVZ. For completeness, we\ninclude the sublattice-symmetry breaking term H∆, which we have neglected in the main text.\nThe wave function is an 8-component spinor\n\u0000\nΨA↑,ΨB↑,ΨA↓,ΨB↓,Ψ′\nA↑,Ψ′\nB↑,Ψ′\nA↓,Ψ′\nB↓\u0001\n, (SE3)\nwhere the unprimed and primed components are the amplitudes at the valley K(τz= +1) and K′(τz=−1),\nrespectively. The Hamiltonian His invariant under the time-reversal operation T=isyτxKup to the inversion of the\nmagnetic field:\nT H(B)T†=H(−B), (SE4)\nand commutes with the total angular momentum operator J=Lz+1\n2(τzσz+sz).\nSinceHis diagonal in valley space, we will focus on a single valley ( τz= +1) and omit the valley index. Then, the\nwave function Ψ is a four-component spinor in sublattice/spin space, ΨT= (Ψ A↑,ΨB↑,ΨA↓,ΨB↓). The single-valley\nHamiltonians are related by the unitary transformation\nHτz=−1(∆, λVZ) =iσyHτz=+1(−∆,−λVZ)(−iσy). (SE5)\nUsing this identity, one can find the eigenstates at the valley τz=−1 once the eigenstates at the valley τz= +1 are\ndetermined.\nBefore closing this section, we notice that we express energy in units of the relativistic cyclotron energy ℏωc, length\nin units of the magnetic length ℓB, and wave vectors in units of ℓ−1\nB, with\nℓB=r\nℏc\neB=26p\nB[T]nm,ℏωc=ℏvF\nℓB= 26p\nB[T] meV ,ℏωZ=gsµB\n2B= 5.8×10−2B[T] meV .\nWe set e=ℏ=vF= 1 unless specified otherwise.9\nII. EXACT MODEL SOLUTION\nIn this section, we provide the exact solution of the problem of graphene’s Landau levels in the symmetric gauge in\nthe presence of SOCs and a constant potential. (See [43, 53] for the solution to this problem in the Landau gauge.)\nSince we work in a given valley, the valley-Zeeman term can be absorbed into the Zeeman term and will be omitted\nbelow. The single-valley Hamiltonian ( τz= +1) in the symmetric gauge commutes with the total angular momentum\nJ=Lz+σz\n2+sz\n2, (SE6)\nwith Lz=−i∂θ, hence the eigenfunctions can be labeled by the eigenvalues of J, which span the set of integers, and\ntake the form\nΨj(r) =ei(j−σz\n2−sz\n2)θ\n√\n2πψj(r), (SE7)\nwhere ( r, θ) are polar coordinates and j∈Z. The radial spinor ψj(r) is a solution of the equation\n(Hj−E)ψj= 0, (SE8)\nwhere\nHj=e−i(j−σz\n2−sz\n2)θHei(j−σz\n2−sz\n2)θ(SE9)\nis the radial Hamiltonian in a fixed jsector:\nHj−E=\n−µ+ −i(d\ndr+j\nr+r\n2) 0 0\ni(−d\ndr+j−1\nr+r\n2) −µ− −iλR 0\n0 iλR −ν− −i(d\ndr+j+1\nr+r\n2)\n0 0 i(−d\ndr+j\nr+r\n2) −ν+\n. (SE10)\nHere, we have introduced the auxiliary symbols\nµ±=E−V−(λZ±λKM±∆), (SE11a)\nν±=E−V−(−λZ±λKM∓∆), (SE11b)\nand we will use the notation\nµ=µ+µ−= (E−V−λZ)2−(λKM+ ∆)2, (SE12)\nν=ν+ν−= (E−V+λZ)2−(λKM−∆)2. (SE13)\nIn terms of the variable ξ=r2/2, we find\nHj−E=\n−µ+ −i√2ξ\u0010\nd\ndξ+1\n2+j\n2ξ\u0011\n0 0\ni√2ξ\u0010\n−d\ndξ+1\n2+j−1\n2ξ\u0011\n−µ− −iλR 0\n0 iλR −ν− −i√2ξ\u0010\nd\ndξ+1\n2+j+1\n2ξ\u0011\n0 0 i√2ξ\u0010\n−d\ndξ+1\n2+j\n2ξ\u0011\n−ν+\n.\n(SE14)\nThe general solution of Eq. (SE8) can be expressed in terms of confluent hypergeometric functions [50]. In the\nfollowing, we will present the solutions separately for j≥0 and j <0.\nA. Case j≥0\nFirst, we assume j >0. The solutions of graphene’s Landau levels problem without SOCs (see, e.g., [22, 51, 52])\nand with SOCs in the Landau gauge [43, 53] suggest the following ansatz:\nψj(ξ) =e−ξ/2ξj/2\nd1ξ−1/2M(a, j, ξ )\nid2M(a, j+ 1, ξ)\nd3M(a, j+ 1, ξ)\nid4ξ1/2M(a, j+ 2, ξ)\n, (SE15)10\nwhere diare constant coefficients (for simplicity, we omit the index jon the coefficients), and M(a, b, ξ ) denotes the\nconfluent hypergeometric function of the first kind [50], regular at the origin. The parameter awill be determined\nbelow. By using recurrence relations between confluent hypergeometric functions, Eq. (SE14) is converted into a\nlinear system for the coefficients di:\n\n−µ+√\n2j 0 0√\n2\u0010\n1−a\nj\u0011\n−µ− −λR 0\n0 −λR −ν−√\n2(j+ 1)\n0 0√\n2\u0010\n1−a\nj+1\u0011\n−ν+\n\nd1\nd2\nd3\nd4\n= 0. (SE16)\nThe existence of a non-trivial solution requires the vanishing of the determinant of the coefficient matrix:\n[2(a−j) +µ][2(a−j−1) +ν]−λ2\nRµ+ν+= 0. (SE17)\nThe solution of the linear system (SE16) is (up to an overall constant)\n\nd1\nd2\nd3\nd4\n=\n√\n2j\nµ+\n−2(a−j)+µ\nλR √\n2(a−j−1)[2(a−j)+µ])\nλRν+(j+1)\n. (SE18)\nA second solution, singular at the origin, is built using the confluent hypergeometric function of the second kind\nU(a, b, ξ ) [50]:\nψj(ξ) =e−ξ/2ξj/2\nd1ξ−1/2U(a, j, ξ )\nid2U(a, j+ 1, ξ)\nd3U(a, j+ 1, ξ)\niξ1/2d4U(a, j+ 2, ξ)\n. (SE19)\nThe corresponding linear system for the coefficients diis\n\n−µ+√\n2(j−a) 0 0√\n2−µ−−λR 0\n0 −λR−ν−√\n2(j+ 1−a)\n0 0√\n2 −ν+\n\nd1\nd2\nd3\nd4\n= 0, (SE20)\nand the determinant equation is the same as in Eq. (SE17). The solution of this linear system gives\n\nd1\nd2\nd3\nd4\n=\n√\n2(j−a)\nµ+\n−2(a−j)+µ\nλR\n−√\n2[2(a−j)+µ]\nλRν+\n. (SE21)\nThe solution for j= 0 can now be obtained by taking the limit j→0 in the previous formulas and using the\nfollowing identities:\nlim\nj→0jM(a, j, ξ ) =aξM(a+ 1,2, ξ), (SE22)\nU(a,0, ξ) =ξU(a+ 1,2, ξ). (SE23)\nB. Case j <0\nIn this case, the correct ansatz for the solution regular at the origin reads\nψj(ξ) =e−ξ/2ξ−j/2\nd1ξ1/2M(a+ 1,−j+ 2, ξ)\nid2M(a,−j+ 1, ξ)\nd3M(a,−j+ 1, ξ)\nid4ξ−1/2M(a−1,−j, ξ))\n. (SE24)11\nThen, the algebraic equation for the coefficients diis\n\n−µ+√\n2a\n1−j0 0√\n2(j−1)−µ−−λR 0\n0 −λR−ν−√\n21−a\nj\n0 0√\n2j−ν+\n\nd1\nd2\nd3\nd4\n= 0. (SE25)\nThe condition of vanishing determinant reads\n(2a+µ)(2a−2 +ν)−λ2\nRµ+ν+= 0, (SE26)\nand the solution of the linear system (up to an overall constant) is\n\nd1\nd2\nd3\nd4\n=\n√\n2λRν+a\n1−j\nλRµ+ν+\n−ν+(2a+µ)\n−√\n2j(2a+µ)\n. (SE27)\nThe second solution, singular at the origin, is given by\nψj(ξ) =e−ξ/2ξ−j/2\nd1ξ1/2U(a+ 1,−j+ 2, ξ)\nid2U(a,−j+ 1, ξ)\nd3U(a,−j+ 1, ξ)\nid4ξ−1/2U(a−1,−j, ξ)\n. (SE28)\nThe associated linear system is\n\n−µ+−√\n2a0 0√\n2−µ−−λR 0\n0−λR−ν−√\n2(1−a)\n0 0√\n2−ν+\n\nd1\nd2\nd3\nd4\n= 0, (SE29)\nwith the same determinant equation as in Eq. (SE26), and the solution given by\n\nd1\nd2\nd3\nd4\n=\n−√\n2λRν+a\nλRµ+ν+\n−ν+(2a+µ)\n−√\n2(2a+µ)\n. (SE30)\nWe note in passing that, by taking the limit j→0 in the formulas above, we recover the solution for j= 0 given\nin Sec. II A.\nC. General solution\nThe two determinant equations (SE17) and (SE26) can be merged into a single equation:\n[2(a−jΘ(j)) +µ][2(a−jΘ(j)−1) +ν]−λ2\nRµ+ν+= 0, (SE31)\nwhere Θ( x) is the Heaviside function. (We adopt the convention Θ(0) = 1.) This condition admits the solutions\na=a±given by\na±=jΘ(j)−1\n4\u0014\nµ+ν−2±q\n(µ−ν+ 2)2+ 4λ2\nRµ+ν+\u0015\n. (SE32)\nWe denote by ψ<\nj(ξ) the wave functions regular at the origin, Eqs. (SE15) and (SE24), and by ψ>\nj(ξ) the wave functions\nsingular at the origin, Eqs. (SE19) and (SE28). The eigenspace of energy Eand total angular momentum jis then\nspanned by the linear combinations of the four solutions obtained by taking ψ<\njandψ>\njwith a=a±in Eq. (SE32):\nψj(ξ) =c1ψ<\nj,a+(ξ) +c2ψ<\nj,a−(ξ) +c3ψ>\nj,a+(ξ) +c4ψ>\nj,a−(ξ).\nAs we will see below, the quantized energy eigenvalues (Landau levels) are obtained by imposing appropriate conditions\non this general solution. In Sec. III we consider the case of a uniform system, where the quantization condition\noriginates simply from the requirement of normalizability. In Sec. IV we consider the case of a circular pnjunction,\nwhere the quantization condition arises from the combination of the requirements of normalizability and continuity\nof the wave function.12\n(d)(c)(b)(a)\nFigure SF1. Landau level spectrum in the symmetric gauge. (a) Spin degenerate case, with λZ=λKM=λR= ∆ = 0. (b)\nCase with only Zeeman splitting, with λZ= 0.3. (c) Case with only intrinsic SOC, with λKM= 0.3. (d) Case with only Rashba\nSOC, with λR= 0.3. In panels (a) to (c), the arrows are associated with the eigenvalues of sz. In all panels, the color refers to\nthe radial quantum number n: red to n= 0, blue to n= 1, green to n= 2 and orange to n= 3.\nIII. UNIFORM SYSTEM\nFor a uniform system ( V= 0) we have to select the solutions regular at the origin, Eqs. (SE15) and (SE24), which\ninvolve the functions M(a, b, ξ ). Then normalizability requires that the first argument of Mis a non-positive integer\n−n, where nis interpreted as the radial quantum number. As a result, we find two sets of Landau levels, obtained\nby solving the equations\n(\na±(E) =−n n = 0,1,2, . . . , forj≥0\na±(E)−1 =−n n = 0,1,2, . . . , forj <0, (SE33)\nwith a±in Eq. (SE32). For the correct counting of the solutions, one should notice the following:\n•Case j= 0: for n= 0 (i.e., a= 0) the solution of Eq. (SE31) with µ+= 0 must be omitted, because all wave\nfunction components vanish, see Eq. (SE18).\n•Case j <0: for n= 0 (i.e., a= 1) only the solution of Eq. (SE31) with ν+= 0 is allowed because the first three\ncomponents of the wave function (which are not normalizable functions if a= 1) have a vanishing coefficient,\nsee Eq. (SE27). For n= 1 (i.e., a= 0) the solution of Eq. (SE31) with µ+= 0 must be omitted for the same\nreason as in the case j= 0 above.\nThe quantization equation (SE31) is quartic in the energy and can be solved explicitly. However, the general\nexpression of the solutions is cumbersome and not particularly illuminating. Below, we briefly discuss few special\ncases and give the explicit formulas for the corresponding energy eigenvalues.\n•If the Rashba SOC vanishes, Eq. (SE31) decouples into two separate equations, each giving a set of spin-polarized\nLandau levels. From 2( a−jΘ(j)) +µ= 0 we find the spin-up levels\nEn,j,α,−=λZ+αp\n2(n+jΘ(j)) + ( λKM+ ∆)2, n = 0,1,2, . . . , j = 0,±1,±2, . . . , α =±1, (SE34)\nwhere for n= 0 and j≤0 only the level E0,j=λZ−λKM−∆ must be kept. From 2( a−jΘ(j)−1) +ν= 0\nwe find the spin-down levels\nEn,j,α, +=−λZ+αp\n2(n+ (j+ 1)Θ( j)) + ( λKM−∆)2, n = 0,1,2, . . . , j = 0,±1,±2, . . . , α =±1,(SE35)\nwhere for n= 0 and j <0 only the level E0,j=−λZ+λKM−∆ must be kept. For λZ=λKM= ∆ = 0,\nthe expressions in Eqs. (SE34) and (SE35) coincide with the Landau level formula in the symmetric gauge (see,\ne.g., [22]) after the replacement j=j′±1\n2for spin up/down states, with j′half-integer.\n•If we have only a finite Rashba SOC and all the other couplings are set to zero, the Landau levels obtained by\nsolving Eq. (SE31) are given by (see also [53])\nEn,j,α,β =α\n2[n+ (j+ 1)Θ( j)]−1 +λ2\nR\n2+βs\u0012\n1−λ2\nR\n2\u00132\n+ 2λ2\nR[n+ (j+ 1)Θ( j)]\n1\n2\n, α, β =±1. (SE36)13\n(a) (b)\nEn,jEn,j\nFigure SF2. (a) Spectrum of the circular pnjunction with ξ0= 5.1 and V0= 0.51 as a function of jfor vanishing SOCs and\nZeeman coupling. (b) The same as in (a) but with the spin-down states translated by 1 along the j-axis, showing that they\ncoincide, as expected because of spin degeneracy.\nFor the correct counting of the states, one should keep in mind the remarks at the beginning of this section. It\nis interesting to observe that at the lowest order in λRone finds\nEn,j,α,β ≈α\u0012\n1 +βλ2\nR\n2\u0013p\n2[n+ (j+ 1)Θ( j)] +β−1. (SE37)\nWe see that the effect of a small Rashba SOC is essentially a renormalization of the cyclotron frequency. We\nnote in passing that the states corresponding to the two sets of levels in Eq. (SE37) are not eigenstates of sz.\nWe illustrate in Fig. SF1 the exact Landau level spectrum in the symmetric gauge for four relevant cases. We\nobserve that, in all cases, at a fixed value of the radial quantum number n, the energy is independent of the angular\nquantum number jforj≤norj≤n−1. In the first three panels, we set λR= 0, hence the spin projection in\nthezdirection is a good quantum number and the eigenfunctions describe spin states polarized along the zaxis. In\npanel SF1(a) we present the spin degenerate case with λZ=λKM=λR= ∆ = 0. The spectra of spin-up and spin-\ndown states appear to have a relative horizontal shift, because we label our states with the total angular momentum\nj. They coincide if we plot the spin-up spectrum versus j′=j−1\n2and the spin-down spectrum versus j′=j+1\n2. This\nshift is the reason why the lowest-energy states with j≥0 appear singly degenerate. If only the Zeeman coupling is\nactive, see panel SF1(b), we observe the usual energy shift, upwards for spin-up states and downwards for spin-down\nstates. In the case that only the intrinsic SOC is active, illustrated in panel SF1(c), we see that the spin degeneracy\nof the zero-energy Landau level is lifted, while all other levels remain spin degenerate.\nFinally, in panel SF1(d) we show the spectrum when only the Rashba SOC is active. In this case, the projection of\nthe spin along the zaxis is no longer a good quantum number, because the SOC mixes spin-up and spin-down states.\nAs a result, the spin degeneracy of all levels is lifted, with the exception of the zero-energy level, which remains doubly\ndegenerate at zero energy. This residual degeneracy is a result of the fact that the zero-energy states have support\non a single sublattice [43, 53].\nIV. LANDAU LEVELS IN A PNJUNCTION\nNext, we discuss the exact solution of the Landau level problem in the case of a pnjunction. We assume that the\npotential has the following profile:\nV(r) =V0sign(R−r), V 0>0, (SE38)\nnamely, V=V0within a disc of radius RandV=−V0outside the disc. We use ξ0=R2/2 as a measure for the size\nof the circular junction. Using the solutions found in Sec. II, we write the radial wave function as\nψ(ξ) =(\nc1ψ<\na+,−V0(ξ) +c2ψ<\na−,−V0(ξ)ξ < ξ 0\nc3ψ>\na+,V0(ξ) +c4ψ>\na−,V0(ξ) ξ > ξ 0. (SE39)14\n(a) (b)\nFigure SF3. (a) The energy spectrum of the zero modes as a function of the total angular momentum jfor several values of\nthe Rashba SOC. (b) The zero-mode splitting defined in Eq. (SE43) as a function of the total angular momentum jfor several\nvalues of the Rashba SOC. For both panels, the values of the Rashba SOC are indicated on the side of the panel (b).\nHere we omit the index j, being understood that we work at fixed angular momentum, and we append two indexes\nto indicate the values of the parameter aand of the potential V. The eigenenergies and the eigenstates are obtained\nby matching the wave functions at the boundary of the disc ξ=ξ0:\nc1ψ<\na+,−V0(ξ0) +c2ψ<\na−,−V0(ξ0) =c3ψ>\na+,V0(ξ0) +c4ψ>\na−,V0(ξ0). (SE40)\nIn analogy to the case of a linear junction [22, 53], we obtain a linear system for the ci, with the matrix of coefficients\ngiven by\nW=\u0002ψ<\na+,−V0(ξ0)ψ<\na−,−V0(ξ0)−ψ>\na+,V0(ξ0)−ψ>\na−,V0(ξ0)\u0003\n. (SE41)\nThe allowed energy eigenvalues are found by solving the equation\ndetW= 0. (SE42)\nOnce the eigenvalues are determined, the corresponding normalized eigenstates can be calculated from Eq. (SE39)\nusing the solution of the linear system (SE40).\nIn Fig. SF2 we show the exact spectrum of the circular pnjunction obtained from the numerical solution of\nEq. (SE42) in the absence of Rashba SOC for the spin-degenerate case. The effect of the potential step is that the\nlevels acquire a dispersion in j. As observed in the uniform case discussed in Sec. III, the spectra for spin-up and\nspin-down states appear horizontally shifted one with respect to the other, which results from labeling the states\nwith the total angular momentum j. As shown in panel SF2(b), when the spin-down spectrum is shifted to the\nright by 1, they do overlap. This observation suggests defining the splitting of the energy levels as the difference\n|En,j,α, +−En,j−1,α,−|, which vanishes in the spin-degenerate case.\nWe now focus on the two lowest-energy levels, which we refer to as the top and bottom modes and denote as ET\n0,j\nandEB\n0,jwith corresponding radial wave functions ψT\nj(ξ) and ψB\nj(ξ). In Fig. SF3(a) we show their dispersion as a\nfunction of the total angular momentum jfor different values of the Rashba SOC. We see that the j-dependence\naround zero energy is linear with good approximation. In Fig. SF3(b) we show the energy splitting of the zero modes,\ndefined as\nδE0,j=|ET\n0,j−EB\n0,j−1|. (SE43)\nWe observe that δE0,jis not constant but depends quite strongly on the value of the angular momentum j. This\ndependence can be rationalized by considering the Rashba SOC as a perturbation. As shown below, the spatial\nlocation of the zero modes is essentially determined by j. For j= 0 and j≫1, the radial wave function is localized\nfar from the pninterface. In this case, the zero modes are supported on only one of the sublattices, so the Rashba15\n(a) (b) (c)\n(d) (e) (f)\n(g) (h) (i)\n(k) (j) (l)0 0 0\n0 0 05 5 5\n5 5 510 10 10\n10 10 1015 15 15\n15 15 15\nFigure SF4. Radial profile of the observable densities defined in Eq. (SE44) for the bottom (solid lines) and top (dashed lines)\nzero modes as a function of ξfor different values of the Rashba SOC λR(λR= 0,0.1,0.2,0.3,0.4,0.5,0.6). Panels (a) to (c)\ndisplay the modulus square of the wave function for the lowest value of j(j=−1/0 for top/bottom mode), an intermediate\nvalue of j(j=−6/−5 for top/bottom mode), and a large value of j(j=−11/−10 for top/bottom mode). Panels (d) to (f),\n(g) to (i), and (k) to (l) are the same but for the densities ⟨σθ⟩,⟨sr⟩, and⟨sz⟩respectively. In all panels, the plane at ξ0= 5.1\nrepresents the position of the pninterface.\nSOC matrix element is very small. For values of jat which the wave function is localized close to the pninterface,\ninstead, the zero modes have support on both sublattices so that the matrix element of the Rashba SOC is largest\nand the splitting reaches a maximum. A similar effect was observed in the case of the linear pnjunction [53].\nIn the panels of Fig. SF4, we present the radial profiles of various observable densities for the top and bottom zero\nmodes for several values of the Rashba SOC and for three different values of the angular momentum j. We denote\nthese quantities as follows:\n⟨ρ⟩=ψi†\njψi\nj,⟨σθ⟩=ψi†\njσθψi†\nj,⟨sz⟩=ψi†\njszψi\nj,⟨sr⟩=ψi†\njsrψi\nj, i= T,B. (SE44)16\nThey describe the radial probability density, the azimuthal probability current density, and the perpendicular and\nradial components of the spin density. (See Eqs. (SE46) and (SE46b) for the definition of σθandsr.) Because of\nsymmetry, the two densities ⟨σr⟩and⟨sθ⟩are identically zero. Panels (a) to (c) in Fig. SF4 clearly show how the\nradial profile of the probability density changes with the angular momentum j: the wave function is localized at the\ncenter of the circular region for {jT, jB}={−1,0}, it is located close to the pninterface for {jT, jB}={−6,−5},\nand finally, it moves outside the circular region and away from the pninterface for {jT, jB}={−11,−10}. A similar\nbehavior is found in all the other observables we have considered. We note that in the absence of Rashba SOC, the\nradial spin density ⟨sr⟩vanishes, and it increases for increasing values of λR. On the contrary, the perpendicular spin\ndensity ⟨sz⟩is finite also in the absence of Rashba SOC and decreases with increasing λR. This can be understood\nsince for increasing λRthe spin will tend to align with the effective magnetic field generated by the Rashba SOC [46].\nV. EFFECTIVE ZERO-MODE HAMILTONIAN\nIn this section, we provide the details of the derivation of the effective one-dimensional (1D) Hamiltonian governing\nthe spin and angular dynamics of the zero modes localized at the junction. We assume that the Fermi energy is at\nthe charge neutrality point, EF= 0, and we aim at an effective model valid in the low-energy region around EF. We\nfollow the approach of [54], where the effective 1D Hamiltonian for the analogous problem of electrons in a mesoscopic\nring in the presence of Rashba SOC was derived. The approach is based on the projection of the full Hamiltonian onto\nthe zero-energy radial state, localized at the interface between the pand the nregions. This projection is justified\nas long as the separation between the zero modes and the first Landau level is much larger than any other relevant\nenergy scale in the problem.\nWe start with the Dirac equation\nHΨ =EΨ,\nwhere the Hamiltonian (SE1) in the symmetric gauge is expressed in polar coordinates as follows:\nH=σr(−i∂r) +σθ\u0012Lz\nr+Aθ\u0013\n+V(r) + ∆ σz+λZsz+λKMσzsz+λR\n2(σrsθ−σθsr), (SE45)\nwith Aθ=r/2,V(r) =V0sign(R−r),Lz=−i∂θ, and we have defined\nσr= cos θ σx+ sin θ σy=\u0012\n0e−iθ\neiθ0\u0013\n, (SE46a)\nσθ=−sinθ σx+ cos θ σy=\u0012\n0−ie−iθ\nieiθ0\u0013\n, (SE46b)\nwith analogous expressions for srandsθ. First, we make a unitary transformation in sublattice and spin space:\n˜H=UHU−1, (SE47a)\n˜Ψ =UΨ, (SE47b)\nwhere U=eiσz\n2(θ+π\n2)eisz\n2θ. The additional π/2 rotation in sublattice space is included in order to obtain a real\nHamiltonian. Using\nUσrU−1=−σy, Uσ θU−1=σx, (SE48)\nUsrU−1=sx, Us θU−1=sy, (SE49)\nULzU−1=Lz−σz\n2−sz\n2, (SE50)\nwe arrive at\n˜H=iσy\u0012\n∂r+1\n2r\u0013\n+σx\u0012Lz\nr+r\n2−sz\n2r\u0013\n+V(r) + ∆ σz+λZsz+λKMσzsz−λR\n2(σysy+σxsx). (SE51)\nNote that under the unitary transformation U, the total angular momentum Jis mapped to UJU−1=Lz. Next, we\nseparate the Hamiltonian into a radial part and an angular/spin part, ˜H=˜Hr+˜Hθ, where the radial part is defined\nas\n˜Hr=iσy\u0012\n∂r+1\n2r\u0013\n+σx\nr\u0012r2\n2−Φ\u0013\n+V(r) + ∆ σz, (SE52)17\nand the angular/spin part as\n˜Hθ=σx\nr(Lz+ Φ) +\u0010\nλZ+λKMσz−σx\n2r\u0011\nsz−λR\n2(σysy+σxsx). (SE53)\nHere, Φ is a parameter whose value is set in such a way that the zero mode of ˜Hris at the Fermi energy EF= 0.\nIn practice, Φ is with good approximation the magnetic flux through the pnjunction in units of the flux quantum,\nΦ≈R2/2ℓ2\nB. The radial Hamiltonian ˜Hrcoincides, up to the π/2 rotation in sublattice space, with the model of a\ncircular pnjunction for spinless graphene solved in [22, 52], with the appropriate identification of the parameter Φ.\nWe now project the full Hamiltonian onto the spin-degenerate zero mode of ˜Hr. To this aim, we write the wave\nfunction as\n˜Ψ(r, θ) =˜ψ0(r)˜χ(θ), (SE54)\nwhere the sublattice spinor ˜ψ0(r) is the zero mode of ˜Hr, which satisfies\n˜Hr˜ψ0(r) = 0 , (SE55)\nand we choose to be real, and ˜ χ(θ) is a two-component angular spinor in spin space. From the equation ˜H˜Ψ = E˜Ψ\nwe find that ˜ χ(θ) satisfies the equation\n˜Heff˜χ(θ) =E˜χ(θ), (SE56)\nwith the effective 1D Hamiltonian\n˜Heff=⟨˜H⟩0=ω0(Lz+ Φ) +\u0010\nωZ−ω0\n2\u0011\nsz−ωRsx. (SE57)\nHere, the brackets ⟨. . .⟩0denote the expectation value in the radial zero mode:\n⟨. . .⟩0=ˆ∞\n0rdr˜ψ†\n0(r). . .˜ψ0(r),\nand we have used ⟨σy⟩0= 0, which holds because ˜ψ0(r) is a real spinor. In Eq. (SE57) we have defined the angular\nvelocity ω0and the Zeeman and Rashba frequencies ωZandωR, as follows:\nω0≡Dσx\nrE\n0, ω Z≡λZ+λKM⟨σz⟩0, ω R≡λR\n2⟨σx⟩0. (SE58)\nSince σxbefore the unitary transformation was σθ, see Eq. (SE48), we recognize the coefficient of Lz,⟨σx/r⟩0, as\nthe angular velocity of the circular motion along the junction, and the coefficient that renormalizes λR,⟨σx⟩0, as the\nazimuthal component of the velocity. Similarly, the vanishing of ⟨σy⟩0expresses the vanishing of the radial velocity.\nWe note that if we undo the unitary transformation in spin space, we obtain the effective Hamiltonian\nHeff=ω0(Lz+ Φ) + ωZsz−ωRsr, (SE59)\nwhich explicitly shows that the Rashba SOC acts as an effective magnetic field that pushes the spin in the in-plane\nradial direction.\nIt is straightforward to diagonalize ˜Heff. Its eigenvalues read\nE0,m,±=ω0(m+ Φ)±r\u0010\nωZ−ω0\n2\u00112\n+ω2\nR, (SE60)\nwhere mis an integer if we impose periodic boundary conditions. The corresponding eigenstates are\n˜χm,+(θ) =eimθ\n√\n2π\u0012\ncosγ\n2\n−sinγ\n2\u0013\n,˜χm,−(θ) =eimθ\n√\n2π\u0012\nsinγ\n2\ncosγ\n2\u0013\n, (SE61)\nwhere we define\nsinγ=ωRq\n(ωZ−ω0\n2)2+ω2\nR,cosγ=ωZ−ω0/2q\n(ωZ−ω0\n2)2+ω2\nR. (SE62)18\nFigure SF5. (a) The angular frequency ω0in Eq. (SE58) (yellow plane) and the effective Zeeman frequency given by Eq. (SE63)\n(green plane) as functions of junction size ξ0and the Kane-Mele SOC λKM. The intersection of the green and yellow planes\ncorresponds to the sweet spots where the resonance condition occurs. For a given value of λKM(λKM= 0.01 in the figure),\nillustrated by the black dashed line, the value of ξ0at which the resonance occurs is indicated by the red dashed line. (b) The\nexact expectation values of the radial and perpendicular spin components in the top and bottom modes as a function of ξ0, for\nλZ= 0.048 and λR= 0.2 and λKM= 0.01. The top curve shows ⟨sr⟩B, the bottom one ⟨sr⟩T, and the two central ones ⟨sz⟩T\nand⟨sz⟩B. In all the panels, V0= 0.51.\nEquations (SE60) and (SE61) provide useful approximation to the SOC coupled zero-mode energies and wave func-\ntions, which hold as long as transitions to higher Landau levels due to ˜Hθcan be neglected, and for angular states\nwith m≈ −Φ, where it predicts a linear dependence of the energy on m.\nWe note that in the uniform system ( V0= 0), the zero mode of ˜Hrhas only one non-vanishing sublattice amplitude\n(the sublattice pseudo-spin is down-polarized). As a consequence, both ω0andωRvanish, and the eigenstates are\nspin-polarized along the zdirection and orbitally degenerate (i.e., the energy is independent of m). In the presence\nof the potential step ( V0̸= 0), both sublattice amplitudes in ˜ψ0are finite. Then ω0andωRare finite, the zero modes\nacquire a dispersion, and the Rashba term is activated and pushes the spin polarization in the planar radial direction.\nThe spin dynamics is therefore controlled by the potential step amplitude V0.\nVI. RABI CONDITION FOR GENERAL SPIN-ORBIT COUPLING\nIn the main text, we have investigated the Rabi condition for the full model in Eq. (SE1) under the assumption\nthat only the Rashba SOC is non-vanishing, and that effects due to the VZSOC can be neglected — single-valley\nmodel. We now relax this condition and include the Kane-Mele ( λKM) SOC terms. In general, the angular frequency\nassociated with the Zeeman term can then be expressed as:\nωZ=λZ+λKM⟨σz⟩0. (SE63)\nNotice that here the brackets ⟨. . .⟩0denote the expectation value in the j-state with energy closest to zero energy at\nthe given value of ξ0. As already mentioned in the main text, in the single-valley approximation, the valley-Zeeman\nSOC just produces a shift of the Zeeman term. The Kane-Mele SOC gives a nontrivial contribution to ωZthat\ndepends on the expectation value of σzover the spinless system. In Fig. SF5, we present the effect of the Kane-Mele\nSOC on the shift of the Rabi condition. From Fig. SF5(a), we observe for fixed λZthat the position of the sweet spot\nξ0for increasing values of λKMmoves at larger values of the radius.\nVII. S-MATRIX APPROACH\nHere we introduce the scattering approach [59] used to obtain the quantum transmission and conductance of the\ninterferometers discussed in the main text. We begin by discussing a spinless model and then generalize it to the\nspin-dependent case. Without any significant loss of generality, we stick to the p-dot-based interferometer depicted\nin Fig. SF6(a).19\nIncoming and outgoing chiral modes are described in Fig. SF6 by fermionic annihilation operators {a1, a3}and\n{b1, b3}, respectively, such that\n\u0012\nb1\nb3\u0013\n=\u0012\nr t′\nt r′\u0013\u0012\na1\na3\u0013\n. (SE64)\nThe conductances G21andG31are determined from the scattering amplitudes randt, respectively, by following the\nLandauer-B¨ uttiker approach.\nThe scattering matrix on the r.h.s. of Eq. (SE64) can be obtained by combining the scattering blocks S1andS2\ncorresponding to the barriers τ1andτ2, as depicted in Fig. SF6(b). These block are connected by channels {a2, b2}\npropagating around the central pdot by accumulating additional phases ϕ±, satisfying\n\u0012\nb1\nb2\u0013\n=\u0012\nr1t′\n1\nt1r′\n1\u0013\u0012\na1\na2\u0013\n, (SE65)\n\u0012\na2\nb3\u0013\n=\u0012\nr2t′\n2\nt2r′\n2\u0013\u0012\nb2\na3\u0013\n, (SE66)\nwith\nr1=√\n1−τ1, t′\n1=eiϕ−√τ1,\nt1=eiϕ+√τ1, r′\n1=−ei(ϕ++ϕ−)√\n1−τ1,\nr2=√\n1−τ2, t′\n2=√τ2,\nt2=√τ2, r′\n2=−√\n1−τ2.\nAfter a little algebra, from (SE64)-(SE66) we find\nr=r1+t′\n1(1−r2r′\n1)−1r2t1, (SE67)\nt=t2(1−r′\n1r2)−1t1. (SE68)\nNotice that expanding (SE67) and (SE68) as geometric series supplies the Feynman paths contributing to the quantum\namplitudes due to multiple reflections between the barriers. Moreover, when the barriers are placed symmetrically on\nopposite sides of the dot we find that ϕ+=ϕ−.\nThe results of Eqs. (SE67) and (SE68) can be generalized to the spin-dependent case by choosing convenient spin\nbases along the linear and circular pnjunctions and calculating their local projection at barriers 1 and 2. For the\ncircular junction, the natural choice is the spin-eigenstate basis, which evaluated at the barriers reads\n|χ+, ℓ⟩=\u0012\ncosγ\n2\nℓsinγ\n2\u0013\n, (SE69)\n|χ−, ℓ⟩=\u0012\nsinγ\n2\n−ℓcosγ\n2\u0013\n, (SE70)\nwith ℓ= 1 for barrier 1 and ℓ=−1 for barrier 2. For linear junctions (acting as incoming and outgoing leads) we can\nsimply choose the canonical z-basis\n| ↑⟩=\u0012\n1\n0\u0013\n,| ↓⟩=\u0012\n1\n0\u0013\n. (SE71)\nThe use of a field-dependent, spin-eigenstate basis has no practical advantage here since the conductance is independent\nof the spin phases gathered along the leads. As for the phases ϕ±, they can be obtained by setting Em,s= 0 in\nEq. (SE60) and finding the corresponding spin-dependent m(which is not necessarily an integer any longer due to\nthe open boundary conditions introduced by the barriers). 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(a) Sketch of the electronic circuit for the case of a p-doped circular region. We present the corresponding circuit\nfor the n-doped circular region in the main text. (b) Generic representation of two scattering regions S1andS2separated by\na region where the carriers can gain a geometrical phase eiϕ±.\nThe spin-dependent scattering amplitudes are determined by the projections ⟨↑ |χs, ℓ⟩and⟨↓ |χs, ℓ⟩, and by the\nphases (SE72), accordingly. In this way, we find\nr1=√\n1−τ1\u0012\n1 0\n0 1\u0013\n, (SE73)\nt1=√τ1 \neiϕ+cosγ\n2−eiϕ+sinγ\n2\neiϕ−sinγ\n2eiϕ−cosγ\n2!\n, (SE74)\nt′\n1=√τ1 \neiϕ+cosγ\n2eiϕ−sinγ\n2\n−eiϕ+sinγ\n2eiϕ−cosγ\n2!\n, (SE75)\nr′\n1=−√\n1−τ1 \nei2ϕ+0\n0ei2ϕ−!\n, (SE76)\nt2=√τ2\u0012\ncosγ\n2sinγ\n2\nsinγ\n2−cosγ\n2\u0013\n, (SE77)\nr2=√\n1−τ2\u0012\n1 0\n0 1\u0013\n. (SE78)\nHere we omit the spin-dependent expressions for t′\n2andr′\n2since they do not contribute to the scattering amplitude\nmatrices tandrin Eqs. (SE67) and (SE68). By following the Landauer-B¨ uttiker approach, we find the expression\nfor the linear conductances in terms of tandrthat we use in the main text:\nG21=e2\nhtr[rr†], (SE79)\nG31=e2\nhtr[tt†]. (SE80)21\n0.51.01.52.02.53.03.5\n<latexit sha1_base64=\"wef6n6ct8YkQOB93xxGTw+A98JE=\">AAACB3icbZBLSgNBEIZ7fMb4irp00xgEV2Em+FoG3biMYB6QjKGnU5M06ekeu3uEMMwBPIFbPYE7cesxPID3sJPMQhN/KPj5q4oqviDmTBvX/XKWlldW19YLG8XNre2d3dLeflPLRFFoUMmlagdEA2cCGoYZDu1YAYkCDq1gdD3ptx5BaSbFnRnH4EdkIFjIKDE28ruhIjSF+2qWDrNeqexW3KnwovFyU0a56r3Sd7cvaRKBMJQTrTueGxs/JcowyiErdhMNMaEjMoCOtYJEoP10+nSGj23Sx6FUtoTB0/T3RkoircdRYCcjYoZ6vjcJ/+0FgeT9ueMmvPRTJuLEgKCz22HCsZF4AgX3mQJq+NgaQhWz72M6JBaNseiKlos3T2HRNKsV77xydntarl3lhAroEB2hE+ShC1RDN6iOGoiiB/SMXtCr8+S8Oe/Ox2x0ycl3DtAfOZ8/yseaNw==</latexit>e2h\n1<latexit sha1_base64=\"gyi46qU2K6X9WHIumjSXNedR4sc=\">AAACA3icbVDLSgMxFL1TX7W+qi7dBIvgqsyIr2XRjctW7EPaoWQymTY0yQxJRihDl36BW/0Cd+LWD/ED/A/TdhZaPXDhcM69nMsJEs60cd1Pp7C0vLK6VlwvbWxube+Ud/daOk4VoU0S81h1AqwpZ5I2DTOcdhJFsQg4bQej66nffqBKs1jemXFCfYEHkkWMYGOl+0Y/6ymBbif9csWtujOgv8TLSQVy1Pvlr14Yk1RQaQjHWnc9NzF+hpVhhNNJqZdqmmAywgPatVRiQbWfzR6eoCOrhCiKlR1p0Ez9eZFhofVYBHZTYDPUi95U/NcLgpiHC+EmuvQzJpPUUEnm2VHKkYnRtBAUMkWJ4WNLMFHMvo/IECtMjK2tZHvxFlv4S1onVe+8etY4rdSu8oaKcACHcAweXEANbqAOTSAg4Ame4cV5dF6dN+d9vlpw8pt9+AXn4xsQfZgf</latexit>QR\n<latexit sha1_base64=\"Cls+3/iyjt5O4GxFST3o8hgOYbI=\">AAACA3icbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cdmCfWg7lEwm04YmkyHJCGXo0i9wq1/gTtz6IX6A/2HazkJbD1w4nHMv53KChDNtXPfLWVpeWV1bL2wUN7e2d3ZLe/tNLVNFaINILlU7wJpyFtOGYYbTdqIoFgGnrWB4M/Fbj1RpJuM7M0qoL3A/ZhEj2Fjpvt7Lukqgh3GvVHYr7hRokXg5KUOOWq/03Q0lSQWNDeFY647nJsbPsDKMcDoudlNNE0yGuE87lsZYUO1n04fH6NgqIYqkshMbNFV/X2RYaD0Sgd0U2Az0vDcR//WCQPJwLtxEV37G4iQ1NCaz7CjlyEg0KQSFTFFi+MgSTBSz7yMywAoTY2sr2l68+RYWSfO04l1Uzutn5ep13lABDuEITsCDS6jCLdSgAQQEPMMLvDpPzpvz7nzMVpec/OYA/sD5/AEdPZgn</latexit>QZ122334\n455\n00\nFigure SF7. Differential conductance G21=G+\n21+G−\n21for the circuit of Fig. 3(b) including a valley-Zeeman QVZ= 0.25. The\nsolid line at QZ= 1/2 corresponds to a symmetry axis in the pattern.\nVIII. FIRST-ORDER EXPANSION\nThe scattering amplitudes (SE67) and (SE68) can be expanded as\nr=r1+t′\n1r2t1+t′\n1r2r′\n1r2t1+..., (SE81)\nt=t2t1+t2r′\n1r2t1+... (SE82)\nThese expansions have a simple interpretation in terms of Feynman paths: each term corresponds to a possible\nscattering history through barriers 1 and 2, comprising a different number of windings around the central dot. Previous\nworks in semiconductor-based rings [46, 60] have shown that the first two terms in the expansions (SE81) and (SE82),\ncorresponding to zero- and single-winding path contributions, are sufficient to capture all relevant features of the\nconductances (SE79) and (SE80). This also facilitates further physical insight by discriminating the role that different\nquantum phases play in the interference. By setting τ1=τ2= 1/2 and neglecting higher order contributions, we find\nG21≈1 + cos ϕABcosϕS, (SE83)\nwith\nϕAB= 2πΦ, (SE84)\nϕS= 2πs\u0012\nQZ−1\n2\u00132\n+Q2\nR, (SE85)\nwhere ϕABandϕSare independent phase contributions with origin in the orbital and spin degrees of freedom,\nrespectively.\nIX. EFFECTS OF VALLEY-ZEEMAN COUPLING\nThe introduction of the valley-Zeeman coupling leads to a valley-dependent shift in the Zeeman term (a valley\nsplitting), such that λZ→λZ+τλVZ, with τ=±1 the valley index. This implies that the sweet spots corresponding\nto in-plane spin eigenstates and πgeometric phases shift as well, in a valley-dependent way. By assuming no valley\nmixing (due to the separation between lattice and pn-junction length scales), the conductance turns valley-dependent.\nThe 1st order expansion of Sec. VIII generalizes to\nGτ\n21≈1 + cos ϕABcosϕτ\nS, (SE86)22\nwith\nϕτ\nS= 2πs\u0012\nQZ+τQVZ−1\n2\u00132\n+Q2\nR, (SE87)\nandQVZ=λVZ/ω0. The valley-resolved conductances for circuits with n−andp−doped central regions would then\nlook like those of Figs. 3(c) and 3(d) with additional ±QVZshifts along the QZaxis, respectively. As for the total\nconductance, it is the sum of the corresponding valley conductances, G21=G+\n21+G−\n21. In Fig. SF7 we illustrate this\nsituation for the p-doped dot circuit of Fig. 3(b), with the same setting used to produce Fig. 3(d) and an additional\nvalley-Zeeman coupling QVZ= 0.25. The composed pattern is symmetric with respect to the axis QZ= 1/2. Although\nthis axis does not correspond any longer to a sweet spot in the strict sense discussed above, we notice that the average\nspin projection along zvanishes at QZ= 1/2, i.e., ⟨sz⟩=⟨sz⟩++⟨sz⟩−= 0, due to the opposite valley-Zeeman pulls."    },    {        "title": "2012.07227v2.Driven_dynamics_of_a_quantum_dot_electron_spin_coupled_to_bath_of_higher_spin_nuclei.pdf",        "content": "Driven dynamics of a quantum dot electron spin coupled to bath of higher-spin nuclei\nArian Vezvaee1,\u0003Gargee Sharma2, Sophia E. Economou1, and Edwin Barnes1y\n1Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA\n2School of Basic Sciences, Indian Institute of Technology Mandi, India\nThe interplay of optical driving and hyper\fne interaction between an electron con\fned in a\nquantum dot and its surrounding nuclear spin environment produces a range of interesting physics\nsuch as mode-locking. In this work, we go beyond the ubiquitous spin 1/2 approximation for\nnuclear spins and present a comprehensive theoretical framework for an optically driven electron\nspin in a self-assembled quantum dot coupled to a nuclear spin bath of arbitrary spin. Using a\ndynamical mean-\feld approach, we compute the nuclear spin polarization distribution with and\nwithout the quadrupolar coupling. We \fnd that while hyper\fne interactions drive dynamic nuclear\npolarization and mode-locking, quadrupolar couplings counteract these e\u000bects. The tension between\nthese mechanisms is imprinted on the steady-state electron spin evolution, providing a way to\nmeasure the importance of quadrupolar interactions in a quantum dot. Our results show that\nhigher-spin e\u000bects such as quadrupolar interactions can have a signi\fcant impact on the generation\nof dynamic nuclear polarization and how it in\ruences the electron spin evolution.\nI. INTRODUCTION\nSpins in self-assembled quantum dots (QDs) are under\nintense investigation for a variety of quantum information\napplications, including quantum information processing,\nquantum communication, and quantum transduction [1{\n4]. The relatively long coherence times, fast control-\nlability [5{7], and good photon emission properties of\nthese systems [8{11] make them promising candidates\nfor achieving high-quality spin-photon interfaces and for\nproducing large-scale multi-photon entangled states [12{\n15]. The deterministic generation of these multi-photon\nentangled states has been demonstrated experimentally\nusing the dark excitonic states of QDs [16].\nWhile optically controlled quantum dot spins o\u000ber a\nwide range of technological possibilities, hyper\fne (HF)\ninteractions between the con\fned spin and its surround-\ning nuclear spin bath have been a major impediment.\nThis interaction is the main source of decoherence in\nthese systems, and it also causes spectral wandering\nand inhomogeneities in quantum dot ensembles, aspects\nthat have been researched extensively over the past two\ndecades [17{52]. However, many works have shown that\nthe state of the bath, and consequently its deleterious\ne\u000bects, can be in\ruenced by driving the electron spin.\nFor example, several experiments have shown that driv-\ning can generate dynamic nuclear polarization (DNP), an\ne\u000bect that has been observed in self-assembled QDs [53{\n64] and also in other systems such as gated QDs [65{\n68], quantum wires [69] and in bulk materials [70, 71],\n\fndings that have been supported by a number of the-\nory works [72{84]. An important example of DNP in\nself-assembled QDs is the mode-locking experiment of\nRef. [54], where an ensemble of QD electron spins be-\ncomes synchronized with a periodic train of optical pulses\n\u0003avezva@vt.edu\nyefbarnes@vt.eduas a consequence of DNP. Continuous-wave laser driving\nof the electron has been shown to create DNP in QDs as\nwell, leading to interesting phenomena such as the line-\ndragging e\u000bect, i.e. the locking of an optical QD transi-\ntion to the frequency of the laser [59, 61, 62, 76, 85]. Ow-\ning to the long coherence times of nuclear spins, DNP has\nbeen proposed for applications such as quantum memo-\nries [86, 87], which has recently been demonstrated ex-\nperimentally [88].\nAlthough most of the fully quantum mechanical the-\noretical treatments of the hyper\fne decoherence prob-\nlem allow for nuclei with spin greater than 1/2 [22{\n24], studies of the driven, hyper\fne-induced genera-\ntion of DNP have mostly focused on spin 1/2 nuclei\nto reduce the computational complexity of the problem\n[54, 72, 74, 81, 82, 89]. The latter works typically rely on\neither stochastic equations or rate equations to solve for\nthe nuclear polarization distribution. While solving the\nfeedback problem for spin 1/2 nuclear baths can yield\nqualitative insights about DNP experiments, the quan-\ntitative accuracy of such models is limited by the fact\nthat the most commonly studied semiconductor QDs are\nin materials such as InAs or GaAs, which contain nu-\nclei of spin I >1=2. In addition to arti\fcially reducing\nthe size of the bath Hilbert space, assuming spin 1/2 nu-\nclei also ignores e\u000bects such as quadrupolar interactions,\nwhich are only present for I > 1=2. There do exist a\nfew theoretical works that allow for I >1=2 [18, 39, 73].\nSpeci\fcally, Huang and Hu [18] studied DNP arising from\nhyper\fne interactions with the spin 3/2 arsenic nuclei\nin InGaAs by making use of Fermi's golden rule; how-\never, only qualitative agreement with experiment was\nachieved due to the need to introduce phenomenologi-\ncal parameters. Yang and Sham [73] presented a gen-\neral framework for nuclei of arbitrary total spin by uni-\nfying the stochastic and rate-equation approaches. In\nthis work they focused on a drift feedback loop (which\nallows for a possible bias in nuclear spin-\rip processes)\nand obtained a Fokker-Planck equation for the polariza-\ntion of the bath. Although this framework captures line-arXiv:2012.07227v2  [cond-mat.mes-hall]  17 Dec 20202\ndragging and other DNP phenomena seen in experiments,\nit has only been established for continuous-wave driving,\nand so it is not immediately applicable to experiments\nwith periodic driving such as the mode-locking experi-\nment of Ref. [54]. Theoretical works that have speci\fcally\nfocused on mode-locking type experiments have either\nassumedI= 1=2 nuclear baths [72, 74] or utilized semi-\nclassical methods [37, 38]. While such approaches have\nbeen successful in reproducing qualitative features seen in\nexperiments including dynamic nuclear polarization and\nmode-locking, it remains an outstanding challenge to de-\nvelop a more quantitatively accurate description of the\ndriven electron-nuclear spin system. Allowing for higher\nspin is also important for capturing additional qualitative\nbehavior that can arise from quadrupolar interactions.\nIn this paper, we develop a quantum, non-perturbative\nframework to solve the dynamics of an optically driven\nelectron spin coupled to a bath of I >1=2 nuclear spins.\nWe focus on DNP feedback mechanisms that arise from\ndriving the electron with a periodic train of optical pulses\nwhile it is subject to hyper\fne interactions with a nuclear\nspin bath, as in the mode-locking experiment [53, 54].\nHere, we also consider the e\u000bect of quadrupolar interac-\ntions. To compute DNP and its e\u000bect on the evolution of\nthe electron spin, we use an approach based on dynamical\nmaps and kinetic equations introduced in Refs. [72, 74],\nbut, importantly, we generalize the formalism to higher\nnuclear spin and treat the problem non-perturbatively.\nOur framework provides a self-consistent description of\nthe feedback loop between the driven electron and DNP.\nWe compute the nuclear spin polarization distribution\nand its in\ruence on the electron spin evolution for spin\n1 and spin 3/2 baths and compare the results to the\nI= 1=2 case. Our approach is able to treat bath sizes of\nup to thousands of nuclear spins in the I= 1=2 andI= 1\ncases, and up to several hundred spins in the I= 3=2\ncase. Although evidence of mode-locking is seen in all\nthree cases, we \fnd that quadrupolar interactions act to\nsuppress mode-locking for I >1=2, especially when the\nangle between the principal strain axis and the applied\nmagnetic \feld is large. We also \fnd that while HF inter-\nactions can produce a signi\fcant bath polarization that\ngrows linearly with the number of nuclei for I > 1=2,\nquadrupolar interactions work to counteract this buildup\nof DNP. We further show that the relative importance of\nquadrupolar e\u000bects grows as the magnitude of the applied\nmagnetic \feld is increased. The competition between HF\nand quadrupolar interactions imprints clear signatures in\nthe steady-state electron spin evolution, providing an ex-\nperimental tool to measure the strength of quadrupolar\ncouplings in a QD. Our results show that accounting for\nhigher nuclear spin is important not only for quantita-\ntive accuracy, but also for capturing important qualita-\ntive features of the DNP process in driven QD systems.\nThe paper is structured as follows. In Sec. II, we\npresent an overview of our framework and brie\ry describe\neach step of the calculation. In Sec. III, we lay out the\napproach in detail for arbitrary nuclear spin Iand con-\nSteady stateof electronicspin:S(1)eJoint evolution:S(1)e⌦Sn\nFlip rateswji:i!jKinetic equation:ddtP(m)=K[P(m)]Bath polarizationP(m)FeedbackAdding a singlenuclear spin\nMultinucleareffectsFIG. 1. Schematic depiction of the self-consistent formalism\nwe use to model DNP with feedback. We exploit a hierarchy of\ntimescales to \frst solve for the joint evolution of the electron\ncoupled to a single nuclear spin. Under a Markovian approxi-\nmation, the electron spin state is reset after each drive period.\nThe resulting nuclear spin evolution yields nuclear spin-\rip\nrates that are then fed into a kinetic equation governing the\ndynamics of the multi-nuclear spin polarization distribution.\nThe \rip rates depend on the e\u000bective electron spin precession\nfrequency, including the Overhauser \feld contribution for self-\nconsistency. The solution to the kinetic equation is then used\nto update the electron steady state, closing the feedback loop.\nstruct the equations that govern DNP for I= 1=2, 1, and\n3/2 nuclear spin baths. We present an analytical solution\nfor the steady-state nuclear spin polarization distribution\nforI= 1=2. In Sec. IV, we numerically compute steady-\nstate polarization distributions for I= 1 and 3/2 and\ncompare the results to the I= 1=2 solution for various\nparameter choices. We also study the e\u000bect of DNP on\nthe electron spin evolution. We conclude in Sec. V.\nII. OVERVIEW OF THE THEORETICAL\nFRAMEWORK\nBefore we describe our approach in detail, we \frst\ngive an overview of the general strategy and main in-\ngredients. This will hopefully better orient the reader\nfor what follows and also highlight the generality of the\napproach, which could potentially be adapted to other\ntime-dependent many-body problems. We still, how-\never, frame this overview in the context of the electron-\nnuclear problem in QDs for the sake of concreteness and\nto make it readily apparent how the discussion here \fts\nwith the rest of the paper. Our framework is summarized\nin Fig. 1. It closely follows the approach introduced in\nRefs. [72, 74], but we outline each step in detail to keep\nthe discussion self-contained and to make it clearer which\nparts must be modi\fed to allow for higher nuclear spin.\nOur focus in this work is on developing a theory that3\ndescribes QD experiments in which a single electron is pe-\nriodically pumped by a train of optical pulses [54, 55, 90].\nEach pulse excites the electron to a trion state (a bound\nstate of an electron and an exciton), which then decays\nback to the electronic ground state manifold via sponta-\nneous emission. The underlying physical mechanism be-\nhind the formation of DNP can be understood as follows.\nImagine that the electron spin starts in a pure (polarized)\nstate and the nuclear spins are in a totally mixed (unpo-\nlarized) state. The HF interaction then transfers angular\nmomentum from the electron onto the nuclei, creating\nDNP. In the absence of driving, this would lead to only\na modest nuclear spin polarization, and this polariza-\ntion would be short-lived because it would eventually be\ntransferred back to the electron via the HF interaction.\nHowever, the laser pulses periodically reset the electron\nspin to a polarized state, enabling a net transfer of angu-\nlar momentum from the laser, through the electron, and\nonto the nuclei. It is this transfer process that we aim to\ndescribe with our framework.\nWe are dealing with a system that is both open and\ndriven. An e\u000ecient way to treat non-unitary evolution is\nto use dynamical maps [72, 74, 91{93]. In this approach,\nthe non-unitary evolution of a system from an initial state\n\u001ato a \fnal state \u001a0is implemented by applying a set of\noperators and summing the results:\n\u001a0=X\nkEk\u001aEy\nk: (1)\nThe operators Ekare known as Kraus operators, and\nthey constitute a generalization of the usual unitary op-\nerators that evolve closed quantum systems to the case\nof non-unitary evolution in open systems. The condi-\ntionP\nkEy\nkEk= 1ensures that the trace of the density\nmatrix is always unity. The advantage of Kraus opera-\ntors is that they allow one to incorporate e\u000bects due to\nthe transient occupation of excited states using opera-\ntors that live purely in the ground space of the system.\nIn the present problem, we use these operators to de-\nscribe the e\u000bect of each optical pulse on the electron spin\nstate. The entire process of optical excitation, subse-\nquent decay, and rotation is captured by an appropriate\nset of Kraus operators (given in the next section) without\nhaving to explicitly include excited states or a photonic\nbath into the formalism. The dynamical map description\nworks well so long as the population returns regularly to\nthe electron spin ground states, as is the case for the\nperiodic driving used in the mode-locking experiments.\nGiven a set of Kraus operators that describe the evolu-\ntion of the driven electron, the next step in our theoretical\nframework is to \fnd the steady-state of the electron spin.\nOf course, we are interested in the case where the electron\nspin is coupled to a nuclear spin bath through HF interac-\ntions (which are described in detail in the next section)\nwhile it is being driven. Under the condition that the\nelectron is being pumped fast enough (which indeed is\nthe case for the mode-locking experiments [54]), the elec-\ntron reaches its steady state on a much faster timescalecompared to the electron-nuclear interaction dynamics\nand the electron spin decoherence time. This allows us\nto use a Markovian approximation in which we \frst solve\nfor the driven electron steady state and then incorporate\nthe e\u000bects due to the electron-nuclear couplings on top\nof this.\nTo bring the nuclei into the framework, we \frst solve\nfor the joint evolution of one nuclear spin coupled to the\ndriven electron spin. Although the HF interaction gen-\nerates unitary dynamics, this is disrupted periodically\nby the pulses, and this in turn leads to an e\u000bective non-\nunitary dynamical map for the nuclear spin that depends\non the electron steady state under the Markovian ap-\nproximation. We extract nuclear spin-\rip rates from this\ne\u000bective nuclear spin evolution operator; these rates pro-\nvide information about the movement of population be-\ntween the di\u000berent nuclear spin levels.\nWe calculate the steady state of the entire nuclear spin\nbath using a rate equation that depends on the spin-\n\rip rates obtained from the single-nucleus solution. A\ncritical step is that we build in self-consistent system-\nenvironment feedback by modifying the \rip rates. To un-\nderstand this, we \frst need to describe the Overhauser ef-\nfect [94], which is the main feedback mechanism between\nthe electron and nuclei. A polarized nuclear spin bath\nacts as an e\u000bective magnetic \feld and therefore shifts\nthe Zeeman frequency of the electron. However, the in-\nteraction between the electron and the nuclear spin bath\nis reciprocal; not only will the state of the electron change\nunder the Overhauser \feld, but the nuclear spins will also\nbe a\u000bected by the Knight \feld [95], i.e., the e\u000bective mag-\nnetic \feld due to polarization of the electron. The Knight\n\feld is given by the electron steady state spin vector, and\nso it enters into the nuclear spin \rip rates, as explained\nabove. The electron steady state (and hence the Knight\n\feld) in turn depends on the total magnetic \feld, which\nincludes the Overhauser \feld due to nuclear polarization.\nThese interdependencies constitute a complete feedback\nloop that must be treated self-consistently. We do this\nby making the nuclear spin-\rip rates depend on the net\nnuclear polarization of the bath. The steady-state of the\nrate equation then gives the polarization distribution of\nthe nuclear spin bath with feedback included. Finally, we\nuse this nuclear polarization distribution to perform the\nOverhauser shift on the Zeeman frequency of the elec-\ntron and update the nuclear-bath-averaged electron spin\nsteady-state self-consistently.\nThe framework we have just outlined can be thought\nof as a self-consistent dynamical mean-\feld approach. In\nthe following section, we describe each step of our for-\nmalism as it applies to the periodically driven electron-\nnuclear problem in full detail. Our method is quite gen-\neral and can be applied to baths of any nuclear spin. We\nfocus on the cases I= 1=2, 1, and 3/2 to illustrate the\nvarious steps.4\n|¯Ti\n|¯xi\u0000\u0000 \u0000e|Ti\n|xi\u0000+ \u0000e\n11BOptical axis: xMagnetic field: z\nFIG. 2. The relevant level structure in the mode-locking ex-\nperiments.jxiandj\u0016xiare the electron spin states along the\noptical axis. These states are coupled by an external magnetic\n\feld along the zdirection. Circularly polarized light excites\nthe ground electron spin states to excited trion levels jTiand\f\f\u0016T\u000b\nwith angular momentum projections +3 =2 and\u00003=2, re-\nspectively. The selection rules are such that each ground state\ncouples to only one excited state. The trion states decay via\nspontaneous emission with rate \re. In this work, we focus on\nleft-circularly polarized driving.\nIII. SELF-CONSISTENT DYNAMICAL\nMEAN-FIELD FORMALISM AND RESULTS\nThe full Hamiltonian of the nuclear spin bath and the\ndriven electron is given by\nH(t) =H0;e+H0;n+Hc(t) +Hres+HHF+HQ:(2)\nHere,H0;edescribes the electronic degrees of freedom in\nthe QD in the absence of driving:\nHe;0=!e^Sz+!\u0016Tj\u0016Tih\u0016Tj; (3)\nwhere!eis the electron spin Zeeman frequency, ^Szis the\nspin operator in the electronic ground space, and !\u0016Tis\nthe energy of the trion state j\u0016Ti. We take the magnetic\n\feld to be oriented along the zdirection, while the optical\naxis lies in the xdirection (see Fig. 2). We neglect the\nsecond trion level jTiinH0;ebecause it is not excited by\nthe laser polarization we are considering. This driving is\ndescribed by the Hamiltonian\nHc(t) = \n(t)j\u0016xih\u0016Tj+h:c:; (4)\nwhere we assume the drive laser is left-circularly polar-\nized (red arrow in Fig. 2) with periodic temporal pro\fle\n\n(t+TR) = \n(t), so that each pulse couples the electron\nspin statej\u0016xito the trion state\f\f\u0016T\u000b\n. The latter decays via\nspontaneous emission with rate \re. This process arises\nfrom interactions with a photonic bath, which is repre-\nsented by the term Hres. We do not give an explicit ex-\npression for this term as it is not explicitly considered in\nwhat follows. The Zeeman splitting of the nuclear spins\nis given by H0;n=!nP\ni^Ii\nz.\nThe HF interaction is given by the contact term:\nHHF=NX\ni=1Ai^Sz^Ii\nz+NX\ni=1Ai=2(^S+^Ii\n\u0000+^S\u0000^Ii\n+);(5)whereNis the number of nuclei that interact appre-\nciably with the electron. The \frst term is referred to\nas the Overhauser term, and it gives rise to an e\u000bec-\ntive magnetic \feld seen by the electron spin in the case\nof nonzero nuclear spin polarization. The second term\ngenerates \rip-\rop interactions under which the electron\nspin \rips with a nuclear spin. These terms are responsi-\nble for transferring angular momentum from the electron\nonto the nuclei, while the Overhauser term is the pri-\nmary mechanism for feedback between the nuclear spin\npolarization and the electron spin evolution. The HF\ncouplingsAiare determined by the magnitude of the elec-\ntronic wave function at the location of the nuclear spin\nIi. However, on timescales short compared to N=A\u0018\u0016s,\nwhereAis the total HF interaction energy, the variations\nin these couplings do not signi\fcantly a\u000bect the electron\nspin evolution [96]. Here, we focus on fast optical driving\nwhere the electron reaches a steady state over a timescale\nof about 100 ns [74], which allows us to make the \\box\nmodel\" approximation in which all the HF couplings are\ntaken equal: Ai=A\u0011A=N[96, 97].\nThe quadrupolar interaction is given by [98, 99]\nHQ=NX\ni=1\u0017i\nQ\n2\u0012\n^Ii\nz02\u0000I(I+ 1)\n3\u0013\n: (6)\nThis interaction occurs due to the coupling of the nuclear\nquadrupole moment to electric \feld gradients caused by\nstrain in the semiconductor lattice, and it is only present\nforI > 1=2. The presence of quadrupolar interac-\ntions has lead to striking phenomena in various types\nof experiments conducted in QDs. A few examples in-\nclude the anomalous Hanle e\u000bect [46] and suppression of\nspin di\u000busion [44]. Line-dragging phenomena have also\nbeen associated with the presence of quadrupolar inter-\nactions [61, 62, 73]. The coupling strength \u0017Qis referred\nto as the nuclear quadrupole resonance frequency, which\nis estimated to be around 2.8 MHz for As [44]. The\nquadrupole resonance frequency generally depends on the\nlocal strain in the vicinity of each nuclear spin, and so\nit generally varies across the material. Here, we assume\nthat the strain remains roughly constant over the QD,\nand so we take all the frequencies to be equal: \u0017i\nQ=\u0017Q.\nThe operator ^Iz0in Eq. (6) is the component of the nu-\nclear spin operator along the principal axis of the electric\n\feld gradient. Our focus will be on the case of QDs with\ncylindrical symmetry in which the electric \feld gradient\nmakes an angle \u0012with the magnetic \feld. Therefore, we\nhave ^Iz0=^Izcos\u0012+^Ixsin\u0012, which then gives [98]:\nHQ=\u0017Q\n2NX\ni=1h\n(^Ii\nz)2cos2\u0012\u0000I(I+ 1)\n3\n+ (^Ii\nz^Ii\nx+^Ii\nx^Ii\nz) sin\u0012cos\u0012+ (^Ii\nx)2sin2\u0012i\n:(7)\nWhen\u0012= 0,HQcreates non-uniform energy spacings\nbetween the nuclear spin levels. For \u00126= 0,HQhas the\nadditional e\u000bect of driving \u0001 mI=\u00061 and \u0001mI=\u000625\nnuclear spin-\rip transitions, where mIis the eigenvalue\nof^Iz. Notice that the rate for \u0001 mI=\u00061 transitions\nis maximal at \u0012=\u0019=4, while the rate for \u0001 mI=\u00062\ntransitions is largest for \u0012=\u0019=2, which is also the value\nof\u0012where the non-uniformity in the energy level spacings\nis zero. Thus, we see that the role of HQchanges as \u0012\nvaries from 0 to \u0019=4, and from \u0019=4 to\u0019=2. Because\nHQis\u0019-periodic in \u0012, it su\u000eces to focus on the range\n0\u0014\u0012\u0014\u0019=2.\nIn the following subsections, we carry out the various\nsteps of the formalism as outlined above in Sec. II. In\nSec. III A, we introduce the Kraus operators that describe\nthe optical pumping process and subsequent spontaneous\nemission generated by each pulse. In Sec. III B, we then\ncombine these with the Larmor precession between pulses\nto construct a dynamical map that evolves the electron\nspin vector over one complete driving cycle (with du-\nrationTR). This dynamical map is used to obtain the\nsteady state of the electron spin vector. In Sec. III C,\nwe derive a dynamical map that describes the e\u000bective\nevolution of a single nuclear spin hyper\fne-coupled to\nthe electron and subject to quadrupolar e\u000bects, where\nwe use the Markovian approximation to freeze the elec-\ntron in its steady state as described above. We use this\ndynamical map to compute the nuclear spin-\rip rates\nin Sec. III D. We obtain non-perturbative analytical ex-\npressions for these \rip rates for an I= 1=2 bath and for\nI= 1 andI= 3=2 baths in the case of no quadrupolar\ninteractions. Results with the quadrupolar interaction\nincluded are obtained numerically. We take into account\nmultinuclear e\u000bects in Sec. III E by constructing kinetic\nequations for I= 1=2, 1, and 3/2 that govern the time\nevolution of the nuclear spin polarization distribution of\nthe entire bath. In these equations, we include Over-\nhauser shifts in the \rip rates to incorporate dynamical\nfeedback e\u000bects between the electron and nuclei. The\nsteady states of these equations describe the DNP that\nis created through the combination of optical pumping\nof the electron and hyper\fne \rip-\rops. These equations\nare then solved in various cases in Sec. IV, where we also\ndescribe how the resulting polarization distributions in\nturn modify the evolution of the electron spin.\nA. Kraus operators for optical pumping of the\nelectron\nThe existence of a hierarchy of timescales in mode-\nlocking experiments allows us to \frst solve for the elec-\ntron spin dynamics without having to include nuclear\nspin e\u000bects. This is due to the fact that the nuclear\nspin dynamics are slow compared to those of the elec-\ntron. Given that the nuclear spins are the main source of\ndecoherence for the electron, this means we can also ne-\nglect electron spin decoherence e\u000bects. In addition, the\noptical pumping and spontaneous emission are fast com-\npared to the pulse period, \reTR\u001d1, which ensures that\nthe excited population returns fully to the ground statebefore the next pulse comes. This allows us to treat the\nevolution of the electron over each period in terms of a\ndynamical map that acts only on the electron spin ground\nstate subspace, as in Eq. (1).\nThe Kraus operators, Ek, that make up the dynami-\ncal map can be found by explicitly computing the non-\nunitary part of the evolution for an arbitrary initial den-\nsity matrix and comparing the initial and \fnal density\nmatrices. To compute the non-unitary part of the evo-\nlution due to the sequence of pulses Hc(t), we only need\nthe electronic parts of the full Hamiltonian in Eq. (2):\nHe(t) =H0;e+Hc(t). The fact that the pulse is much\nshorter than the spin precession period allows us to ignore\nthe precession during the action of the pulse. Therefore\nj\u0016xiand\f\f\u0016T\u000b\ncan be considered as an e\u000bective two-level\nsystem, where the evolution operator due to the pulse in\nthejxi,j\u0016xi,\f\f\u0016T\u000b\nbasis is\nUp=2\n41 0 0\n0u\u0016x\u0016x\u0000u\u0003\n\u0016T\u0016x\n0u\u0016T\u0016xu\u0003\n\u0016x\u0016x3\n5: (8)\nAfter the pulse, a fraction ju\u0016T\u0016xj2of the population re-\nmains in the trion state. We can describe the decay of\nthis population due to spontaneous emission using the\nLiouville-von Neumann equation with appropriately cho-\nsen Lindblad operators L: _\u001a=i[\u001a;H 0;e]+L(\u001a), where the\n\frst term includes the Larmor precession of the ground\nspin states during the decay. Solving this equation for\nan arbitrary initial state then yields the following Kraus\noperators in thejxi,j\u0016xibasis [74]:\nE1=\u0014\n1 0\n0q\u0015\n; E 2=\u0014\n0a1\n0\u0000a2\u0015\n; E 3=\u0014\n0 0\n0\u0014\u0015\n;(9)\nwhereq=u\u0016x\u0016x\u0011qoei\u001e,a1=!ep\n(1\u0000q2o)=2(4\r2e+!2e),\na2 =i\rep\n2p\n(1\u0000q2o)=(4\r2e+!2e), and\u0014 =p\n1\u0000q2o\u0000a2\n1\u0000ja2j2. These Kraus operators guar-\nantee the unity of the trace of the density matrix by\nsatisfyingP\nkEy\nkEk= 1. The parameter qoquanti\fes\nthe amount of population remaining in the spin state j\u0016xi\nafter the pulse is applied, and \u001eis the angle about the x\naxis by which the pulse rotates the electron spin. These\ntwo parameters can be computed given a speci\fc pulse\nshape, but in the following we leave these parameters\narbitrary.\nB. Electron spin steady state\nWe can use the Kraus operators from above to compute\nthe electron spin steady state. Rather than work directly\nwith the Kraus operators, it is more convenient to switch\nto the spin vector (SV) representation, especially since\n\fnding the steady state requires applying the Kraus op-\nerators an in\fnite number of times. In general, a SV S\ntransforms under non-unitary evolution as follows:\nS0=YS+K; (10)6\nwhereYis a matrix that generally both rotates and\nshrinks the SV, while Kcorresponds to the non-unital\npart of the evolution (i.e., a loss or gain of population in\nthe subspace described by S). IfKis nonzero, then a\nnontrivial steady state is possible. As shown in Ref. [74],\nfor spin 1/2 these quantities can be obtained from the\nKraus operators using the following formulas:\nKi=1\n2TrX\nk^\u001biEkEy\nk; (11)\nYij=1\n2TrX\nk^\u001biEk^\u001bjEy\nk; (12)\nwhere the ^\u001biare Pauli matrices. In the case of the mode-\nlocking experiment, the Kraus operators Ekevolve the\nelectron spin over one period, that is, they include both\nthe non-unitary dynamics ( Ek) generated by a pulse and\nalso the unitary precession under the magnetic \feld overtimeTR:Ek=Eke\u0000i!eTR^Sz. To \fnd the steady state,\nit is convenient to combine both YandKinto a single\n4\u00024 matrix:\nYe=2\n641 0 0 0\nKxYxxYxyYxz\nKyYyxYyyYyz\nKzYzxYzyYzz3\n75; (13)\nwhere the evolution of the electron SV over one period\nis now given byS0\ne=YeSe. Here, the \frst component\nof the 4-component SV Seis always \fxed to 1, while the\nremaining three components constitute the usual spin 1/2\nSV. In this representation it is easy to see that the steady\nstateSss\ne= (1;Sss\ne;x;Sss\ne;y;Sss\ne;z) is the eigenvector of 1\u0000Ye\nwith eigenvalue zero. Transforming the Kraus operators\nof Eq. (9) from the xbasis to the zbasis, plugging the\nresult into Eq. (13), and computing the null vector of Ye\nleads to the following steady state electron SV [74]:\nSss\ne;x=a1(a1qo(qo\u0000cos\u001e) cos (!eTR)\u0000ia2(qocos\u001e\u00001) sin (!eTR)\u0000a1qocos\u001e+a1)\n(a2\n1+q2o\u00001) cos (!eTR)\u0000a1qocos\u001e[ia2sin (!eTR) +a1cos (!eTR) +a1] +ia1a2sin (!eTR) + (a2\n1\u00001)q2o+ 1;\nSss\ne;y=a1(a1qo(cos\u001e\u0000qo) sin (!eTR)\u0000ia2(qocos\u001e\u00001) (cos (!eTR)\u00001))\n(a2\n1+q2o\u00001) cos (!eTR)\u0000a1qocos\u001e[ia2sin (!eTR) +a1cos (!eTR) +a1] +ia1a2sin (!eTR) + (a2\n1\u00001)q2o+ 1;\nSss\ne;z=a1qosin\u001e(a1sin (!eTR)\u0000ia2(cos (!eTR)\u00001))\n(a2\n1+q2o\u00001) cos (!eTR)\u0000a1qocos\u001e[ia2sin (!eTR) +a1cos (!eTR) +a1] +ia1a2sin (!eTR) + (a2\n1\u00001)q2o+ 1:\n(14)\nThese are the components of the electron SV immedi-\nately after each pulse. The steady state at other times\nduring the driving period can be obtained by rotating\nthis vector about the zaxis by angle !eTR(to account\nfor the Larmor precession).\nC. E\u000bective dynamical map for one nuclear spin\nNow that we have the electron spin steady state\n(Eq. (14)), we can proceed to construct an e\u000bective dy-\nnamical map for a single nuclear spin. We do this by \frst\nconstructing the evolution operator in the SV represen-tation that describes the joint evolution of the electron\nand nuclear spins over one driving period. We then apply\nthe Markovian approximation and reset the electron spin\nto its steady state at the end of the period. Tracing out\nthe electron then leaves an e\u000bective dynamical map for\nthe nuclear spin.\nTo start, we de\fne the nuclear SV using a basis of\nHermitian matrices ^\u0015kof dimension 2 I+ 1, where k=\n1;:::; (2I+ 1)2. We choose the \frst 2 I+ 1 of these ma-\ntrices to be diagonal, each with a single nonzero compo-\nnent equal to one. The remaining 2 I(2I+ 1) matrices\neach have two nonzero components, and these matrices\nare purely real or purely imaginary. For example, in the\ncase ofI= 3=2, we have 16 basis matrices:\n^\u0015k;ab=\u000eak\u000ebk; k = 1:::4;\n^\u00155;ab=1p\n2(\u000ea1\u000eb2+\u000ea2\u000eb1);^\u00156;ab=\u0000ip\n2(\u000ea1\u000eb2\u0000\u000ea2\u000eb1);^\u00157;ab=1p\n2(\u000ea1\u000eb3+\u000ea3\u000eb1);\n^\u00158;ab=\u0000ip\n2(\u000ea1\u000eb3\u0000\u000ea3\u000eb1);^\u00159;ab=1p\n2(\u000ea1\u000eb4+\u000ea4\u000eb1);^\u001510;ab=\u0000ip\n2(\u000ea1\u000eb4\u0000\u000ea4\u000eb1);\n^\u001511;ab=1p\n2(\u000ea2\u000eb3+\u000ea3\u000eb2);^\u001512;ab=\u0000ip\n2(\u000ea2\u000eb3\u0000\u000ea3\u000eb2);^\u001513;ab=1p\n2(\u000ea2\u000eb4+\u000ea4\u000eb2);\n^\u001514;ab=\u0000ip\n2(\u000ea2\u000eb4\u0000\u000ea4\u000eb2);^\u001515;ab=1p\n2(\u000ea3\u000eb4+\u000ea4\u000eb3);^\u001516;ab=\u0000ip\n2(\u000ea3\u000eb4\u0000\u000ea4\u000eb3): (15)\nThese matrices are normalized such that Tr[ ^\u0015j^\u0015k] =\u000ejk. Denoting the nuclear spin density matrix as \u001an, the com-7\nponents of the nuclear SV Snare then given by\nSn;k= Tr[\u001an\u0015k]: (16)\nNote that the populations, \u001an;ii, are the \frst four com-\nponents ofSn. We will see that this feature simpli\fes the\nprocess of computing \rip rates.\nLet us denote the density matrix that describes the\ntotal electron-nuclear spin state at the beginning of a\ndriving period by %. We expand this in terms of an op-\nerator basis formed from tensor products of the nuclear\nspin operators ^\u0015kwith the electron spin Pauli matrices\n^\u001bj:\n^G(2I+1)2j+k= ^\u001bj\n^\u0015k; (17)\nwithj= 0;::;3,k= 1;:::;(2I+ 1)2, and where we de\fne\n^\u001b0= 12\u00022. We use this set of 4(2 I+ 1)2operators as a\nbasis for the SV of the joint system: S`= Tr(%^G`). This\nSV evolves over one driving period according to S0=YS,\nwhere the SV evolution operator Yis given by\nY``0=1\n2Trh\n^G`U^G`0Uyi\n; (18)\nwhereU= expf\u0000i(!e^Sz+!n^Iz+HN=1\nHF+HN=1\nQ)TRgde-\nscribes the joint evolution of the electron spin and single\nnuclear spin under precession and the HF and quadrupo-\nlar interactions. At this point, we invoke the Markovian\napproximation: Because the electron reaches its steady\nstate,Sss\ne, quickly compared to the timescales for nuclear\nspin and HF dynamics, we reset the electron SV to its\nsteady state value at the beginning/end of each period:\nS=Sss\ne\nSn. We then obtain an e\u000bective nuclear spin\ndynamical map,Yn, by acting with the full evolution op-\nerator,Y, on the tensor product Sss\ne\nSnand reading o\u000b\nthe coe\u000ecients of the components of the nuclear SV, Sn,from the resulting S0:\nYn;jk=d\ndSn;k[Y(Sss\ne\nSn)]j: (19)\nHere,j;k= 1;:::;(2I+ 1)2, that is, we only retain the\ncomponents ofS0that correspond to the basis opera-\ntors ^Gk= 12\u00022\n^\u0015k, i.e., the components that corre-\nspond to purely nuclear spin degrees of freedom. Note\nthat although the joint evolution operator Ydescribes\nunitary evolution, the nuclear spin dynamical map, Yn,\nimplements non-unitary evolution. This non-unitarity is\na consequence of the Markovian approximation, which is\nitself due to the non-unitary driving of the electron spin.\nD. Single-nucleus \rip rates\nWe can use the nuclear spin dynamical map, Yn, that\nwe found in the previous subsection to \fnd the \rip rates\nfor a single nuclear spin interacting with the electron\nspin. These \rip rates govern the movement of population\nfrom one nuclear spin state to another. Such processes\nare described by the following kinetic equation:\ndpm\ndt=X\nn6=mwm\nnpn\u0000X\nn6=mwn\nmpm; (20)\nwherepmis the population of level m, and wm\nnis the\nrate to \rip from state ntom, which in general di\u000bers\nfrom the rate to \rip from mton, wn\nm. Which transitions\nare allowed depends on the type of interactions present\nin the Hamiltonian. For instance, the HF \rip-\rop terms\nonly cause \u0001 mI=\u00061 transitions, while the quadrupolar\ninteraction also drives \u0001 mI=\u00062 transitions. We can\ncombine the rate equations (20) into a matrix equation.\nWe exemplify this in the I= 3=2 case, where we denote\nthe four statesj+3=2i,j+1=2i,j\u00001=2i,j\u00003=2i, by the\nshorthandf++;+;\u0000;\u0000\u0000g. The matrix equation is then\n_P=MP, whereP= (p++;p+;p\u0000;p\u0000\u0000), and\nM=2\n6664\u0000(w+\n+++ w\u0000\n+++ w\u0000\u0000\n++) w++\n+ w++\n\u0000 w++\n\u0000\u0000\nw+\n++\u0000(w++\n++ w\u0000\n++ w\u0000\u0000\n+) w+\n\u0000 w+\n\u0000\u0000\nw\u0000\n++ w\u0000\n+\u0000(w++\n\u0000+ w+\n\u0000+ w\u0000\u0000\n\u0000) w\u0000\n\u0000\u0000\nw\u0000\u0000\n++ w\u0000\u0000\n+ w\u0000\u0000\n\u0000\u0000(w+\n\u0000\u0000+ w\u0000\n\u0000\u0000+ w++\n\u0000\u0000)3\n7775: (21)\nIt is clear that this equation satis\fes the condition that\nthe sum of the components of the probability vector P\nshould be unity at all times. This is guaranteed by the\nproperty that the sum of the rows of Mvanishes.\nTo determine the \rip rates, we need to connect the\ngeneric kinetic equation, Eq. (20), to the nuclear spin\nevolution operator, Eq. (19), derived earlier. This can\nbe done by starting from the evolution over one drivingperiod:\nSn(t+TR) =YnSn(t): (22)\nThe fact that the nuclear spin evolution is much slower\nthan the driving period TRallows us to coarse-grain this\nequation to arrive at a continuous evolution equation:\nd\ndtSn=1\nTR(Yn\u0000 1)Sn: (23)8\n+-,-+0.00.20.40.60.81.0Flip rates(MHz)\n-+-+--70-60-50-40-30-4-2024\nMagnetization mFlip rate differences(kHz)+0,0+-0,0-0.00.51.01.5Flip rates(MHz)\n0+-+0-0-0--70-60-50-40-30-10-50510\nMagnetization mFlip rate differences(kHz)+++,+++-+,+----,---0.00.51.01.52.02.53.0Flip rates(MHz)\n+++-+++--------+-+--70-60-50-40-30-15-10-5051015\nMagnetization mFlip rate differences(kHz)(a)\n(b)(c)\n(d)(e)\n(f)\nFIG. 3. Single-nucleus spin-\rip rates for (a,b) I= 1=2, (c,d)I= 1, (e,f)I= 3=2 as a function of the magnetization mof the\nnuclear spin bath. Flip rates are shown in (a), (c), (e), while \rip rate di\u000berences are shown in (b), (d), (f). The parameter\nvalues areTR= 13:2 ns,NA= 10 GHz, N= 1000,!e0= 0:5 GHz,!n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2. For\n(c-f), we set the quadrupolar parameters to \u0017Q= 2:8 MHz and \u0012= 0. Only the nonzero \rip rates are shown.\nBecause we have de\fned Snsuch that its \frst four compo-\nnents are just the populations of the nuclear spin states,\nwe can identify this equation with _P=MP and there-\nfore read o\u000b the \rip-rate matrix components from the\nnuclear spin evolution matrix:\nMij=1\nTR(Yn\u0000 1)ij; i;j = 1:::2I+ 1:(24)\nThis allows us to read o\u000b the \rip rates from the nuclear\nspin dynamical map. It is worth noting that Yncontains\nnot only terms that mix the populations of the di\u000berent\nnuclear spin levels but also terms that mix populations\nand nuclear spin coherences. Here, we are neglecting the\nin\ruence of the latter on the late-time populations. In\nnumerical simulations, we \fnd that these terms have a\nnegligible e\u000bect on the \rip rates. Moreover, they will\nbe further suppressed by nuclear spin dephasing [7, 32],\nwhich happens quickly compared to nuclear spin \rips.\nIn the case of I= 1=2 nuclei, the \rip rates can be\nobtained analytically following the above procedure:\nw\u0006=A2(1\u0006Sss\ne;z) sin2(TRp\n(!e\u0000!n)2+A2=2)\n2TR[(!e\u0000!n)2+A2];(25)\nwhere we use the shorthand notation w +\u0011w+1=2\n\u00001=2and\nw\u0000\u0011w\u00001=2\n+1=2. The \rip rates for I= 1 andI= 3=2\ncan also be obtained analytically in the case of zero\nquadrupolar coupling, \u0017Q= 0. In this case, there arefour nonzero \rip rates for I= 1:\nw0\n\u00001=A2(1 +Sss\ne;z) sin2(TR\n(1)\n\u0000=2)\nTR(\n(1)\n\u0000)2;\nw\u00001\n0=A2(1\u0000Sss\ne;z) sin2(TR\n(1)\n\u0000=2)\nTR(\n(1)\n\u0000)2;\nw+1\n0=A2(1 +Sss\ne;z) sin2(TR\n(1)\n+=2)\nTR(\n(1)\n+)2;\nw0\n+1=A2(1\u0000Sss\ne;z) sin2(TR\n(1)\n+=2)\nTR(\n(1)\n+)2; (26)\nwith\n\n(1)\n\u0006=p\n(!e\u0000!n)2\u0006A(!e\u0000!n) + 9A2=4; (27)\nwhile there are six nonzero \rip rates for I= 3=2:\nw++\n+=3A2(1 +Sss\ne;z) sin2(TR\n(3=2)\n+1=2)\n2TR(\n(3=2)\n+1)2;\nw+\n++=3A2(1\u0000Sss\ne;z) sin2(TR\n(3=2)\n+1=2)\n2TR(\n(3=2)\n+1)2;\nw+\n\u0000=A2(1 +Sss\ne;z) sin2(TR\n(3=2)\n0=2)\nTR(\n(3=2)\n0)2;\nw\u0000\n+=A2(1\u0000Sss\ne;z) sin2(TR\n(3=2)\n0=2)\nTR(\n(3=2)\n0)2;\nw\u0000\n\u0000\u0000=3A2(1 +Sss\ne;z) sin2(TR\n(3=2)\n\u00001=2)\n2TR(\n(3=2)\n\u00001)2;\nw\u0000\u0000\n\u0000=3A2(1\u0000Sss\ne;z) sin2(TR\n(3=2)\n\u00001=2)\n2TR(\n(3=2)\n\u00001)2; (28)9\nwith\n\n(3=2)\n\u0011 =p\n(!e\u0000!n)2+ 2\u0011A(!e\u0000!n) + 4A2:(29)\nIn the absence of quadrupolar interactions, only \u0001 mI=\n\u00061 transitions (i.e., transitions between adjacent spin\nlevels) are allowed, as follows directly from the form of\nthe HF \rip-\rop interaction. When the quadrupolar cou-\npling is nonzero, we can no longer obtain an analytical\nexpression for the \rip rates, but these are still easily ob-\ntained numerically by computing Ynfor speci\fc param-\neter values.\nFig. 3 shows the dependence of the \rip rates on the\nnet magnetization mof the entire nuclear spin bath for\nI= 1=2, 1, and 3/2. This dependence comes from the\nOverhauser e\u000bect in which nuclear spin polarization acts\nas an e\u000bective magnetic \feld seen by the electron spin.\nWe incorporate this e\u000bect by adding a magnetization-\ndependent shift to the precession frequency of the elec-\ntron:\nwj\ni(m) = wj\ni(!e!!e0+mA); (30)\nwhere!e0denotes the contribution to the precession fre-\nquency due purely to the external magnetic \feld, and\nwhere we use wj\ni(m) to denote the rate to \rip from state\nito statejin the presence of nuclear spin magnetization\nm. For nuclei of spin I, we can express this magnetiza-\ntion in terms of occupation numbers, N`, for each of the\nnuclear spin states:\nm=IX\n`=\u0000I` N`: (31)\nIn Fig. 3, results for zero quadrupolar angle, \u0012= 0, are\nshown in the I >1=2 cases. Even though the quadrupo-\nlar coupling is nonzero, \u0017Q>0, only \u0001mI=\u00061 transi-\ntions are permitted in this case because when \u0012= 0, the\nonly e\u000bect of the quadrupolar interaction is to modify\nthe energy splittings between nuclear spin levels, and so\nthe selection rules are still determined solely by the HF\ninteraction. We discuss the e\u000bect of nonzero \u0012below.\nSeveral salient features are evident in Fig. 3. First of\nall, the \rip rates are strongly peaked at magnetization\nm\u0019 \u0000!e0=A. In the spin 1/2 case, the precise loca-\ntion of the peak is the value of mat which the argument\nof the sine in Eq. (25) vanishes since the \rip rates are\nessentially given by squared sinc functions. For low to\nmoderate external magnetic \feld strengths and large N,\nthe terms involving !nandA2can be neglected, leaving\nm\u0019\u0000!e0=A. Similar statements hold for I= 1 and\nI= 3=2 in the absence of quadrupolar e\u000bects, as is clear\nfrom Eqs. (26) and (28). The fact that the \rip rates\nare maximal at m\u0019\u0000!e0=Acan be understood from\nenergy conservation: At these values, the e\u000bective Zee-\nman energy of the electron is almost zero, and thus so is\nthe energy mismatch between the electron and nucleus.\nThis in turn reduces the energy penalty for \rip-\rops, ac-\ncelerating the transfer of polarization. Conversely, theoverall decay of the \rip rates away from m\u0019!e0=Ais\ndue to the HF interaction becoming ine\u000ecient at over-\ncoming the large energy mismatch between the electronic\nand nuclear spin splittings.\nIt is also evident in Fig. 3 that the \rip rates vanish\nperiodically as a function of m. The periodicity is also\ncontrolled by the arguments of the sine functions in the\n\rip rates. These zeros correspond to values of !efor\nwhich complete \rip-\rops between the electronic and nu-\nclear spins occur|polarization is transferred back and\nforth between the electron and nucleus an integer number\nof times within a single drive period TR. Because there\nis no net polarization transfer, the \rip rate vanishes. For\nI > 1=2, the locations of these zeros depend on which\npair of adjacent spin levels we consider, although this de-\npendence fades away in the large Nlimit, where A!0.\nIn the next section, we show that these \rip-rate zeros\nplay a central role in the phenomenon of mode-locking.\nEach pair of \rip rates describing transitions between\nthe same two spin levels are almost equal [see panels (b),\n(d), (f) of Fig. 3]. As can be seen from Eqs. (25)-(28), the\ndi\u000berences of these \rip rates are proportional to Sss\ne;z(m),\nand this component of the electron steady state is sup-\npressed near m\u0019\u0000!e0=Abecause it is proportional to\n!e(see Eq. (14)). This is a re\rection of the fact that\nwhen!e= 0, the electron steady state becomes polar-\nized along the optical pulse axis (the xdirection), where\nit is no longer a\u000bected by the pulses and is thus sta-\nbilized. In the \fgure, we see that this combination of\naccelerated \rip-\rops and the suppression of Sss\ne;z(m) near\nm\u0019\u0000!e0=Aresults in \rip rate di\u000berences that are more\nthan two orders of magnitude smaller than the \rip rates\nthemselves.\nThe e\u000bect of a nonzero quadrupolar angle \u0012on the \rip\nrates is shown in Figs. 4 and 5 for I= 1 and 3/2, re-\nspectively. In the case I= 1, it is evident that \u0012has\na negligible e\u000bect on the \u0001 mI=\u00061 \rip rates. On the\nother hand, su\u000eciently large values of the angle, \u0012&\u0019=4,\ngive rise to \u0001 mI=\u00062 transitions that are not otherwise\npresent. Although the rates for these transitions are two\norders of magnitude smaller than those of the \u0001 mI=\u00061\ntransitions, they are still large enough to a\u000bect the polar-\nization distribution of the nuclear spin bath, as we show\nin Sec. IV. Similar but somewhat more prominent e\u000bects\nare evident for I= 3=2 in Fig. 5. Here, larger values of \u0012\nproduce small but noticeable changes in \u0001 mI=\u00061 \rip\nrates, signi\fcant \u0001 mI=\u00062 transition rates, and even\n\u0001mI=\u00063 transitions. A striking feature evident in both\nFigs. 4 and 5 is that the \rip rates for \u0001 mI=\u00062 transi-\ntions do not decay as mmoves away from m=\u0000!e0=A.\nThis is consistent with the fact that spin \rips caused by\nthe quadrupolar interaction do not require the electron\nand nuclear spin Zeeman energies to be equal. Unlike HF\nspin \rips, quadrupolar spin \rips depend weakly on the\nbath magnetization. On the other hand, the \u0001 mI=\u00063\n\rip rates are sensitive to m(see Fig. 5(b)), because these\narise from a higher-order process that combines HF and\nquadrupolar spin \rips.10\n\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-70-60-50-40-300.00.51.01.5\nMagnetization m0+(MHz)\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-70-60-50-40-30-10-505\nMagnetization m0+-+0(kHz)\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-70-60-50-40-300.000.010.020.030.04\nMagnetization m-+(MHz)(a)(b)(c)\nFIG. 4. Single-nucleus spin-\rip rates as a function of nuclear spin bath magnetization mforI= 1 and for di\u000berent values\nof the quadrupolar angle \u0012. (a) Flip rate for the \u0001 mI= 1 transitionj0i!j +1i. (b) Flip rate di\u000berence for the j0i$j +1i\ntransitions. (c) Flip rate for the \u0001 mI= 2 transitionj\u00001i!j +1i. The parameter values are TR= 13:2 ns,NA= 10 GHz,\nN= 1000,!e0= 0:5 GHz,!n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\n\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u000180.00.51.01.52.02.53.0-+(MHz)\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u000180.00.51.01.52.02.5---(MHz)\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-10-505101520-------(kHz)\n\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-70-60-50-40-300510152025\nMagnetization m--++(kHz)\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-70-60-50-40-300.000.020.040.060.080.100.12\nMagnetization m--+(MHz)\u0001=0\u0001=\u00012\u0001=\u00014\u0001=\u00018-70-60-50-40-30-0.4-0.20.00.20.40.60.8\nMagnetization m--+-+--(kHz)(a)\n(b)(c)\n(d)(e)\n(f)\nFIG. 5. Single-nucleus spin-\rip rates as a function of nuclear spin bath magnetization mforI= 3=2 and for di\u000berent values of\nthe quadrupolar angle \u0012. (a) Flip rate for the \u0001 mI= 1 transitionj\u00001=2i!j +1=2i. (b) Flip rate for the \u0001 mI= 3 transition\nj\u00003=2i!j +3=2i. (c) Flip rate for the \u0001 mI= 1 transitionj\u00003=2i!j\u0000 1=2i. (d) Flip rate for the \u0001 mI= 2 transition\nj\u00003=2i!j +1=2i. (e) Flip rate di\u000berence for the j\u00003=2i$j\u0000 1=2itransitions. (f) Flip rate di\u000berence for the j\u00003=2i$j +1=2i\ntransitions. The parameter values are TR= 13:2 ns,NA= 10 GHz, N= 1000,!e0= 0:5 GHz,!n=\u00000:5 MHz,\re= 0:5 GHz,\nq0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\nE. Kinetic equations for multi-nuclear spin\npolarization distributions\nIn this section, we use the \rip rates obtained in the\nprevious section to construct kinetic rate equations that\ngovern the evolution of the polarization distribution of\nthe entire nuclear spin bath. We do this for each of the\nthree values of nuclear total spin Iconsidered in this\nwork. Although the kinetic equation for I= 1=2 has been\ndiscussed in detail elsewhere [72, 74], here we present an\nanalytical solution to this equation that was not previ-\nously known. The kinetic equations for I= 1 and 3/2\nwill be solved numerically in the next section to obtain\nnuclear spin polarization distributions in these cases. De-\ntailed comparisons of the polarization distributions that\nresult in all three cases for various parameter values are\ngiven below in Sec. IV. In that section, these distributionsare then used to compute the e\u000bect on the electron spin\nevolution with and without quadrupolar interactions.\n1. Kinetic equation for spin I= 1=2nuclei\nThe polarization of a spin 1/2 nuclear bath in a de\f-\nnite con\fguration with occupation numbers N+andN\u0000\n(the number of spins in the j+1=2iandj\u00001=2istates,\nrespectively) is given by m= (N+\u0000N\u0000)=2. The total\nnumber of spins is N=N++N\u0000. Knowledge of the\npolarization mis su\u000ecient to determine the two occupa-\ntion numbers, N+andN\u0000. This in turn means that the\nprobability of each bath con\fguration is equal to the po-\nlarization probability distribution P(m). We may write\ndown a kinetic equation governing the dynamics of this11\ndistribution [72, 74]:\nd\ndtP(m) =\u0000X\n\u0006\u0014\nw\u0006(m)N\u00072m\n2\u0015\nP(m) (32)\n+X\n\u0006w\u0007(m\u00061)\u0014N\u00062m\n2+ 1\u0015\nP(m\u00061):\nA close look at this kinetic equation reveals that the\nright-hand side is comprised of two terms that are re-\nlated to each other by shifting m!m+ 1:\nd\ndtP(m) =F(m+ 1)\u0000F(m); (33)\nwhereF(m) =w\u0000(m)(m+N=2)P(m)\u0000w+(m\u00001)(\u0000m+\n1 +N=2)P(m\u00001). Therefore, in the steady state where\ndP(m)=dt= 0, we \fnd F(m) =F(m+ 1) = constant.\nSince we must have P(N+ 1) = 0, it follows that this\nconstant is zero. The equation F(m) = 0 then yields a\ntwo-term recursion relation [72, 74]:\nP(m) =N\u00002m+ 2\nN+ 2mw+(m\u00001)\nw\u0000(m)P(m\u00001): (34)\nThis relation can easily be solved iteratively starting from\nan arbitrary value for P(\u0000N) and then imposing the nor-\nmalization conditionP\nmP(m) = 1. This approach was\nused to produce numerical results for the polarization\ndistribution in Refs. [72, 74].\nHere, we obtain an analytical solution for P(m) by\nexploiting the explicit, non-perturbative expressions we\nobtained for the \rip rates in Eq. (25). First of all, an\nexpression for P(m) follows immediately from Eq. (34):\nP(m) =N\u00001mY\nk=1\u0000N=2N\u00002k+ 2\nN+ 2kw+(k\u00001)\nw\u0000(k)\n=N\u00001N!\n(N=2 +m)!(N=2\u0000m)!mY\nk=1\u0000N=2w+(k\u00001)\nw\u0000(k);\n(35)\nwhereNis a normalization factor. Next, we use the fact\nthat the two \rip rates only di\u000ber by the sign in front\nofSss\ne;z(m), which leads to a cascade of cancellations be-\ntween the numerator and denominator in the product.\nWe are left with\nP(m) =N\u00001\n(N=2 +m)!(N=2\u0000m)!mY\nk=1\u0000N=21 +Sss\ne;z(k\u00001)\n1\u0000Ssse;z(k)\n\u0002(!e0\u0000!n+Am)2+A2\nsin2(TRp\n(!e0\u0000!n+Am)2+A2=2);(36)\nwhere we have absorbed additional constants into N.\nThe \frst, combinatoric factor in P(m) corresponds to a\nGaussian-like envelope that quickly approaches a Gaus-\nsian asNincreases: [( N=2)!]2=[(N=2+m)!(N=2\u0000m)!]!e\u00002m2=NasN!1 . The second factor in Eq. (36) pro-\nduces sharp spikes at values of mthat correspond to the\nzeros of the \rip rates. These values of msatisfy\np\n(!e0\u0000!n+Am)2+A2\u00192\u0019p\nTR; (37)\nwherepis an integer. The concentration of proba-\nbility near these special values of mproduces mode-\nlocking: Nuclear polarization shifts the electron Zeeman\nfrequency to values where HF \rip-\rops stop transferring\npolarization between the electronic and nuclear spins.\nThis happens because an integer number of \rip-\rops oc-\ncur during each drive period. Using that !n\u001c!e0and\nassumingNis su\u000eciently large that A\u001c!e0, these val-\nues ofmcorrespond to the electron precession becoming\ncommensurate with the pulse train: !e=!e0+Am\u0019\n2\u0019p=TR, which is the primary signature of mode-locking\nseen in experiments [54].\nThe middle factor (the product) in Eq. (36) is primarily\nresponsible for the average magnetization of the nuclear\nspin bath,hmi=P\nmmP(m). This factor is also where\nadditional pulse parameters such as the rotation angle \u001e\nand the residual ground state population q0in\ruence the\npolarization distribution. If \u001eis equal to 0 or \u0019or ifq0\nis zero, then Sss\ne;z(k) = 0 for all k, in which case the \fnal\nfactor in Eq. (36) reduces to 1. In this case, the combi-\nnatoric factor, which is centered about m= 0, ensures\nthat the average magnetization will be small, hmi\u00190.\nOn the other hand, if \u001e6= 0 and the external magnetic\n\feld is su\u000eciently large, then hmican be signi\fcant, and\nits sign depends on the sign of \u001eand on the orientation\nof the external \feld. If \u001e>0, thenSss\ne;z(m) is more often\npositive than negative for m <\u0000!e0=A, which in turn\nmeans that1+Sss\ne;z(m\u00001)\n1\u0000Ssse;z(m)is biased toward values larger\nthan 1, and so the product grows as mincreases. Once\nmpasses\u0000!e0=A,Sss\ne;z(m) now tends to more negative\nvalues, and the product shrinks as mincreases. Thus,\nwe see that for \u001e>0, the product in Eq. (36) is peaked\natm\u0019\u0000!e0=A, and so the average magnetization will\nlie between 0 and \u0000!e0=A. On the other hand, if \u001e<0,\nthen the same reasoning leads to the conclusion that the\nproduct in Eq. (36) has a dip at m\u0019\u0000!e0=A, and thus\nthe net magnetization is driven away from this point and\nwill have a sign that coincides with that of !e0. These\nfeatures are borne out in plots of Eq. (36), as shown be-\nlow in Sec. IV.\n2. Kinetic equation for spin I= 1nuclei\nBefore we write down the kinetic equation for I= 1 nu-\nclei, we \frst introduce the notation we use to distinguish\ndi\u000berent bath con\fgurations. We denote the occupation\nnumbers of the three spin states by N\u00001,N0, andN1.\nThe bath polarization for a given con\fguration is then\nm= +1\u0002N1+ 0\u0002N0\u00001\u0002N\u00001. We see immediately\nthat there is an important di\u000berence compared to the12\nI= 1=2 case considered above: The polarization does\nnot uniquely specify a con\fguration of the bath. For in-\nstance, in the case of two I= 1 spins with m= 0, we can\nhave either N1= 1 =N\u00001andN0= 0 orN1= 0 =N\u00001\nandN0= 2. This is in contrast to the I= 1=2, where\neach value of mcorresponds to a unique con\fguration.\nAs the number of spins increases, the number and orders\nof such \\degeneracies\" grow quickly. Because the po-\nlarization does not uniquely specify a con\fguration, we\nmust combine it with one of the occupation numbers to\nuniquely label di\u000berent con\fgurations. We choose to use\nN0and express the probability of a given con\fguration\nbyP(m;N 0). Unlike in the spin 1/2 case, this quantity is\nnow distinct from the polarization probability distribu-\ntion; the latter is obtained by summing over all possible\nvalues ofN0that are consistent with the given value of\nm:\nP(m) =X\nN0P(m;N 0): (38)\nWe can write down a kinetic equation for P(m;N 0):\nd\ndtP(m;N 0) =F(m;N 0) +G(m+ 1;N0\u00001)\n\u0000G(m;N 0)\u0000F(m+ 1;N0+ 1);(39)\nwhere\nF(m;N 0) =\u0000w\u00001\n0P(m;N 0)N0 (40)\n+w0\n\u00001(m\u00001)P(m\u00001;N0\u00001)N\u0000(m\u00001;N0\u00001);\nG(m;N 0) =w0\n1P(m;N 0)N+(m;N 0)\n\u0000w1\n0(m\u00001)P(m\u00001;N0+ 1)(N0+ 1): (41)HereN\u0006(m;N 0)\u0011(1=2)(N\u0006m\u0000N0). In the kinetic\nequation above we have only considered the \u0001 mI=\u00061\ntransitions. Including transitions that change the an-\ngular momentum by more than 1 (for instance due to\nquadrupolar interactions) leads to additional terms not\nshown above. Such terms are illustrated for the case of\nI= 3=2 nuclei in the next section. Returning to the spin\n1 case, the steady state of the above kinetic equation,\nF(m;N 0)\u0000G(m;N 0) =F(m+1;N0+1)\u0000G(m+1;N0\u00001);\n(42)\ndoes not yield a recursion relation as in the I= 1=2 case.\nWe solve this equation (and its generalization for nonzero\nquadrupolar interactions) numerically in Sec. IV.\n3. Kinetic equation for spin I= 3=2nuclei\nWe again adopt the notation f++;+;\u0000;\u0000\u0000g to la-\nbel quantities associated with the four spin quantum\nnumbersmI=f+3=2;+1=2;\u00001=2;\u00003=2gof a spin 3/2\nnucleus. For a nuclear spin bath comprised of N=\nN+++N++N\u0000+N\u0000\u0000spins, the magnetization of the\nsystem (Eq. (31)) is m= (3N+++N+\u0000N\u0000\u00003N\u0000\u0000)=2.\nIn theI= 3=2 case, we need two more quantities in\naddition to mto uniquely label di\u000berent multi-spin con-\n\fgurations. We choose these to be N++andN\u0000\u0000. The\nremaining two occupation numbers are then determined\nby these three quantities for a \fxed total number of spins:\nN+=1\n2(2m+N\u00004N+++ 2N\u0000\u0000); (43)\nN\u0000=1\n2(\u00002m+N+ 2N++\u00004N\u0000\u0000): (44)\nThe probabilities P(m;N ++;N\u0000\u0000) that the nuclear spin\nbath is in the various con\fgurations labeled by m,N++,\nandN\u0000\u0000obey the following set of kinetic equations:\nd\ndtP(m;N ++;N\u0000\u0000) =F(m;N ++;N\u0000\u0000) +G(m;N ++;N\u0000\u0000) +H(m;N ++;N\u0000\u0000)\n+I(m;N ++;N\u0000\u0000) +J(m;N ++;N\u0000\u0000)\n\u0000F(m+ 1;N +++ 1;N\u0000\u0000)\u0000G(m+ 1;N ++;N\u0000\u0000\u00001)\u0000H(m+ 1;N ++;N\u0000\u0000)\n\u0000I(m\u00002;N ++\u00001;N\u0000\u0000)\u0000J(m+ 2;N ++;N\u0000\u0000\u00001); (45)\nwhere13\nF(m;N ++;N\u0000\u0000) = +w++\n+(m\u00001)P(m\u00001;N ++\u00001;N\u0000\u0000)N+(m\u00001;N ++\u00001;N\u0000\u0000)\n\u0000w+\n++(m)P(m;N ++;N\u0000\u0000)N++; (46)\nG(m;N ++;N\u0000\u0000) = +w\u0000\n\u0000\u0000(m\u00001)P(m\u00001;N ++;N\u0000\u0000+ 1)(N\u0000\u0000+ 1)\n\u0000w\u0000\u0000\n\u0000(m)P(m;N ++;N\u0000\u0000)N\u0000(m;N ++;N\u0000\u0000); (47)\nH(m;N ++;N\u0000\u0000) = +w+\n\u0000(m\u00001)P(m\u00001;N ++;N\u0000\u0000)N\u0000(m\u00001;N ++;N\u0000\u0000)\n\u0000w\u0000\n+(m)P(m;N ++;N\u0000\u0000)N+(m;N ++;N\u0000\u0000); (48)\nI(m;N ++;N\u0000\u0000) = +w\u0000\n++(m+ 2)P(m+ 2;N +++ 1;N\u0000\u0000)(N+++ 1)\n\u0000w++\n\u0000(m)P(m;N ++;N\u0000\u0000)N\u0000(m;N ++;N\u0000\u0000); (49)\nJ(m;N ++;N\u0000\u0000) = +w+\n\u0000\u0000(m\u00002)P(m\u00002;N ++;N\u0000\u0000+ 1)(N\u0000\u0000+ 1)\n\u0000w\u0000\u0000\n+(m)P(m;N ++;N\u0000\u0000)N+(m;N ++;N\u0000\u0000): (50)\n123456789101\n2\n3\n4\n5\n6\n7\n8\n9\n1012345678910\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n1 50 100 150 2002311\n50\n100\n150\n200\n2311 50 100 150 200231\n1\n50\n100\n150\n200\n231\nFIG. 6. Structure of the matrix Rde\fning the linear sys-\ntem of equations governing the steady-state solution of the\nmulti-nuclear kinetic equation for spin I= 1 for (left) N= 3\nspins and (right) N= 20 spins in the absence of quadrupolar\ninteractions.\nHere, we have included \u0001 mI=\u00061 and \u0001mI=\u00062\ntransitions. Although \u0001 mI=\u00063 transitions cannot\nbe directly driven by either the HF interaction or the\nquadrupolar interaction to \frst order in their respec-\ntive coupling strengths, they can potentially arise from\nhigher-order e\u000bects as we saw from the \rip rates in Fig. 5.\nNow that we have the kinetic equations governing the nu-\nclear polarization, the next step is to solve them.\nIV. NUCLEAR POLARIZATION\nDISTRIBUTION AND FEEDBACK\nA. Steady-state polarization distributions\nForI= 1 andI= 3=2, we solve the respective kinetic\nequations numerically to obtain steady-state polariza-\ntion distributions. This is done by \frst setting the time\nderivatives to zero:d\ndtP(m;N ++;N\u0000\u0000) = 0. The result-\ning algebraic equations are then collected together and\nwritten as a matrix Racting on a vector Vof the prob-\nabilitiesP(m;N ++;N\u0000\u0000) such thatRV= 0. Thus, thesteady-state polarization distribution is the unique null\nvector ofR. The matrixRdepends on the Overhauser-\nshifted \rip rates and occupation numbers for each con-\n\fguration. The linear dimension of this matrix is equal\nto the number of distinct multi-spin con\fgurations. For\nNspins of total spin I, the number of con\fgurations is\ngiven by the simplicial polytopic numbers\u0000N+2I\n2I\u0001\n. For\nI= 1=2, 1, and 3/2, this gives N+ 1, (N+ 1)(N+ 2)=2,\nand (N+ 1)(N+ 2)(N+ 3)=6, respectively. Therefore, in\nthe case of I= 1, we must compute the null vector of a\nmatrix that grows quadratically with the number of nu-\nclei, while for I= 3=2, we must do the same for a matrix\nthat grows like N3. The matrixRis quite sparse in both\ncases (see Fig. 6), especially in the absence of quadrupo-\nlar interactions. This allows us to employ the Arnoldi\nmethod to compute the steady-state polarization distri-\nbution for hundreds of spins with I= 3=2 and thousands\nof spins with I= 1.\nFig. 7 compares results for the steady-state nuclear\nspin polarization for N= 200 for all three values of I.\nIn theI > 1=2 cases, we set the quadrupolar angle to\nzero,\u0012= 0; however, the nonzero quadrupolar interac-\ntion\u0017Q>0 still modi\fes the energy splittings between\nthe nuclear spin levels. In all three cases, the polarization\ndistribution exhibits multiple narrow peaks at values of\nmthat correspond to the mode-locking frequencies, i.e.,\nthese values of mare such that !e0+Am = 2\u0019p=TR\nwherepis an integer (for an analytical derivation of\ntheI= 1=2 case see Section III E 1). As discussed in\nSec. III D, the \rip rates approximately vanish at these\nvalues ofm. (Note that the spacing of the peaks in Fig. 7\nis \fve times smaller than the spacing of the \rip-rate zeros\nin Figs. 3, 4, and 5 because this spacing is proportional to\n1=A=N=A, andNis \fve times smaller in Fig. 7.) The\nsteady-state probabilities P(m;N ++;N\u0000\u0000) are largest at\nthese magnetization values because they are multiplied\nby nearly vanishing \rip rates in the kinetic equations;\nthe probabilities must compensate for the smallness of\nthe \rip rates such that the product of the two is \fnite\nand comparable to terms of similar size in the kinetic\nequations. This trend can be seen explicitly from the an-14\n-40-20020400.00.10.20.30.40.50.6\nMagnetization mProbablity P(m)Spin-1/2<m>=2.99\n-60-40-2002040600.000.010.020.030.040.050.060.07\nMagnetization mProbablity P(m)Spin-1<m>=17.85\n-500500.0000.0050.0100.0150.0200.0250.0300.035\nMagnetization mProbablity P(m)Spin-3/2<m>=25.44Spin-1/2Spin-1Spin-3/2(a)(b)(c)\nFIG. 7. Steady-state nuclear spin polarization distribution of a bath with N= 200 nuclear spins for (a) I= 1=2, (b)I= 1,\nand (c)I= 3=2. The parameter values are TR= 13:2 ns,NA= 10 GHz, !e0= 0:5 GHz,!n=\u00000:5 MHz,\re= 0:5 GHz,\nq0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz. In the case of I= 3=2 andI= 1 the quadrupolar angle is \u0012= 0.\nI=1,<m>=5.15+0.06 N\nI=3/2,<m>= -1.23+0.13 N\n0 200 400 600 800 1000020406080100120140\nNuclear bath size NPolarization<m>I=1,<m>=3.26+0.076 N-9.39×10-6N2\nI=3/2,<m>= -1.27+0.13 N-4.14×10-6N2\n0 200 400 600 800 1000050100150\nNuclear bath size NPolarization<m>\nFIG. 8. Extrapolation of the average nuclear spin bath po-\nlarizationhmito larger bath sizes Nfor two values of total\nspin:I= 1 (red circles) and I= 3=2 (blue diamonds). The\npoints are obtained by solving the respective kinetic equa-\ntions, Eqs. (39) and (45). The lines are linear \fts. The param-\neter values are TR= 13:2 ns,NA= 10 GHz, !e0= 0:5 GHz,\n!n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\n\u0017Q= 2:8 MHz,\u0012= 0.\nalytical solution in the I= 1=2 case, Eq. (36), where it\nis evident that P(m) depends inversely on the \rip rates.\nIn Fig. 7, we see that this also occurs for I >1=2. For all\nvalues ofI, we can physically understand the formation\nof probability peaks at \rip-rate zeros as resulting from\nthe fact that, at these magnetization values, the joint\nelectron-nuclear spin evolution under the HF interaction\nbecomes commensurate with the driving pulses. Conse-\nquently, the pulses do not cause a net polarization trans-\nfer between the electron and nuclear spins. Thus, these\nvalues of the magnetization mprovide a point of stability\nin the electron-nuclear feedback mechanism. We also see\nfrom Fig. 7(a), and to some degree from Fig. 7(b), that\nthe polarization distribution is suppressed in the vicinity\nofm=\u0000!e0=A(which corresponds to m=\u000010 for the\nparameters used in the \fgure). This is due to the fact\nthat the \rip rates are largest near these magnetization\nvalues and therefore drive population away from thesevalues.\nAnother striking feature of the polarization distribu-\ntions in Fig. 7 is that the distributions for I >1=2 exhibit\nbroad envelopes in addition to the mode-locking peaks.\nThis is a consequence of the fact that there are multiple\ndistinct \rip rates for I >1=2, as shown in Eqs. (26) and\n(28). These \rip rates oscillate with !eat distinct frequen-\ncies that di\u000ber from each other by an amount propor-\ntional toA. Therefore, they do not all vanish at the same\nvalues of!e, dulling the sharpness of the mode-locking\npeaks. This e\u000bect becomes diminished at larger N, be-\ncause in this limit Adecreases, and all the \rip-rate zeros\napproach the values of mat which!e0+Am= 2\u0019p=TR,\nwherepis an integer, producing a more comb-like dis-\ntribution. The broadening of the distribution at smaller\nvalues ofNis an important feature that is missed when\nI= 1=2 spins are used to model I > 1=2 spin baths.\nIn the example of Fig. 7, we see that it also leads to an\nincrease in the average magnetization hmidue to the en-\nhanced weight of the distribution at positive magnetiza-\ntions. This enhancement is more pronounced for I= 3=2\ncompared to I= 1. Fig. 8 examines the behavior of hmi\nas a function of N. The points are obtained by solving\nthe respective kinetic equations, Eqs. (39) and (45). In\ntheI= 1 case, it is possible to obtain results for much\nlarger bath sizes because the Rmatrix is much smaller\nin this case. For both I= 1 andI= 3=2, the points are\nwell described by a linear relationship between hmiand\nN, as shown in the \fgure. We \fnd that for the parame-\nters considered and for large N, the average polarization\nforI= 3=2 is approximately two times larger compared\nto that of an I= 1 bath, with the net polarization in this\ncase approaching 9%.\nThe e\u000bects of nonzero quadrupolar angle on the polar-\nization distribution for I= 1, 3/2 are illustrated in Fig. 9.\nHere, we set N= 150, because nonzero \u0012reduces the\nsparsity of theRmatrix, making the numerical compu-\ntation more intensive than before, especially for I= 3=2.\nFrom Fig. 9(a), we see that for I= 1, nonzero \u0012leads to\nquantitative changes in the heights of the mode-locking\npeaks, along with a slight redistribution of the proba-\nbility to negative magnetizations for intermediate values15\n\u0001=0\u0001=\u00018\u0001=\u00014\u0001=\u00012\n-40-20020400.000.050.100.15\nMagnetization mProbablity P(m)\u0001=0\u0001=\u00018\u0001=\u00014\u0001=\u00012\n-60-40-2002040600.000.020.040.060.08\nMagnetization mProbablity P(m)Spin-1Spin-3/2(a)(b)\nFIG. 9. Steady-state nuclear spin polarization distribution of a bath with N= 150 nuclear spins for four di\u000berent values of the\nquadrupolar angle \u0012for (a)I= 1 and (b) I= 3=2. The parameter values are TR= 13:2 ns,NA= 10 GHz, !e0= 0:5 GHz,\n!n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\nθ=0\nθ=π\n8θ=π\n4θ=π\n2\n-100 -50 0 50 100 1500.000.020.040.060.080.10\nMagnetization mProbablity P(m)Spin-1\nFIG. 10. Steady-state nuclear spin polarization distribution\nof a bath with N= 1000I= 1 nuclear spins for four di\u000berent\nvalues of the quadrupolar angle \u0012. The other parameter values\nareTR= 13:2 ns,NA = 10 GHz, !e0= 0:5 GHz,!n=\n\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\nof\u0012. Similar behavior occurs for I= 3=2, as shown in\nFig. 9(b). The redistribution can be understood from\nthe fact that, in the absence of the HF interaction, the\nquadrupolar coupling produces a Gaussian distribution\ncentered around m= 0. This is discussed in more detail\nbelow. The fact that this redistribution is strongest near\n\u0012=\u0019=4 suggests that the \u0001 mI=\u00061 quadrupolar-driven\ntransitions play an important role in this process. This\ne\u000bect constitutes another way in which the quadrupolar\ninteraction can make the DNP process for I >1=2 depart\nsigni\fcantly from what is predicted for an I= 1=2 bath.\nAlso notice that in both panels of Fig. 9, the polarization\ndistributions are still suppressed near m=\u0000!e0=Aeven\nfor\u0012 >0. This indicates that the HF contributions to\nthe \rip rates remain an important factor in shaping the\noverall distribution.\nFIG. 11. The average polarization hmiof a nuclear spin bath\nwithN= 1000 nuclei of total spin I= 1 for several values of\nthe quadrupolar angle in the range of 0 \u0014\u0012\u0014\u0019=2. The inset\ncolor map shows the steady-state nuclear spin polarization\ndistribution over the same range of quadrupolar angles. The\nother parameter values are TR= 13:2 ns,NA = 10 GHz,\n!e0= 0:5 GHz,!n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\n\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\nFig. 10 again shows the e\u000bect of nonzero \u0012forI= 1,\nbut now for a bath of size N= 1000. For \u0012= 0, there\nis a distinct comb-like structure that is the hallmark of\nmode-locking. However, for \u0012>0, this structure quickly\ndisappears and is replaced by an almost Gaussian distri-\nbution centered around zero magnetization. A Gaussian\ndistribution is in fact what occurs in the absence of the\nHF interaction, because the \rip rates are then purely\ndue to the quadrupolar coupling, which means that they\nare independent of mand are equal for \u0001 mI>0 and\n\u0001mI<0. This shows that the quadrupolar interaction\nplays a much more important role compared to the HF\ninteraction for the case considered in Fig. 10. This is16\n\u0001=0\u0001=\u0002/2024681012-0.50.00.5\nt(ns)Electron steady-stateSe,xss\u0003e0=0.5 GHz\u0001=0\u0001=\u0002/2024681012-0.50.00.5\nt(ns)\u0003e0=2.45 GHz\u0001=0\u0001=\u0002/2024681012-0.6-0.4-0.20.00.20.40.6\nt(ns)\u0003e0=15.19 GHz(a)(b)(c)\nFIG. 12. The feedback e\u000bect of N= 1000I= 1 nuclear spins on the xcomponent of electron spin steady state as a function\nof time over one drive period TR= 13:2 ns. Here the quadrupolar angles \u0012= 0 and\u0012=\u0019=2 are considered for di\u000berent bare\nelectron Zeeman frequencies of (a) 0.5 GHz, (b) 2.45 GHz and (c) 15.19 GHz. The electron Zeeman frequencies chosen for\n(b) and (c) correspond to the local minima shown in Fig. 13 and the nuclear spin polarization distribution for (a) is shown in\nFig. 10. The parameter values are NA= 10 GHz, !n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\nbecause the larger value of Ncorresponds to a reduc-\ntion in the HF coupling A, and hence in the magnitude\nof the \rip rates (see Eq. (26)). This in turn increases\nthe relative importance of the quadrupolar interaction.\nThis can be seen from Fig. 4, where it is evident that\nas\u0012increases, the \rip rate for the \u0001 mI= 2 transition\nquickly surpasses the di\u000berence in the \rip rates for the\n\u0001mI=\u00061 transitions. As a consequence, the proba-\nbility distribution is no longer sensitive to the detailed\nfeatures of the \u0001 mI=\u00061 transitions, which are respon-\nsible for both the comb-like mode-locking structure and\nthe suppression near m=\u0000!e0=A. This shows that even\nsmall values of \u0012can have a dramatic e\u000bect on the DNP\nprocess for large numbers of nuclei. This is quanti\fed in\nFig. 11, which shows how the nuclear spin polarization\ndistribution and average magnetization, hmi, depend on\n\u0012. The latter quickly decays with increasing \u0012. As is\nevident from the inset in Fig. 11, the distribution it-\nself exhibits mode-locking fringes at small \u0012that become\nblurred at larger \u0012. The sensitivity of mode-locking to the\nquadrupolar interaction suggests that it could be used as\na diagnostic tool to estimate the size of the quadrupolar\ncoupling strength and angle in experiments. This is fur-\nther supported in the next section, where we show how\nthe steady-state electron spin vector in the presence of\nDNP feedback depends on the quadrupolar angle.\nB. Feedback on electron spin\nOnce we obtain the steady-state polarization distribu-\ntion of the nuclear spin bath, the \fnal step is to update\nthe steady state of the electron by applying the Over-\nhauser shift to the Zeeman frequency:\nSss\ne;i(t;!e0) =X\nmP(m)Sss\ne;i(t;!e0+mA):(51)\nHere the summation is over all possible values of m, and\ntis the time elapsed since the last pulse. We obtain the\ntime-evolved electron steady state by starting from the\nSe,xssSe,xssLocal MinimaSe,xsswithθ=0\nSe,xsswithθ=π\n2Se,xsswithθ=π\n4Se,xsswithθ=π\n8\n0 5 10 15 20-0.50.00.51.0\nωe0(GHz)Electron steady-state x componentFIG. 13. The e\u000bect of the I= 3=2 nuclear feedback on the\nxcomponent of the steady-state electron spin vector. The\nred \flled circles indicate local minima of Sss\ne;x(shown in gray)\nfor several values of the electron Zeeman frequency !e0with-\nout nuclear feedback. The other points indicate the values of\nSss\ne;x(!e0) at the same values of !e0, but now with feedback\nincluded as in Eq. (51). Results for four di\u000berent values of\nthe quadrupolar angle \u0012are shown. Other parameter values\nareN= 150,TR= 13:2 ns,NA= 10 GHz, !n=\u00000:5 MHz,\n\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\nexpression for the steady state immediately after a pulse,\nEq. (14), and evolving it under Larmor precession with\nfrequency!e0+mAfor timet. Fig. 12 shows the result-\ning DNP-modi\fed electron steady state over one drive\nperiod for six di\u000berent N= 1000,I= 1 polarization\ndistributions. Two of these are distributions shown in\nFig. 10|the ones corresponding to \u0012= 0 and\u0012=\u0019=2.\nThe modi\fed steady states for these two cases are shown\nin Fig. 12(a), where it is evident that a large quadrupolar\nangle suppresses oscillations, both in the vicinity of the\ndriving pulses and in the \\echo\" that occurs in the mid-\ndle of the drive period near t=TR=2, which is 6.6 ns for\nthe chosen parameter values. Similar behavior occurs for\nother values of the external magnetic \feld, as is demon-\nstrated in Figs. 12(b), (c). It should be noted that the17\n\u0001e0=2.45(GHz)\u0001e0=6.17(GHz)\u0001e0=11.25(GHz)\u0001e0=15.19(GHz)\u0001e0=18.22(GHz)\n-40-2002040600.000.010.020.030.040.050.060.07Probability P(m)\u0001=0\u0001e0=2.45(GHz)\u0001e0=6.17(GHz)\u0001e0=11.25(GHz)\u0001e0=15.19(GHz)\u0001e0=18.22(GHz)\n-40-20020400.0000.0050.0100.0150.0200.0250.0300.035Probability P(m)\u0001=\u00028\n\u0001e0=2.45(GHz)\u0001e0=6.17(GHz)\u0001e0=11.25(GHz)\u0001e0=15.19(GHz)\u0001e0=18.22(GHz)\n-40-2002040600.0000.0050.0100.0150.0200.0250.0300.035\nMagnetization mProbability P(m)\u0001=\u00024\u0001e0=2.45(GHz)\u0001e0=6.17(GHz)\u0001e0=11.25(GHz)\u0001e0=15.19(GHz)\u0001e0=18.22(GHz)\n-40-2002040600.000.010.020.030.040.05\nMagnetization mProbability P(m)\u0001=\u00022(a)\n(d)(b)\n(c)\nFIG. 14. The nuclear spin polarization distributions corresponding to \fve of the electron Zeeman frequency values from Fig. 13\nfor quadrupolar angles (a) \u0012= 0, (b)\u0012=\u0019=8, (c)\u0012=\u0019=4, and (d)\u0012=\u0019=2 for anI= 3=2 nuclear bath. Other parameter\nvalues areN= 150,TR= 13:2 ns,NA= 10 GHz, !n=\u00000:5 MHz,\re= 0:5 GHz,q0= 0:3,\u001e=\u0000\u0019=2,\u0017Q= 2:8 MHz.\namplitude of these oscillations are used to identify the\npresence of mode-locking [54], and so the suppression of\nthese oscillations can provide an experimental indicator\nof substantial quadrupolar e\u000bects.\nThe electron steady state, Eq. (14), is a rapidly oscilla-\ntory function of the applied magnetic \feld. In Ref. [74],\nit was found using perturbation theory that for I= 1=2,\nnuclear feedback suppresses the amplitudes of these oscil-\nlations. In particular, it was shown that the xcomponent\nof the electron steady-state SV approaches unity for all\nvalues of the external magnetic \feld as a consequence of\nmode-locking: The SV becomes synchronized with the\npulses such that it lies parallel to the optical axis at the\npulse times. Here, we examine how this e\u000bect is modi-\n\fed by the presence of quadrupolar interactions. This is\nillustrated in the case of I= 3=2 in Fig. 13, where we\nshow thex-component of the electron steady state imme-\ndiately after a pulse, Sss\ne;x, for ten di\u000berent values of the\nelectron Zeeman frequency with and without feedback.\nWe are primarily interested in the amplitude of the elec-\ntron steady-state oscillations, so we choose the ten dif-\nferent Zeeman frequencies that correspond to minima of\nthe oscillations in the absence of feedback (red dots in\nFig. 13). To \fnd how the envelope of the electron spin\noscillations is a\u000bected by the feedback process, we com-pute the nuclear spin polarization distributions for each\nof these minima. These distributions then alter the val-\nues of these minima according to Eq. (51) (with t= 0).\nAs can be seen from Fig. 13, the amplitude of the elec-\ntron steady-state oscillations is suppressed (i.e., the min-\nima increase up toward unity) in the presence of DNP,\nand the degree of this suppression varies weakly and non-\nmonotonically with the quadrupolar angle \u0012. To under-\nstand this behavior better, in Fig. 14 we show the polar-\nization distributions for \fve of the minima from Fig. 13\nfor four di\u000berent quadrupolar angles. It is clear that for\nall values of \u0012, as the electron spin Zeeman frequency due\nto the external magnetic \feld, !e0, is increased, the po-\nlarization distributions gravitate toward m= 0. This is\nbecause larger values of the electron Zeeman frequency\nsuppress HF \rip-\rops, as the violation of energy conser-\nvation becomes more pronounced in this case. This is\nwhy the\u0012= 0 curve in Fig. 13 monotonically decreases\nwith increasing !e0. On the other hand, quadrupole-\ninduced nuclear spin \rips do not depend on the electron\nZeeman frequency, and so these gradually begin to dom-\ninate as both \u0012and!e0increase. This in turn causes the\ncurves in Fig. 13 to become essentially independent of\n!e0as\u0012increases. This is another manifestation of how\nquadrupolar interactions suppress mode-locking e\u000bects.18\nV. CONCLUSIONS\nIn this work, we developed a general theoretical frame-\nwork to describe the dynamics of an electron trapped in\na self-assembled quantum dot that is driven by a periodic\ntrain of optical pulses and coupled to a nuclear spin bath.\nUsing a dynamical, self-consistent, mean-\feld type ap-\nproach, we calculated the steady-state dynamic nuclear\npolarization, as well as its in\ruence on the evolution of\nthe electron spin. Our framework is non-perturbative,\napplies to nuclei of arbitrary total spin I, and includes\nquadrupolar e\u000bects that arise for I >1=2.\nWe showed that the phenomenon of mode-locking, or\nDNP-induced frequency-focusing, seen in experiments\n[53{57] emerges naturally from our formalism. It can\nbe understood as originating from the structure of the\nrates for the electron and nuclear spins to \rip with one\nanother under the hyper\fne interaction. The \rip rates\nvanish when the e\u000bective electron precession frequency\n(including the DNP-driven Overhauser shift) becomes\ncommensurate with the optical pulse train, because in\nthis case the pulses do not interrupt the joint electron-\nnuclear evolution, and so no polarization is transferred\nfrom the electron spin to the nuclei. The vanishing of\nthe \rip rates then leads to sharp peaks in the nuclear po-\nlarization distribution at magnetization values that sat-\nisfy the commensurability condition. Our exact result for\nthe nuclear spin probability distribution in the I= 1=2\ncase makes this connection explicit, since the distribution\ndepends inversely on the \rip rates. In addition to mode-\nlocking, we showed that hyper\fne \rip-\rops also give rise\nto a net nuclear spin polarization that appears to grow\nlinearly with the number of nuclei.\nOur formalism includes not only hyper\fne-driven phe-\nnomena, but also quadrupolar e\u000bects that can arise for\nI > 1=2. We found that the importance of quadrupo-\nlar interactions depends sensitively on the quadrupolar\nangle\u0012between the applied magnetic \feld and the prin-\ncipal axis of strain in the dot. For \u0012 < \u0019= 8, hyper\fne\ninteractions tend to dominate, leading to clear signatures\nof mode-locking. However, for \u0012\u0015\u0019=8, quadrupole-\ninduced nuclear spin \rips begin to dominate, which leads\nto a suppression of mode-locking and a reduction of the\nnet nuclear polarization. We also showed that quadrupo-lar e\u000bects become more pronounced when the applied\nmagnetic \feld is increased, because hyper\fne \rip-\rops\nare suppressed by the increasingly large Zeeman energy\nmismatch between the electron and nuclei. These e\u000bects\nare clearly visible in the nuclear spin polarization distri-\nbutions for both I= 1 andI= 3=2, and they translate\nto experimentally detectable signatures that are encoded\nin the presence or absence of electron spin oscillations\nin the steady state. Hyper\fne \rip-\rops lead to coherent\noscillations in the vicinity of each pulse and halfway be-\ntween pulses, while quadrupolar interactions act to sup-\npress these oscillations. These signatures o\u000ber a potential\nmethod to measure the strength of quadrupolar interac-\ntions in quantum dots.\nThe framework we have presented constitutes an e\u000e-\ncient, quantitative approach to describing the dynamics\nof a driven spin coupled to a spin bath. Going forward,\nit would be interesting to see if some of the simplify-\ning assumptions made here can be relaxed to enhance\nquantitative accuracy. For example, can we go beyond\nthe box model limit and allow for non-uniform hyper-\n\fne couplings, perhaps using a \\wedding cake\" model in\nwhich the electronic wavefunction envelope is approxi-\nmated by a piecewise-constant function? Such a gener-\nalization would also allow for the inclusion of multiple\nnuclear species, which is relevant for common semicon-\nductor QD compounds such as InGaAs. It would also be\ninteresting to extend this method beyond the indepen-\ndent nuclei approximation, perhaps using a cluster-based\napproach in which inter-nuclear interactions are included\ngradually within clusters of increasing size [20, 100]. In\nterms of applications, our framework could be employed\nto design driving protocols to achieve desired bath po-\nlarization states to either mitigate decoherence or utilize\nthe bath as a quantum memory [86{88]. 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Verstraten1, and Rembert Duine1,6\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands;\n2Department of Engineering Sciences, Universitetet i Agder,\nPostboks 422, 4604 Kristiansand, Norway;\n3Department of Philosophy, Institute of Technology Futures,\nKarlsruhe Institute of Technology, Douglasstraße 24, 76133 Karlsruhe, Germany;\n4Department of Quantum Nanoscience, Kavli Institute of Nanoscience,\nDelft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands;\n5Institute of Molecular Physics, Polish Academy of Sciences,\nul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland; and\n6Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: January 23, 2024)\nSpin dynamics is usually described as massless or, more precisely, as free of inertia. Recent\nexperiments, however, found direct evidence for inertial spin dynamics. In turn, it is necessary to\nrethink the basics of spin dynamics. Focusing on a macrospin in an environment (bath), we show\nthat the spin-to-bath coupling gives rise to spin inertia. This bath-induced spin inertia appears\nuniversally from all the high-frequency bath modes. We expect our results to provide new insights\ninto recent experiments on spin inertia. Moreover, they indicate that any channel for spin dissipation\nshould also be accompanied by a term accounting for bath-induced spin inertia. As an illustrative\nexample, we consider phonon-bath-induced spin inertia in a YIG/GGG stack.\nIntroduction. —Spin dynamics is usually considered to\nbe free of inertia and, in turn, it is described by first-order\ndifferential equations; for example, the Bloch equation\n[1] for small quantum-mechanical spins or the Landau-\nLifshitz-Gilbert (LLG) equation [2–5] for larger quasi-\nclassical spins. Based on the broad success of those de-\nscriptions, one might be tempted to conclude that spin\ndynamics is free of any inertia, as spin inertia would be\ndescribed by second-order time derivatives. Recent ex-\nperiments, however, find direct evidence for spin inertia\nclose to THz-frequencies [6, 7]; see also [8]. First and\nforemost, the discovery of spin inertia is of immense inter-\nest for our fundamental understanding of spin dynamics.\nFor example, spin inertia leads to a redshift in the ferro-\nmagnetic resonance peak [9, 10] and, more interestingly,\ngives rise to nutation spin waves [11–13]. In addition,\nspin inertia might also prove relevant in applications as\nthe related nutation dynamics takes place on short time\nscales [6–8, 11, 14]; for example, spin inertia might al-\nlow for faster and more efficient spin switching similar\nto antiferromagnets [15]. For a recent review on inertial\neffects, see reference [16].\nSpin inertia and spin nutation have already been de-\nrived for several systems with various approaches: from\nan adiabatic expansion of a dissipation kernel [17]; for a\nspin in a superconducting Josephson junction [18]; from\nmesoscopic nonequilibrium thermodynamics [19]; with\na classical Lagrangian approach [20] analogous to the\nderivation of Gilbert damping [5]; for the spins coupled\nto an electron bath [21–24]; from higher-order relativistic\nterms [25, 26]; for environments with a Lorentzian bathspectral density [27]; and by modeling a magnetization in\nterms of a current-carrying loop [28]. Despite the many\ncrucial insights provided by these approaches, a complete\nunderstanding of spin inertia is arguably still missing [7].\nHere, in the quest to contribute to a more general and\nunified understanding of spin inertia, we show that spin\ninertia arises universally from the interaction with an en-\nvironment.\nIn this letter, we consider a macrospin that is ex-\nposed to an effective magnetic field and coupled to its\nFIG. 1. The bath spectral density J(ϵ) contains the infor-\nmation about how the environment (bath) affects the spin\ndynamics. We separate J(ϵ) into a low-frequency approxima-\ntionJlf(ϵ) (green solid line) and the remaining high-frequency\npart J(ϵ)−Jlf(ϵ) (red shaded area). This separation sets an\nenergy scale, ϵlf, up to which Jlf(ϵ) is a good approximation\ntoJ(ϵ). Assuming the typical frequencies of the spin dynam-\nics to be small, ω≪ϵlf, the low-frequency bath modes give\nrise to damping, whereas all the high-frequency bath modes\ngive rise to spin inertia. While our results are general, for the\nfigure we assumed Jlf(ϵ) to be linear, which leads to Gilbert\ndamping.arXiv:2310.05621v2  [cond-mat.mes-hall]  22 Jan 20242\nenvironment. The interaction with the effective mag-\nnetic field is described by the Zeeman energy. Using the\nCaldeira-Leggett approach [29, 30], analogous to [31], we\nmodel the environment as a bath of harmonic oscillators\nand, for simplicity, assume a linear coupling between the\nmacrospin and the bath modes. In the quasi-classical\nlimit, after integrating out the bath modes, we find an\ninertial LLG equation. More explicitly, we show that\nthe environmental degrees of freedom (bath modes) af-\nfect spin dynamics in two ways: the low-frequency bath\nmodes give rise to Gilbert damping (or fractional Gilbert\ndamping if the bath is non-ohmic), while all the high-\nfrequency bath modes give rise to spin inertia (indepen-\ndent of the bath details); see Fig. 1.\nFinally, to show that the Caldeira-Leggett approach\nwith linear coupling is not just a toy model but of real\nuse, we discuss its application to a macrospin that is\ncoupled to a phonon bath and we derive the phonon-bath-\ninduced spin inertia for a magnetic plane (YIG) that is\nsandwiched between two nonmagnetic insulators (GGG).\nModel Hamiltonian. —We consider a macrospin that is\ncoupled to its environment and exposed to an (effective)\nmagnetic field. The corresponding Hamiltonian can be\nseparated into three parts, ˆH=ˆHs+ˆHc+ˆHb, where the\nsystem Hamiltonian ˆHsdescribes the macrospin in the\nmagnetic field, the bath Hamiltonian ˆHbdescribes the en-\nvironmental degrees of freedom, and the coupling Hamil-\ntonian ˆHcdescribes the coupling between the macrospin\n(system) and its environment (bath). Explicitly, the\nmacrospin in the magnetic field is described by the Zee-\nman energy with the Hamiltonian ˆHs=−B·ˆS, where\nˆS= (ˆSx,ˆSy,ˆSz) is the macrospin operator with standard\nspin operators ˆSx,ˆSy,ˆSzandBdenotes the (effective)\nmagnetic field that may dependent on time and can point\nin any direction. Using the Caldeira-Leggett approach\n[29, 30], we model the environment as a bath of harmonic\noscillators; in turn, the bath Hamiltonian is given by\nˆHb=P\nn(ˆ p2\nn/2mn+mnω2\nnˆ q2\nn/2), where mn,ωn,ˆ qn,ˆ pn\nare mass, eigenfrequency, position operator, and momen-\ntum operator of the n-th bath oscillator. For simplicity,\nwe assume a linear system-to-bath coupling, which is de-\nscribed by the coupling Hamiltonian ˆHc=−P\nnγnˆ qn·ˆS,\nwhere γnis the coupling coefficient describing the cou-\npling strength between the macrospin and the n-th bath\noscillator respectively. The rationale behind the linear\ncoupling is that, close to the ground state of the full sys-\ntem (macrospin + environment), a Taylor expansion is\nlikely to lead to a linear coupling as the leading order\nterm. Thinking in more physical terms, the linear cou-\npling might, for example, be an sd-like coupling between\nthe macrospin and electron-hole pairs in metallic mag-\nnets. However, as we will show below, even in the case of\nmagnetoelastic coupling, which is quadratic in the spin\nvariables, the linear-coupling approach is still useful to\nunderstand the inertial macrospin dynamics close to the\nmagnetic ground state.Effective action for spin dynamics. —Starting from the\nmodel, we use the Keldysh formalism in its path-integral\nversion [32–34] with spin coherent states |g⟩, as in refer-\nence [33], to derive an action for the combined dynamics\nof the macrospin and the bath oscillators. Because we\nmodel the environment as a bath of harmonic oscilla-\ntors and assume linear coupling, the action is (at most)\nquadratic in ˆ pnandˆ qn. In turn, we can integrate out\nthe bath modes by performing Gaussian path integrals;\nfirst in pn, then in qn. As result, we obtain the Keldysh\npartition function Z=R\nDg eiSwith the effective action\nfor the macrospin dynamics\nS=I\nKdt[−i⟨˙g|g⟩+B·S] +1\n2I\nKdtI\nKdt′S(t)α(t−t′)S(t′),\n(1)\nwhere |˙g⟩=∂t|g⟩andS=⟨g|ˆS|g⟩. Information about\nthe coupling to the bath is contained in the kernel func-\ntionα(t−t′) that, when represented in Keldysh space,\nis a matrix containing as elements the retarded and\nadvanced parts αR/A(ω) and a Keldysh part αK(ω).\nThe noiseless effects (damping and inertia) are de-\nscribed by the retarded and advanced parts αR/A(ω) =\n−P\nnγ2\nn/{mn[(ω±iη)2−ω2\nn]}, where ηis an infinites-\nimal level broadening or decay rate. The information\nabout fluctuations (noise) is contained in the Keldysh\npartαK(ω) = coth( ω/2kBT) [αR(ω)−αA(ω)] with Boltz-\nmann constant kBand bath temperature T, which is\nsimply the fluctuation-dissipation theorem, because we\nassume the bath to be in (local) equilibrium [34].\nTo be more explicit, as in reference [33], we use\nthe Euler-angle representation of spin coherent states,\n|g⟩= exp( −iϕˆSz) exp(−iθˆSy) exp(−iψˆSz)|⇑⟩, where\n|⇑⟩is the eigenstate to ˆSzwith the maximal eigen-\nvalue S; that is, ˆSz|⇑⟩=S|⇑⟩. This representa-\ntion is convenient, as the macrospin takes the intu-\nitive form of a vector in spherical coordinates S=\nS(sinθcosϕ,sinθsinϕ,cosθ), where Sis its length and\nθandϕdescribe its orientation. The Berry-phase term\nbecomes −i⟨˙g|g⟩=S(˙ψ+˙ϕcosθ), where the angle ψis\na gauge freedom [33]. Properly fixing this gauge on the\nKeldysh contour is nontrivial and can be crucial [35, 36].\nHere, however, a simple choice of ˙ψ=−˙ϕon the Keldysh\ncontour will be just fine; for a detailed discussion, see the\nSupplemental Material.\nGeneralized Landau-Lifshitz-Gilbert equation. —From\nthe effective action, we derive an effective quasi-classical\nequation of motion for the macrospin along the following\nlines: first, to retain information about fluctuations, we\nuse the Schmid trick [33, 34, 37–39] and “decouple” the\npart of the action that is quadratic in quantum compo-\nnents by a Hubbard-Stratonovich transformation; then,\nwe vary the action with respect to the quantum com-\nponents θqandϕq; finally, the resulting quasi-classical\nequations of motion for θcandϕccan be recast into a\nsingle vectorial equation of motion [40]. As result, we3\nobtain a generalized LLG equation\n˙S=S×(B+ξ) +S×Z∞\n−∞dt′˜α(t−t′)S(t′),(2)\nwhere ξis a fluctuating field with ⟨ξm(t)⟩= 0 and\n⟨ξm(t)ξm′(t′)⟩=−(i/2)δmm′αK(t−t′); the indices\nm, m′∈ {x, y, z}denote Cartesian components corre-\nsponding to S= (Sx, Sy, Sz). Furthermore, we defined\n˜α(ω) =αR(ω)−αR(0), where the ω= 0 part is sub-\ntracted for regularization [41]. More explicitly, the kernel\nfunction is given by\n˜α(ω) =−2\nπZ∞\n−∞dϵω2J(ϵ)\n[(ω+iη)2−ϵ2]ϵ, (3)\nwhere J(ϵ) = πP\nn(γ2\nn/2mnωn)δ(ϵ−ωn) is the bath\nspectral density that contains two pieces of information:\nin the delta function δ(ϵ−ωn), it contains the informa-\ntion at which energies the bath modes can be found; in\nthe pre-factor ( γ2\nn/2mnωn) it contains the information of\nhow strongly the macrospin couples to the bath modes.\nSeparating low- and high-frequency bath modes. — In an\nenvironment with many bath modes that are close in en-\nergy, the bath spectral density J(ϵ) will be a continuous\nfunction; see Fig. 1. For a general bath spectral den-\nsity, it is hard to proceed analytically. However, inspired\nby reference [39] but without introducing a cutoff, we\nmake analytical progress by separating the contributions\nof low-frequency and high-frequency bath modes. Explic-\nitly, we rewrite J(ϵ) =Jlf(ϵ)+[J(ϵ)−Jlf(ϵ)], where Jlf(ϵ)\nis a low-frequency approximation of J(ϵ) and we refer to\nJ(ϵ)−Jlf(ϵ) simply as high-frequency contribution, even\nthough non-low-frequency contribution might be a more\naccurate name. Accordingly, we split the kernel function\n˜α(ω) = ˜αlf(ω) + ˜αhf(ω) into a low-frequency contribu-\ntion ˜αlf(ω) and the high-frequency contribution ˜ αhf(ω).\nExplicitly, the low-frequency contribution is given by\n˜αlf(ω) =−2\nπZ∞\n−∞dϵω2Jlf(ϵ)\n[(ω+iη)2−ϵ2]ϵ, (4)\nwhile the high-frequency contribution is given by\n˜αhf(ω) =−2\nπZ∞\n−∞dϵω2[J(ϵ)−Jlf(ϵ)]\n[(ω+iη)2−ϵ2]ϵ. (5)\nNow, the key point to realize is that we are able to\ndetermine the high-frequency contribution under a quite\ngeneral assumption: we assume that typical frequency of\nthe spin dynamics ωis much smaller than ϵlf, which is the\nenergy up to which Jlf(ϵ)≈J(ϵ); see Fig. 1. Under this\nassumption, we can disregard the ωdependence in the\ndenominator of Eq. (5). The reason is that ϵ < ϵ lfthe\nnumerator vanishes, J(ϵ)−Jlf(ϵ)≈0, while for ϵ > ϵ lf\nthe denominator can be approximated, ( ω+iη)2−ϵ2≈\n−ϵ2. In turn, we can approximate the high-frequency\ncontribution to the kernel function,\n˜αhf(ω)≈I ω2, (6)and we arrive at our central result: the bath-induced spin\ninertia\nI=2\nπZ∞\n−∞dϵJ(ϵ)−Jlf(ϵ)\nϵ3. (7)\nSo, if the bath spectral density J(ϵ) is known, it can be\nused to determine its low-frequency approximation Jlf(ϵ)\nand, in turn, to find the spin inertia Iby straightforward\n(potentially numerical) integration. Note that the sign\nof the spin inertia is not fixed; depending on the form of\nJ(ϵ), spin inertia can be positive or negative.\nInertial Landau-Lifshitz-Gilbert equation. — To recover\nthe inertial LLG equation used in experimental analysis\n[6, 7], we assume the bath spectral density to be approxi-\nmately linear at low frequencies. This assumption allows\nus to use an Ohmic bath, which is a bath with linear bath\nspectral density, for the low-frequency approximation\nJlf(ϵ); for non-Ohmic low-frequency approximations, see\nreference [31]. Explicitly, we assume Jlf(ϵ) =α0ϵΘ(ϵ),\nwhere Θ( ϵ) is the Heaviside Θ-function and α0is some\nconstant that will turn out to be the Gilbert-damping\ncoefficient. We then find ˜ αlf(ω) = iα0ωfor the low-\nfrequency kernel function and I= (2/π)R∞\n0dϵ[J(ϵ)−\nα0ϵ]/ϵ3for the bath-induced spin inertia, where we used\nthat bath spectral densities vanish for negative energies.\nIt is now straightforward to derive the inertial LLG\nequation. First, transforming the low- and high-\nfrequency parts of the kernel function back into time\nspace, we find ˜ αlf(t−t′) =−α0δ′(t−t′) and ˜ αhf(t−t′) =\n−I δ′′(t−t′), where δ′(t) and δ′′(t) are respectively the\nfirst and second derivative of the Dirac δ-function. Then,\ninserting them back into the generalized LLG equation\n(2), we obtain the inertial LLG equation\n˙S=S×h\nB+ξ−α0˙S−I¨Si\n, (8)\nwhere α0is the Gilbert-damping coefficient, Iis the spin\ninertia, and ξis a fluctuating field with ⟨ξm(t)⟩= 0 and\n⟨ξm(t)ξm′(t′)⟩ ≈2kBTα0δmm′δ(t−t′) for which we as-\nsumed the temperature to be large [42]. Note that only\nGilbert damping—but not spin inertia—contributes to\nfluctuations. On the formal level, spin inertia does not\ncontribute to fluctuations, as even-frequency contribu-\ntions of αR(ω) and αA(ω) cancel out in αK(ω). In more\nphysical terms, spin inertia is not dissipative and, based\non the general idea of the fluctuation-dissipation theorem\n[39, 43], should therefore also not contribute to fluctua-\ntions.\nThe Gilbert-damping coefficient α0and the spin inertia\nIdepend on the bath spectral density J(ϵ); see Eqs. (4)\nand (5) respectively. Next, to illustrate how our general\nresults can be applied, we consider a macrospin that is\ncoupled to a phonon bath.\nA macrospin coupled to a phonon bath. — The coupling\nbetween spins and phonons can be described by magne-\ntoelastic coupling, as in reference [44], where they derived4\na generalized LLG equation analogous to Eq. (2) but for\na spin lattice. So, to identify the bath-induced spin in-\nertia for phonon baths, we do not need to rederive the\ngeneralized LLG equation. Instead, we start from their\ngeneralized LLG equation [44], recast it into the form of\nour Eq. (2), identify the bath spectral density J(ϵ), and\nfinally determine the spin inertia from Eq. (7).\nNote that magnetoelastic coupling is derived in the\ncontinuum limit of long-wavelength phonons. So, strictly\nspeaking, one would have to rederive the spin-phonon\ncoupling to include short-wavelength (high-frequency)\nphonon modes, which are responsible for spin inertia.\nNevertheless, we believe that the following illustrative\nexample based on magnetoelastic coupling provides valu-\nable insights into the physics of spin inertia and paves the\nway for more detailed derivations from microscopic mod-\nels.\nMagnetoelastic coupling is quadratic in the spin vari-\nables [44]. Close to the ground state, however, we are\nallowed to linearize the spin dynamics and with it the\nmagnetoelastic coupling. Combining this linearization\nwith a macrospin approximation, we relate the phonon\nbath [44] to the Caldeira-Leggett approach above; for de-\ntails, see Supplemental Material. Explicitly, we find that\nthe kernel function (3) becomes a tensor ˜ αmm′(ω), as also\nthe bath spectral density takes a tensorial form [45],\nJmm′(ϵ) =πX\niX\nkλeikRiBmzBm′z\n2MNS 1S ωkλ\n×(kmz·ekλ) (km′z·e−kλ)δ(ϵ−ωkλ),(9)\nwhere S1is the effective spin length on an individual\nlattice site with spin, Bm˜mandBm′˜mare the magne-\ntoelastic coupling coefficients (here ˜ m=z, as we chose\nthez-direction for the ground state), the scalars Nand\nMare respectively the number and effective (oscillat-\ning) mass of lattice sites, the function ωkλis the phonon\ndispersion relation with the phonon momentum k, the\nvector ekλis the polarization vector and λis the index\ndenoting longitudinal and transversal polarization, the\nvector kmm′is defined by kmm′=kmem′+km′emwith\nCartesian unit vectors emandem′, and the sum iruns\nover the lattice-positions Riof the individual spins; the\nnotation is analogous to reference [44].\nKnowing the phonon-bath spectral density (9), we then\nfind a low-frequency approximation Jlf,mm′(ϵ) and, in\nturn, determine the phonon-bath-induced spin inertia\nfrom the difference Jmm′(ϵ)−Jlf,mm′(ϵ) as in Eq. (7).\nTo be specific, let us consider a layer of yttrium iron\ngarnet (YIG) that is sandwiched between two bulk lay-\ners of gadolinium gallium garnet (GGG); afterwards, we\nconsider implications for a layer of YIG on top of a bulk\nof GGG. We model the system as a planar spin lat-\ntice (in the z= 0 plane) that is embedded in a three-\ndimensional lattice with lattice constant a. In this geom-\netry, only phonons travelling in z-direction contribute tothe macrospin damping and inertia. Focusing on acous-\ntic phonons, we use the dispersion relation for a simple\ncubic lattice ωkzλ= (2/a)vλ|sin(kza/2)|, where vλis the\nsound velocity for transversal ( λ∈ {x, y}) or longitudinal\n(λ=z) polarization. At low energies ϵthe bath spec-\ntral density (9) is governed by long wavelength (small k)\nacoustic phonons, which have an approximately linear\ndispersion relation, ωkzλ=vλ|kz|. In turn, Jmm′(ϵ) be-\ncomes linear in ϵat low energies and we can use the low-\nfrequency approximation Jlf,mm′(ϵ) =αmm′ϵΘ(ϵ) with\na tensorial Gilbert-damping coefficient αmm′; for YIG on\nGGG, a tensorial Gilbert damping has been found before\n[46]. The phonon-bath-induced spin inertia is now found\nfrom the high-frequency modes as in Eq. (7); it also\ntakes a tensorial form Imm′= (2/π)R∞\n−∞dϵ[Jmm′(ϵ)−\nJlf,mm′(ϵ)]/ϵ3. Explicitly, we find the Gilbert-damping\ncoefficient αmm′=δmm′(1 +δmz)2B2\nmza/2MS 1Sv3\nmand\nthe phonon-bath-induced spin inertia\nImm′=0.9\nπa\nvmαmm′, (10)\nwhere, after making the ϵ-integral dimensionless, we eval-\nuated and approximated it numerically to 0 .9; a detailed\ncalculation is provided in the Supplemental Material. For\nYIG on top of GGG (not sandwiched) only about half\nthe phonon modes are present, such that we expect the\nGilbert-damping coefficient, and with it the spin inertia,\nto be only half as large.\nBath-induced spin inertia and Gilbert damping are\nclosely related, as becomes clear from Eq. (10); namely,\nthe spin inertia is proportional to the Gilbert-damping\ncoefficient. Thus, since the Gilbert-damping tensor αmm′\nis diagonal, also the spin-inertia tensor Imm′is diago-\nnal. The proportionality constant τm=Imm/αmm=\n0.9a/πv mis typically used in experimental works to char-\nacterize spin inertia or nutation dynamics [6–8]. For the\nGGG phonon bath, with lattice spacing a= 1.2383 nm\n[47] and phonon velocities vz= 6411 m /s (longitudinal)\nandvx=vy= 3568 m /s (transversal) [46, 48], we find\nτz≈55 fs and τx=τy≈99.4 fs. These results, com-\npared to previous measurements on metallic magnets, are\nroughly in the same order of magnitude [7, 8] or about\ntwo to three orders of magnitude lower [6].\nNote that the frequency of the nutation resonance\npeak may deviate from previous experimental results\non metallic magnets, as it also depends on the Gilbert-\ndamping coefficient [49]. Using the effective (oscillating)\nmass of GGG M=ρ a3with the GGG density ρ=\n7.07·103kg/m3[46, 48], the magnetoelastic coupling coef-\nficients Bzz= 6.6·10−22J and Bxz=Byz= 13.2·10−22J\nwith the YIG-lattice-site spin length S1= 14.2ℏ[50], we\nfindαzz= 2·10−7/Sandαxx=αyy= 1.2·10−6/Sfor\nthe elements of the tensorial Gilbert-damping coefficient.\nFor those numbers, using the “weak coupling” approxi-\nmation of reference [49], the nutation resonance peaks\nwould be in the order of X-ray frequencies. While proba-5\nbly impractical in for experiments, it does not affect the\nillustration purposes of our simple example.\nDiscussion and Conclusion. —Using the Caldeira-\nLeggett approach, we have shown that the\nhigh-frequency modes of an environment (bath)\nshould—universally—lead to bath-induced spin inertia.\nThe low-frequency bath modes, if Ohmic, lead to the\nusual Gilbert-damping term. This has two important\nconsequences. First, our results suggest that the ap-\npearance of bath-induced spin inertia is more robust (or\nuniversal) than the form of Gilbert damping; while for\nnon-Ohmic baths Gilbert damping turns into fractional\nGilbert damping [31], the bath-induced spin inertia\nretains its form. Second, our results suggest that,\nwhenever a dissipation channel (environment/bath) is\nadded to some spin system, the spin dynamics will also\nacquire an additional contribution to its spin inertia.\nIn short, spin relaxation is always accompanied by spin\ninertia.\nTo show how our general derivation based on the\nCaldeira-Leggett approach applies to more realistic mod-\nels, we considered the dynamics of a macrospin in a\nphonon bath; specifically, we considered a heterostruc-\nture based on YIG and GGG. We expect that our ap-\nproach can be applied, along similar lines, to many other\nspin environments as well; for example, to electron baths\nin metallic magnets and to metallic leads in contact with\ninsulating magnets. Furthermore, we believe that our ap-\nproach can be generalized from the macrospin case con-\nsidered here to the more general case of spin lattices and,\nin turn, also to continuum theories of spin textures.\nBecause the high-frequency bath modes affect the ac-\ntion, Eq. (1), already before the quasi-classical approx-\nimation, we also expect small-spin systems, as investi-\ngated in [51, 52], to be affected by bath-induced spin\ninertia from high-frequency bath modes.\nAcknowledgements. —We thank M. Cherkasskii,\nE. Di Salvo, A. Kamra, A. Semisalova, A. Shnirman,\nD. Thonig, R. Willa, and X. R. Wang for fruitful dis-\ncussions. This work is part of the research programme\nFluid Spintronics with project number 182.069, financed\nby the Dutch Research Council (NWO). R. C. V. is\nsupported by (NWO, Grant No. 680.92.18.05). R. A. 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Greilich, Spin inertia of resident and photoexcited car-\nriers in singly charged quantum dots, Physical Review B\n98, 121304 (2018).\n[52] D. Smirnov, E. Zhukov, E. Kirstein, D. Yakovlev,\nD. Reuter, A. Wieck, M. Bayer, A. Greilich, and\nM. Glazov, Theory of spin inertia in singly charged quan-\ntum dots, Physical Review B 98, 125306 (2018).1\nSUPPLEMENTAL MATERIAL: BATH-INDUCED SPIN INERTIA\nA SIMPLE CHOICE OF GAUGE IS SUFFICIENT\nBefore choosing a gauge, the action, Eq. (1) in the main text, still contains the gauge freedom ψin the Berry-phase\nterm, which becomes −i⟨˙g|g⟩=S(˙ψ+˙ϕcosθ) in the Euler-angle representation. In principle, we can fix ψas we like.\nHowever, we should respect “boundary conditions” on the Keldysh contour; otherwise, we could choose ˙ψ=−˙ϕcosθ,\nwhich would eliminate the Berry-phase term/contintegraltext\nKdt S(˙ψ+˙ϕcosθ) from the action. It can be crucial to be careful\nwith the boundary conditions of the chosen gauge [1, 2]. Here, however, a simple choice of ˙ψ=−˙ϕon the Keldysh\ncontour will be just fine; as shown next, it leads to the same quasi-classical dynamics as the more elaborate choice of\nreferences [1, 2].\nTo compare the gauge choice of references [1, 2] with the simple choice ˙ψ=−˙ϕ, we rewrite ψ=χ−ϕ. Then, the\ngauge choices differ only in χ; references [1, 2] choose ˙ χc=˙ϕc(1−cosθc) and χq=ϕq(1−cosθc), while the simple\nchoice corresponds to χ= 0. For the Berry-phase term, there is no relevant difference at finite times:\n/contintegraldisplay\nKdt(˙ψ+˙ϕcosθ) =−/contintegraldisplay\nKdt˙ϕ(1−cosθ) +/contintegraldisplay\nKdt˙χ; (1)\nbut with χ±=χc±χq/2 and χq=ϕq(1−cosθc), we find\n/contintegraldisplay\nKdt˙χ=/integraldisplay\ndt[˙ϕq(1−cosθc) +ϕq˙θcsinθc] = 0 , (2)\nwhere, in the last step, we used integration by parts and disregarded potentially arising boundary terms that are\ndynamically irrelevant in the quasiclassical approximation.\nDERIVATION OF BATH SPECTRAL DENSITY FOR A PHONON BATH\nSpins on a lattice will be coupled to the phonons that arise from oscillations of the atoms/ions at the lattice sites. In\nspintronics, this coupling is often described by the magnetoelastic coupling. For example, see [3], where they derived\nthe coupled equations of motions for spins Sion a lattice (with lattice sites denoted by i) that are coupled to a phonon\nbath,\n˙Si=Si×H+Si×hi+Si×/summationdisplay\njKijSj−Si×/integraldisplayt\n0dt′/summationdisplay\njGij(t, t′)˙Sj(t′), (3)\nwhere His the external magnetic field, Kijaccounts for dipolar and nearest-neighbor exchange interactions, and the\ncoupling to the phonon system gives rise to the induced magnetic field hiand the generalized Gilbert damping with\nthe kernel function Gij(t, t′). For the components of that kernel function, they find\nGmm′\nij(t, t′) =1\nNS4\n1/summationdisplay\n˜m˜m′Bm˜mBm′˜m′S˜m\ni(t)S˜m′\nj(t′)/summationdisplay\nkλeik·(Ri−Rj)(km˜m·ekλ)(km′˜m′·e−kλ)cos[ωkλ(t−t′)]\nMω2\nkλ,(4)\nwhere S1is the spin length of the effective spin on a single lattice site with spin, Nis the number of lattice sites,\nkdenotes the phonon momentum, λits polarization (longitudinal/transversal), and the corresponding dispersion\nrelation and polarization vector are given by ωkλandekλrespectively. Further, Nis the number of lattice sites, Mis\ntheir effective ionic mass, and Bm˜mare the magnetoelastic coupling coefficients. We also adopt their short notation\nforkm˜m=kme˜m+k˜memwith Cartesian unit vectors emform∈x, y, z .\nNow, to find the bath spectral density for the phonon bath, our intermediate goal is to relate the spin-lattice\nequation of motion (3) to the macrospin equation of motion of the main text (2); more specifically, we need to relate\nthe generalized Gilbert-damping terms of both equations. This immediately leads us to a key problem: the kernel\nfunction for the phonon bath, Gmm′\nij(t, t′), depends on the lattice spins via S˜m\ni(t)S˜m′\nj(t′), whereas the kernel function\nfor the harmonic-oscillator bath of the main text, ˜ α(t−t′), does not depend on the macrospin. This key difference\narises from the different spin-to-bath couplings. While the magnetoelastic coupling is nonlinear in the spin [3], in the\nmain text we used the Caldeira-Leggett approach with a simple linear spin-to-bath coupling. However, if the spins2\nremain close to the z-direction, such that we can linearize the equation of motion in deviations from the z-direction,\nthen we may simply approximate Si≈S1ezinGmm′\nij(t, t′). In turn, the kernel function simplifies to\nGmm′\nij(t−t′) =1\nNS2\n1BmzBm′z/summationdisplay\nkλeik·(Ri−Rj)(kmz·ekλ)(km′z·e−kλ)cos[ωkλ(t−t′)]\nMω2\nkλ. (5)\nWith this simplification, we can now Fourier-transform equation (3) with/summationtext\nie−iqRi...and use the macrospin approx-\nimation with Sq=0=SandSq̸=0= 0. For the generalized Gilbert damping, we then find\n−Si×/integraldisplayt\n0dt′/summationdisplay\njGij(t, t′)˙Sj(t′) −→ −1\nNsS×/integraldisplayt\n−∞dt′G0(t−t′)˙S(t′), (6)\nwhere Nsis the number of lattice sites with spin and, disregarding transient effects, we set the lower integral boundary\nfrom 0 to −∞. The kernel function G0(t−t′) is the q= 0 case of the Fourier-transform\nGq(t−t′) =/summationdisplay\nieiq(Ri−Rj)Gij(t−t′). (7)\nNow, the generalized Gilbert damping, Eq. (6), can be integrated by parts to find\n−1\nNsS×/integraldisplayt\n−∞dt′G0(t−t′)˙S(t′) =−1\nNsS×/bracketleftbigg/integraldisplayt\n−∞dt′˙G0(t−t′)S(t′) + [G0(t−t′)S(t′)]t\n−∞/bracketrightbigg\n, (8)\nwhere the boundary term [ G0(t−t′)S(t′)]t\n−∞only contains a term that renormalizes the magnetic field and some\ninformation about the initial state, which would only affect transient dynamics that we are not considering anyways.\nSo, we can finally relate the generalized Gilbert damping for the phonon bath to the generalized Gilbert damping\nfor the Caldeira-Leggett approach in the main text. We find that the relation between the two kernel functions is\ngiven by\n˜α(t−t′) =−1\nNs˙G0(t−t′) Θ(t−t′). (9)\nSubtracting the zero-frequency part ˜ α(ω= 0) for regularization, as in the main text, and using the relation\n/integraldisplay∞\n−∞dω\n2πe−iωt 1\n(ω+iη)2−ϵ2=−sin(ϵ t)\nϵΘ(t), (10)\nwe find the (tensorial) bath spectral density for the phonon bath,\nJmm′(ϵ) =π/summationdisplay\ni/summationdisplay\nkλeikRiBmzBm′z\n2MNS 1S ωkλ(kmz·ekλ) (km′z·e−kλ)δ(ϵ−ωkλ), (11)\nwhere we used that the macrospin length Sis given by the sum over all the individual spin lengths S1; so,S=NsS1.\nThe resulting bath spectral density, Eq. (11), is used in the main text.\nEXPLICIT CALCULATION FOR A PLANAR SIMPLE-CUBIC SPIN LATTICE IN A\nTHREE-DIMENSIONAL SIMPLE-CUBIC LATTICE\nWe consider a three-dimensional simple cubic lattice with site length a, where the z= 0-plane (YIG) contains spins\nof length S1on every site, while there is no spin on all the other sites (GGG).\nStarting from the bath spectral density, Eq. (11), we can now use that the sum over lattice sites with spin yields/summationtext\niexp[ikRi] =/summationtext\nnxnyexp[i(kxa nx+kya ny)] =Nxδkx,0Nyδky,0, where nxandnynumerate, respectively, the lattice\nsites in x-direction and y-direction; and NxandNyare the number of lattice sites in x-direction and y-direction\nrespectively. Consequently, the bath spectral density becomes\nJmm′(ϵ) =π/summationdisplay\nkλNxδkx,0Nyδky,0BmzBm′z\n2MNS 1S ωkλ(kmz·ekλ) (km′z·e−kλ)δ(ϵ−ωkλ), (12)3\nand, after carrying out the sums over kxandky, it simplifies to\nJmm′(ϵ) =δmm′π(1 +δmz)2B2\nmz\n2MS 1S/summationdisplay\nkzλk2\nz\nNzωkzλδmλδ(ϵ−ωkzλ), (13)\nwhere we used that the number of lattice sites is given by N=NxNyNzwith the number of lattice sites in z-direction\nNz. Carrying out the summation over the phonon polarization yields,\nJmm′(ϵ) =δmm′π(1 +δmz)2B2\nmz\n2MS 1S/summationdisplay\nkzk2\nz\nNzωkzmδ(ϵ−ωkzm). (14)\nInstead of directly carrying out the sum over kz, which runs over all kz-states in the Brillouin zone (BZ), we switch\nto the continuum case and turn it into an integral; namely, we replace/summationtext\nkz∈BZ...= (Lz/2π)/integraltextπ/a\n−π/adkz..., where Lz\nis the length of the system in z-direction. Further, using that Lz=aNz, we find\nJmm′(ϵ) =δmm′(1 +δmz)2B2\nmza\n4MS 1S/integraldisplayπ/a\n−π/adkzk2\nz\nωkzmδ(ϵ−ωkzm). (15)\nTo carry out the remaining integral, we need to know the dispersion relation. For a simple cubic lattice, focusing\non acoustic phonons, the dispersion relation is given by ωkzm= (2/a)vm|sin(kza/2)|, where vmis the sound velocity\nfor transversal ( m∈ {x, y}) or longitudinal ( m=z) polarization; see [4, 5] for example. In terms of microscopic\nparameters, the sound velocities are given by vm=/radicalbig\nKm/Mwith the force constants Kx=Ky=K2andKz=\nK1+ 2K2, where K1andK2are respectively the force constants for nearest neighbor and next-nearest neighbor\nrestoring forces [4, 5]. Now, the momentum integral becomes\n/integraldisplayπ/a\n−π/adkzk2\nz\nωkzmδ(ϵ−ωkzm)=/integraldisplayπ/a\n−π/adkzk2\nz\n2vm\na/vextendsingle/vextendsinglesin/parenleftbigkza\n2/parenrightbig/vextendsingle/vextendsingleδ/parenleftbig\nϵ−2vm\na/vextendsingle/vextendsinglesin/parenleftbigkza\n2/parenrightbig/vextendsingle/vextendsingle/parenrightbig\n=\n\n24\na2arcsin2a ϵ\n2vm\nvmϵ/radicalig\n1−(a ϵ\n2vm)2forϵ∈/bracketleftbig\n0,2vm\na/bracketrightbig\n0 otherwise.\n(16)\nIn turn, we find the bath spectral density in explicit form\nJmm′(ϵ) =δmm′(1 +δmz)2B2\nmz\nMS 1S4\na2arcsin2a ϵ\n2vm\n2vm\naϵ/radicalig\n1−/parenleftbiga ϵ\n2vm/parenrightbig2Θ(ϵ) Θ/parenleftbig2vm\na−ϵ/parenrightbig\n. (17)\nAt low energies, ϵ≪2vm\na, we can approximate arcsin/parenleftbiga ϵ\n2vm/parenrightbig\n≈a ϵ\n2vmand/radicalig\n1−/parenleftbiga ϵ\n2vm/parenrightbig2≈1, such that we find the\nlow-energy or low-frequency approximation for the bath spectral density\nJlf,mm′(ϵ) =δmm′(1 +δmz)2B2\nmza\n2MS 1S v3mϵΘ(ϵ), (18)\nwhich is linear in ϵ. So, following the derivation of the main text, we can simply read off the (tensorial) Gilbert-damping\ncoefficient as\nαmm′=δmm′(1 +δmz)2B2\nmza\n2MS 1S v3m. (19)\nKnowing the bath spectral density Jmm′(ϵ) and its low-frequency approximation Jlf,mm′(ϵ), we can now determine\nthe bath-induced spin inertia from Eq. (7) from the main text,\nImm′=2\nπ/integraldisplay∞\n−∞dϵJmm′(ϵ)−Jlf,mm′(ϵ)\nϵ3. (20)\nInserting Eqs. (17) and (18), and using Eq. (19), we find\nImm′=2\nπαmm′/integraldisplay∞\n0dϵ\n4v2\nm\na2arcsin2a ϵ\n2vm\nϵ4/radicalig\n1−/parenleftbiga ϵ\n2vm/parenrightbig2Θ/parenleftbig2vm\na−ϵ/parenrightbig\n−1\nϵ2\n. (21)4\nTheϵ-integral can be made dimensionless by substituting ϵ= 2vmϵ′/a, which yields\nImm′=1\nπa\nvmαmm′/integraldisplay∞\n0dϵ′/bracketleftbiggarcsin2ϵ′\nϵ′4√\n1−ϵ′2Θ(2vm\na−2vm\naϵ′)−1\nϵ′2/bracketrightbigg\n. (22)\nTo deal with the Θ-function, it is now convenient to split the integral from 0 to ∞into two parts: one from 0 to 1\nand another one from 1 to ∞. Then, we find\n/integraldisplay∞\n0dϵ′/bracketleftbiggarcsin2ϵ′\nϵ′4√\n1−ϵ′2Θ(2vm\na−2vm\naϵ′)−1\nϵ′2/bracketrightbigg\n=/integraldisplay1\n0dϵ′/bracketleftbiggarcsin2ϵ′\nϵ′4√\n1−ϵ′2−1\nϵ′2/bracketrightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n≈1.9+/integraldisplay∞\n1dϵ′/bracketleftbigg\n−1\nϵ′2/bracketrightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=−1≈0.9, (23)\nwhere we evaluated the integral from 0 to 1 numerically. So, overall, for a layer of YIG sandwiched between two bulk\nlayers of GGG, we find the phonon-bath-induced spin inertia\nImm′=0.9\nπa\nvmαmm′, (24)\nas given in the main text.\n[1] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Geometric quantum noise of spin, Physical\nReview Letters 114, 176806 (2015).\n[2] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman, Thermally driven spin transfer torque system far from equilibrium:\nEnhancement of thermoelectric current via pumping current, Physical Review B 99, 045429 (2019).\n[3] A. R¨ uckriegel and P. Kopietz, Rayleigh-jeans condensation of pumped magnons in thin-film ferromagnets, Physical review\nletters 115, 157203 (2015).\n[4] O. Madelung, Introduction to solid-state theory , Vol. 2 (Springer Science & Business Media, 1996).\n[5] J. S´ olyom, Fundamentals of the Physics of Solids (Springer, 2007)."    },    {        "title": "1112.0459v1.Initialization_and_Readout_of_Spin_Chains_for_Quantum_Information_Transport.pdf",        "content": "arXiv:1112.0459v1  [quant-ph]  2 Dec 2011Initialization and Readout of Spin Chains for Quantum Infor mation Transport\nGurneet Kaur and Paola Cappellaro∗\nDepartment of Nuclear Science and Engineering,\nMassachusetts Institute of Technology, Cambridge, Massac husetts 02139, USA\nLinear chains of spins acting as quantum wires are a promisin g approach to achieve scalable\nquantum information processors. Nuclear spins in apatite c rystals provide an ideal test-bed for the\nexperimental study of quantum information transport, as th ey closely emulate a one-dimensional\nspin chain. Nuclear Magnetic Resonance techniques can be us ed to drive the spin chain dynamics\nand probe the accompanying transport mechanisms. Here we de monstrate initialization and read-\nout capabilities in these spin chains, even in the absence of single-spin addressability. These control\nschemes enable preparing desired states for quantum inform ation transport and probing their evo-\nlution under the transport Hamiltonian. We further optimiz e the control schemes by a detailed\nanalysis of19F NMR lineshape.\nPACS numbers: 03.67.Hk, 03.67.Lx, 75.10.Pq, 76.90.+d\nI. INTRODUCTION\nControl over small quantum systems and the ability to\nperform simple quantum algorithms have been demon-\nstrated on a variety of physical systems ranging from\ntrapped ions [ 1] and electrons [ 2], to neutral atoms and\nmolecules in optical lattices [ 3], to superconducting cir-\ncuits [4] and semiconductor quantum dots [ 5], to nuclear\nand electronic spins [ 6–8]. Although algorithms involv-\ning more than one qubit have been executed [ 9–14], a\nvital requirement for a quantum computer – scalability\nwhile preserving ﬁdelity – has not yet been achieved in\nany physical system. The use of linear chains of spins as\nquantum wires to couple basic memory units is a promis-\ning approach to address this issue [ 15,16]. These spin\nchains have the ability to transmit quantum information\nvia the free evolution of the spins under their mutual\ninteraction [ 17–22]. While advances in fabrication tech-\nniques have made physical implementation of spin wires\npossible [ 23–26], the level of precision available is not\nyet adequate. Therefore, natural systems such as crys-\ntals where spin position is precisely set by nature are a\npreferred choice for exploring such applications.\nOwing to their unique geometry, nuclear spin systems\nin apatite crystals have emerged as a rich test-bed to\nprobe quasi-one-dimensional (1D) dynamics, including\ntransport and decoherence [ 27–30]. The crystal wherein\n19F (or1H) nuclei are aligned along one axis, emulates\na collection of 1D chains. The dynamics of these spin\nchains have been studied by various nuclear magnetic\nresonance (NMR) techniques [ 31–33]. In our previous\nwork [20,34], we have shown that the natural dipolar\ninteraction among the spins can be manipulated via the\navailable collective control to simulate the Hamiltonian\ndriving quantum transport. The lack of single-spin ad-\ndressabilityinthisensemblesystemshoweverwouldseem\nto prevent creating and measuring a single-spin excita-\n∗Electronic address: pcappell@mit.edution as required to study transport. Still, we demon-\nstrated experimentally [ 27,28] in the Fluorapatite (FAp)\nsystem that one can prepare the spin system in an initial\nstate in which polarization is localized at the ends of the\nspin chain, a state that well simulates the conditions for\nspin-excitation transport [ 34].\nIn this paper we take further steps toward enabling\nthe experimental study of quantum transport in spin\nchains: we introduce an experimental technique to read\nout the spins at the chain extremities and we show\nhow to prepare a two-spin encoded state that is able\nto transfer quantum information. We use these initial-\nization and readout techniques to study the dynamics\nunder the transport Hamiltonian (the Double Quantum\n(DQ) Hamiltonian [ 30,35]). Additionally, we probe\nthe spin chain dynamics by creation and evolution of\nmultiple quantum coherences [ 36], which present well-\ncharacterized state dependent signatures. We use both\nthese techniques to demonstrate preparation as well as\nreadout of spins at the chain ends, thus verifying impor-\ntant required tasks towards simulation of quantum infor-\nmationtransfer. Wefurthervalidatetheaddressabilityof\nends of the chains by a detailed analysis of19F lineshape\nin FAp.\nThese techniques will make it possible to explore er-\nrors aﬀecting the transport ﬁdelities as well as control\nschemes to mitigate them, in an experimental setting,\nwhere the interactions among spins are not limited to\nthe ones tractable by solvable models.\nII. TRANSPORT IN MIXED-STATE SPIN\nCHAINS\nA. Spin chain dynamics\nLinear chains of spin-1/2 particles have been proposed\nas quantum wires to transport quantum information be-\ntween distant nodes of a distributed quantum computing\narchitecture. Transport can occur even in the absence\nof individual control of the chain spins, as it is medi-2\nated by the spins mutual interactions. In the most stud-\nied model, energy conserving spin ﬂip-ﬂops (mediated by\nthe isotropic XY Hamiltonian) drive the transport of a\nsingle-spin excitation [ 17–22]. This model has been re-\ncently extended to the case where the initial state of the\nchain cannot be perfectly controlled, and thus it is found\nin a mixed-state rather than its ground state [ 34,37–40].\nSpin chains that are in a maximally mixed-state are\nparticularly interesting from the point of view of exper-\nimental study of quantum information transfer. This\nstate, corresponding to the inﬁnite temperature, can be\neasily achieved experimentally and has been shown to\nprovide a direct simulation of pure state transport [ 34].\nAdditionally, extension to mixed-state chain enables us-\ning the so called double quantum (DQ) Hamiltonian:\nHDQ=/summationdisplay\nj<ℓbjℓ\n2(σx\njσx\nℓ−σy\njσy\nℓ), (1)\nto drive transport, although it does not conserve the\nnumber of spin excitations. This Hamiltonian can be\neasily obtained from the natural dipolar Hamiltonian\nwith only collective control [ 20,34] and it is related to\nthe isotropic XY Hamiltonian (which instead cannot be\ngenerated from the dipolar interaction) via a similarity\ntransformation. The extension to mixed-states and to\nDQ Hamiltonian open the possibility to study experi-\nmentally quantum information transport in nuclear spin\nchains with NMR techniques. Under our experimental\nconditions (strong external magnetic ﬁeld, B 0=7T and\nroom temperature), the initial equilibrium state is the\nZeeman thermal state,\nρ′\nth(0)∝exp(−εσz)≈11−εσz, (2)\nwhereσz=/summationtext\njσz\njandǫ=γB0/kBT. Since the identity\ndoes not evolve and does not contribute to the signal, we\nwill focus on the deviation of the density operator from\nthe maximally mixed state, δρ∼ρ−11. In the absence\nof individual spin addressing, transport within a chain\ncan be studied by preparing a polarization excess at one\nend of the chain, that is, a state where one spin at the\nchain extremities is polarized while the remaining spins\nare fully mixed, δρ∼σ1\nz. Because of the symmetry be-\ntweenthe twochainends, thestatewecanprepareexper-\nimentally [ 27,28], which we call “end-polarized state”, is\ngiven by\nδρend(0) =σz\n1+σz\nN. (3)\nThe end-polarized state simulates the dynamics of a\nsingle-spin excitation in a pure-state spin chain. This\nstate can transfer a bit of classical information by en-\ncoding it in the sign of the polarization. This encoding\nis, however, not enough to transfer quantum informa-\ntion, which requires additional transfer of information\nabout the phase coherence of a state. A two-qubit en-\ncoding allows for the transport of a bit of quantum in-\nformation [ 34,39]. For transport via the DQ Hamilto-\nnian, this encoding is given by the basis |0∝angbracketrightdq\nL=|00∝angbracketrightand|1∝angbracketrightdq\nL=|11∝angbracketright. The operator basis for transport via mixed\nstates under DQ Hamiltonian is thus given by:\nσdq\nxL=σ1\nxσ2\nx−σ1\nyσ2\ny\n2, σdq\nyL=σ1\nyσ2\nx+σ1\nxσ2\ny\n2,\nσdq\nzL=σ1\nz+σ2\nz\n2, 11dq\nL=11+σ1\nzσ2\nz\n2.(4)\nStarting from any of the above initial states, the evo-\nlution under the DQ Hamiltonian directly simulates the\ntransport dynamics within a chain.\nIn the limit of nearest neighbor (NN) coupling only,\nthe evolution under DQ Hamiltonian is exactly solvable\nby invoking a Jordan-Wigner mapping onto a system of\nfree fermions [ 30,41,42]. The resulting dynamics of the\nobservables we analyzed in our experiments have been\nreported in the literature (see e.g. [ 30,34]) and are re-\nviewed in Appendix Afor completeness. Isolated, lin-\near spin chains with NN couplings is an accurate model\nfor the experiments, given the experimental timescales\nused [28]. Comparison of the theoretical model with the\nexperimental results thus allows us to validate our ini-\ntialization and readout methods.\nTo gather more insights into the states generated by\nthe evolution under the DQ Hamiltonian, we experimen-\ntally measured multi-spin correlationsvia multiple quan-\ntum NMR experiments. This experimental technique\n(see Appendix B) allowsmeasurement of multi-spin state\ndynamicsbyindirectly encodingtheirsignaturesinto sin-\ngle spin terms. The dynamics of quantum coherence in-\ntensitiescanaswellbe calculatedanalyticallyin thelimit\nof NN couplings [ 20,34,42] and we review these results\nin Appendix B.\nB. Preparing and reading out desired spin states\nTo probe the quantum transport dynamics it is nec-\nessary to prepare the spins at the ends of the chain in\na perturbed state (such as δρend), which is then left to\nevolve under the transport Hamiltonian. Furthermore,\nto observe the transport, measurement of the end chain\nspins would be desirable. Unfortunately, in a system of\ndipolarly coupled homonuclear spins (such as FAp) it is\nnotpossibletoachievefrequencyorspatialaddressability\nof individual spins. Still, here we show that we can ap-\nproximate these preparation and readout steps with the\ncontrol at hand, combining unitary and incoherent spin\nmanipulation. The key observation is that even in the\nabsence of frequency addressability, the dynamics of the\nend-chainspins under the internal dipolarHamiltonian is\ndiﬀerent from the rest of the spins in the chain. Indeed,\nthe spins at the ends of the chain are coupled to only one\nnearest neighbor whereas spins in the rest of the chain\nhave two nearest neighbors. This fact can be exploited\nto experimentally prepare the spins at the chain ends in\na desired state [ 27,28] as well as subsequently read out\nthis state as explained below.\nWhen the initial thermal equilibrium state is rotated\nto the transverse plane by a π/2 pulse, we create a state3\n/summationtextN\nk=1σk\nxwhich evolvesunder the internal dipolar Hamil-\ntonian. Due to fewer numbers of couplings with neigh-\nboringspins, the spins at the end ofthe chain haveslower\ndynamics ascomparedto the rest ofthe chain. Thus, one\ncanselectaparticulartime t1suchthat whereasthe state\nof the spins at the ends is still mainly σx, the rest of the\nspins have evolved to many-body correlations. A second\nπ/2 pulse brings the magnetization of the end spins back\nto the longitudinal axis while an appropriate phase cy-\ncling scheme cancels out other terms in the state. We\nused the following pulse sequence and appropriate phase\ncycling scheme to select the ends of the chains,\nπ/2|α—t1—π/2|−α, (P1)\nwithα=x,y to average out terms that do not com-\nmute with the total z-magnetization. For FAp crystals\nwe found that the optimal t1time (which we will call\n“end-selection time”) is given by 30.3 µs [27,28] . Fur-\nther details on how we optimized this time are given in\nSectionIIID.\nThe end-selection scheme forms the basis for a strat-\negy to prepare other states, presented in Eq. 4, required\nfor quantum information transport. In order to prepare\nthese encoded states experimentally, we use the follow-\ning scheme. We ﬁrst prepare the end polarized state\nδρend(0) =σz\n1+σz\nNand then let the system evolve un-\nder DQ Hamiltonian for a short time. Applying a double\nquantum ﬁlter by an appropriate phase cycling scheme\nselects the desired state: δρyL∝σdq\nyL/vextendsingle/vextendsingle/vextendsingle\n1,2+σdq\nyL/vextendsingle/vextendsingle/vextendsingle\nn−1,n.\nSimilarly, a π/4 collective rotation around z, prior to the\ndouble-quantum ﬁlter, is needed to select the δρL\nxopera-\ntor.\nA similar combination of unitary and incoherent spin\ncontrol can be used to read out the spins at the end of\nthe chain. In inductively measured NMR, the observable\nis the collective magnetization of the spin ensemble. To\nsimulate the readout of a diﬀerent observable, the de-\nsired state must be prepared prior to acquisition. The\nsequence P1can be used for this purpose: we call this\nthe “end readout step”. We note that the sequence used\nfor readout cannot in general be a simple inversion of the\nend-selection step (which is not a unitary –reversible–\noperation). A proper phase cycling should ensure that\nthe state prior to the end-selection sequence has con-\ntributions mainly from population terms ( ∝σk\nz). This\nproperty is veriﬁed for the states produced from evolu-\ntion under the DQ Hamiltonian in 1D systems, making\ntheP1sequenceeﬀectivefortheend-readoutstep; amore\ncomplex phase cycling would be needed for more general\nstates.\nIII. EXPERIMENTAL METHODS\nExperiments were performed in a 7 T widebore mag-\nnet with a 300 MHz Bruker Avance Spectrometer and a\nprobe tuned to 282.4MHz for19F measurement. A pure,singlecrystalofFluorapatite[Ca 5(PO4)3F]grownbyﬂux\nmethod was used for the measurements [ 43,44]. High\npurity of the crystal is conﬁrmed by long relaxation time\n(T1=1100 s) of19F spins. FAp crystals have a hexagonal\ngeometry with space group P63/m. The19F nuclei form\nlinear chains along the caxis, each surrounded by 6 other\nchains. The intra-nuclear spacing within a single chain is\nd=0.3442nm and chains are separated by D=0.9367nm.\nWhen placed in a strong magnetic ﬁeld, the nuclear spins\ninteract via the secular dipolar Hamiltonian,\nHdip=n/summationdisplay\nj<lbjl/bracketleftbigg\nσj\nzσl\nz−1\n2(σj\nxσl\nx+σj\nyσl\ny)/bracketrightbigg\n,(5)\nwhere the couplings depend on the relative positions as\nbjl= (µ0/16π)(γ2/planckover2pi1/r3\njl)(1−3cos2θjl), withµ0being the\nstandard magnetic constant, γthe gyromagnetic ratio,\nrjlthe distance between nucleus jandl, andθjlthe\nangle between /vector rjland thez-axis, respectively. Due to\n1/r3dependence of dipolar coupling, the largest ratio of\nin-chain to cross chain coupling is close to 40. For our\nexperiments, the crystal was aligned parallel to the ex-\nternal magnetic ﬁeld in order to maximize this ratio. It\nhas been shown that under these settings, and for short\nevolution times, couplings across diﬀerent chains can be\nneglected and the system can be considered as a collec-\ntion of 1D chains [ 28].\nWe performed two sets of experiments for each of the\ndiﬀerent initial states and readouts. First we probed the\ntransport dynamics driven by the DQ Hamiltonian. For\nthis purpose, the collective or end-chain magnetization\nwas measured as we increased the evolution time tun-\nder the DQ Hamiltonian. We used a standard 8-pulse\nsequence [ 45] to implement the DQ Hamiltonian. The\nlength of the π/2 pulse was 1 .45µs. The evolution time\nwas incremented by varyinginter-pulse delay from 1 µs to\n6.2µs and the number of loops was increased from 1 to\n12. A recycle delay of 3000 s was used for these measure-\nments. We restricted the evolution to a timescale where\nthe ideal model applies and errors arising from leakage\nto other chains and next-nearest neighbor couplings are\nsmall [28]. In this timescale, the initial perturbation\ntravels across ≈17 spins [ 30], a distance much shorter\nthan the average length of the chains. Thus, within the\ntimescaleusedin thiswork,onlythepolarizationstarting\nfrom one end of the chain and moving away towards the\nother end was observed: polarization reaching the other\nend could not be observed.\nIn the secondset ofexperiments, welet the initial state\nevolveunderthe DQ Hamiltonianand measuredthe mul-\ntiple quantum coherences (MQC) to gather more infor-\nmation on the evolved state (see Appendix Bfor details\non the experimental method). The inter-pulse delay was\nvaried from 1 µs to 6µs and the number of loops was in-\ncreased from 1 to 3. We encoded coherences up to order\n4 with aK= 4 step phase cycling. Since these measure-\nments involve phase cycling, a shorter recycle delay of\n1000s could be used.4\n00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4 −0.2 00.2 0.4 0.6 0.8 1.0 \nTime (ms)Signal Intensity (a.u.)(a)\n00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 \nTime (ms)Signal Intensity (a.u.)\n1(d)\n00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.2 0.4 0.6 0.8 1.0 \nTime (ms)Signal Intensity (a.u.)(b)\n00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nTime (ms)00.2 0.4 0.6 0.8 1.0 Signal Intensity (a.u.)(c)\nFIG. 1: (Color online) Evolution under the DQ Hamiltonian: ( a) Initial state: δρth. Readout: collective magnetization. (b)\nInitial state: δρend. Readout: collective magnetization. (c) Initial state: δρth. Readout: end readout. (d) Initial state: δρend.\nReadout: end readout. Data points are the experimental data and lines are the ﬁts using the analytical model described in\nappendix A. Error bars are given by the oﬀset of the signal from zero. The ﬁtting gives the following values for the dipolar\ncoupling: 8.165 (a), 8.172(b), 8.048 (c) and 8.63 (d) ×103rad/s.\nA. Experimental Results: Spin Transport\nFig.1(a) shows the observed evolution of the collec-\ntive magnetization under the DQ Hamiltonian starting\nfrom the thermal initial state (Eq. 2). The data points\nwere ﬁtted with the analytical function (Eq. A1) de-\nscribed in appendix A. In ﬁgure 1(b) we plot the sys-\ntem dynamics when starting from an end-polarized state,\nEq.3(where polarization is localized at the ends of the\nchain) and reading out the collective magnetization. Fig-\nure1(c) shows a complementary measurement where we\nstart from thermal initial state, given by the collective\nmagnetization, and read out the ends of the chains after\nevolution under the DQ Hamiltonian. Both these data\nsets were ﬁtted by the analytical expression ( A2). As it\nis evident from near perfect ﬁtting, the analytical model\nexplains the experimental data quite precisely.\nFigs.1(a) and (b) show very diﬀerent chain dynamics\nfor the two initial states (with and without end selec-\ntion), giving an experimental validation of our initial-\nization method. Furthermore, the data and ﬁttings for\nend selection, Fig. 1(b), and end readout measurements,\nFig.1(c), are very similar. This indicates the robustness\nof the readout step. Finally, Fig. 1(d) shows the evo-\nlution of the end polarized initial state under the DQ-\nHamiltonian, measured using the “end readout” scheme.\nThe ﬁtting function used is given in Eq. A3, which has\nthe same form as the transport of a single excitation in a\npure state chain [ 19,34]: this experiment is thus a direct\nsimulation of spin transport.\nIn all the above ﬁttings, we used the following ﬁttingparameters: a scalar multiplier, a baseline constant, the\nnearest-neighbor dipolar coupling and a shift of the time\naxis. The shift of the time axis is needed since there is a\ndelayofafew µsbetweenthe endofthemultiple pulsese-\nquence (ideally t=0) and the measurement of the signal.\nFrom the ﬁt we obtain a value for the dipolar coupling of\n8.161,8.172,8.048and8.63 ×103rad/sforthe fourexper-\niments, respectively. Except for the last experiment, the\nother values agree well with those obtained from similar\nmeasurements done on a diﬀerent FAp crystal [ 28] and\nalso with the theoretical value b=8 .17×103rad/s result-\ning from the known structure of FAp.\nThe small discrepancy in the ﬁtting parameter for\nthe last experiment, where we are initializing and read-\ning out the chain ends, can be explained by imper-\nfections of the end-select and readout schemes. Un-\nfortunately, the phase cycling scheme does not cancel\nout zero-quantum terms. Thus, residual polarization on\nspins 2 and N-1 as well as correlated states of the form\nσz\nj(σ+\nj−1σ−\nj+1+σ−\nj−1σ+\nj+1) contribute to errors, lowering\nthe ﬁdelity with the desired state. This eﬀect is more\nimportant for the last experiment, since errors in the\ntwo selection steps accumulate. Moreover, the end read-\nout scheme works well only if applied to the ideal state\nwhich is expected after transport. The deviation of the\nprepared initial state from the ideal state further con-\ntributes to the error in the ﬁnal measured data. Still,\nthe agreement of the experimental data with the analyt-\nical model indicates that these errors are small and do\nnot invalidate the scheme5\nTime (ms) 0 0.1 0.2 0.2 0.4 0.6 0.8 (a)\n0.0 MQC Intensity \n0 0.1 0.2 0.2 0.4 0.6 0.8 1.0 (d) \nTime (ms) 0.0 MQC Intensity \n0 0.1 0.2 0.0 0.2 0.4 0.6 0.8 1.0 (b)\nTime (ms) MQC Intensity \n0 0.1 0.2 0.2 0.4 0.6 0.8 1.0 (c) \nTime (ms) 0.0 MQC Intensity \nFIG. 2: (Color online) Evolution of multiple quantum cohere nces (0Q blue squares, 2Q red triangles, 4Q green circles): ( a)\nInitial state: δρth. Readout: collective magnetization. (b) Initial state: δρend. Readout: collective magnetization. (c) Initial\nstate:δρthReadout: end readout. (d) Initial state: δρendReadout: end readout. The experimental data points are ﬁtte d by\nanalytical functions (blue, red and green lines) obtained f rom DQ Hamiltonian with NN couplings (equations B1,B2,B2andB3\nfor ﬁgures (a), (b), (c) and (d) respectively). The error bar s are estimated from the deviation of 1st order quantum coher ence\nfrom zero. Fitting of the data gives dipolar coupling: 7.971 (a), 8.077(b), 8.031 (c) and 8.492 (d) ×103rad/s.\nB. Experimental Results: Multiple Quantum\nCoherences\nWe present the results of the second set of experiments\nin Fig.2, which shows the evolution of the 0, 2nd and\n4th order coherence intensities experimentally measured\nfor diﬀerent initial states and readouts. Fig. 2(a) shows\nthe usual MQC signal, obtained measuring the collec-\ntive magnetization and starting from an initial thermal\nstate (Eq. 2). The data points are ﬁtted by analyti-\ncal functions ( B1), given in appendix B. The only vari-\nable used in these ﬁttings was the dipolar coupling con-\nstant,for which we obtained the value b= 7.971×103\nrad/s. The data shown in Fig. 2(b) has been measured\nby ﬁrst selecting the ends of the chain and then perform-\ning the MQC measurement where collective magnetiza-\ntion is read out. Fig. 2(b), on the other hand, shows the\ndata for MQC measurements starting from thermal ini-\ntialstatebut readingoutonlythe spinsatthechainends.\nBoth these data were ﬁtted by the analytical functions\n(B2), givingb= 8.077and 8.031 ×103rad/srespectively.\nThis is in very good agreement with the values obtained\nfrom the quantum transport measurements. We re-\nmark again that the results obtained from end selection,\nFig.2(a), and end readout measurements, Fig. 2(b),\nare very similar, thus validating the eﬀectiveness of the\nreadout step. The data for the case where we initialize\nthe ends of the chains before letting the system evolve\nunder DQ Hamiltonian and then read out the ends is\nshown in Fig. 2(d). Good ﬁtting of the data with Eq.(B3) was obtained for b=8.492 ×103rad/s and shifting\nthe time axis by 15 µs. As mentioned earlier, we expect\nthe errors in selecting the ends of the chains to add up in\nthe experimentaldata, resultingin a slightly highervalue\nof the ﬁtting parameter b. The ﬁrst two data points in\nthe above mentioned ﬁgures were measured using a 4\npulse sequence to implement DQ Hamiltonian (instead\nof a standard 8 pulse sequence), leading to greater error\nbars for these data points.\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 \nTime (ms)Signal Intensity (a.u.)\nFIG. 3: (Color online) Logical initial state and readout of\ncollective magnetization. a) Evolution under DQ Hamilto-\nnian. Data points are the experimental data and lines are\nthe ﬁts using the analytical model described in appendix\nA.(equation A4. Fitting of the data gives dipolar coupling:\n7.551×103rad/s.6\n−20 −10 0 10 20 30 \nFrequency (kHz)  0.5 μs\n10 μs\n20 μs\n30 μs\n35 μs\n50 μs(b) t1\n−30 −20 −10 0 10 20 30 23.5 μs\n27.0 μs\n30.6 μs\n34.0 μs\n37.6 μs\n41.0 μst1\nFrequency (kHz)  (a)\nFIG. 4: (Color online) (a) Simulated19F lineshapes for a chain of 11 spins, for thermal initial stat e (circles), ideal end-polarized\nstate (triangles) and for an initial state prepared via a sim ulatedP1sequence, with varying t1times. (b) Experimentally\nmeasured19F NMR linehsape after state initialization performed with t heP1sequence for various t 1time. Solid lines are for\nt1= 30µs and 35 µs, which give the narrowest linewidth and the best state prep aration.\nC. Quantum Information Transport\nTo demonstrate our ability to experimentally simulate\nnot only the transport of classical information, as en-\ncoded in the spin polarization, but also of quantum in-\nformation, we prepared and studied the evolution of one\nof the logical states in Eq. ( 4). The logical state δρL\nywas\nprepared by the scheme described in section IIB(simi-\nlar schemes could be used to prepare the other states in\nthe operator basis). Figure 3shows the evolution of this\nstate under the DQ Hamiltonian and readout of collec-\ntive magnetization. The data points were ﬁtted by the\nexpression in Eq. ( A4) giving the value of dipolar cou-\npling constant as 7.551 ×103rad/s. We note that the\nscheme for preparing this logical state involves selecting\nthe ends of the chain and then creating MQC and ﬁlter-\ning out all the terms except the double quantum terms.\nThe errorsinvolved in both these steps add up and result\nin deviation of measured data points from the analytical\nfunction. The experimental results, however, follow the\nexpected analytical models and the dipolar coupling con-\nstants obtained from these experiments agree very well\nwith those obtained from other measurements.\nIn all the above described measurements where ends\nof the chains were selected and initialized, we used t 1=\n30.3µs in the pulse sequence P1. As described above,\nthis value was obtained by selecting the time when po-\nlarization of the spins at the ends of the chains is non\nzero while it has decayed to zero for the other spins, as\na result of evolution under the internal dipolar Hamilto-\nnian. In order to conﬁrm this value experimentally, we\nrepeated the transport and MQC measurements at dif-\nferent values of t 1(not shown here). We observed that\nt1= 30.3µs gave the best ﬁttings with the analytical\nfunctions, pointing towards the fact that the ﬁdelity of\nend selection is highest for t 1= 30.3µs. This is further\nconﬁrmed by a detailed19F NMR lineshape analysis as\ndescribed in the next section.D.19F NMR lineshape analysis\nA system comprising linear chains of spins, such as\nFluorapatite, is of immense interest for NMR lineshape\ncalculations by virtue of the simplicity it oﬀers as com-\nparedtoa3-dimensionalsystem[ 46,47]. The19Fspinsin\nFAp have a characteristic 3 peak lineshape, which shows\na strong angular dependence [ 48]. In our study, we uti-\nlize this angular dependence to align the crystal parallel\nto the external magnetic ﬁeld in order to minimize inter-\nchain couplings.\nThe measured lineshapes for thermal initial (t 1= 0µs)\nand end polarized (t 1= 30.3µs) using sequence P2are\nshown in Fig. 6. Due to long T 1of19F nuclei, it was\n010 20 30 40 50 60 70 −400 −300 −200 −100 0100 \nt1 [μs]Amplitude of first peak (c) −50−30−1010 30 50 \nFrequency (kHz) (a) \n−50 −30 −10 10 30 50 \nFrequency (kHz) (b) \nFIG. 5: (Color online) (a)The19F NMR lineshape (for\nt1=10µs) can be ﬁtted by 3 Gaussian lines. Individual Gaus-\nsian lines are shown as thin dotted lines, sum of the lines is\nthick red. (b) The19F NMR lineshape (for t 1=60µs) can be\nﬁtted by 3 Gaussian lines out of which 2 lines have negative\namplitude. (c) Amplitude of 1st Gaussian line as a function\nof t1used to ﬁt the experimental data (red squares) and sim-\nulated spectra (black line).7\nnot possible to measure as many points as in the stan-\ndard FID measurements. This resulted in low resolution\nof the Fourier transformed signal and hence the three\ncharacteristic peaks of FAp lineshape are not resolved\nproperly. Nonetheless, the diﬀerences in the lineshape\nand line width for the two signals are distinctly visible.\nThe lineshape provides not only signatures of the sys-\ntem dimensionality, but ofits initialstateaswell, thus we\nexpect to see qualitative diﬀerences in the NMR spectra\nfor the thermal and end polarized states. We calculated\ntheFreeInductionDecay(FID) forachainof N= 1119F\nnuclear spins evolving under the secular dipolar Hamil-\ntonian (Eq. 5) with coupling b = 8 .1×103rad/s. These\ncalculationswereperformed alsofor end-polarizedstates,\nobtained by simulating the sequence P1for diﬀerent val-\nues of end selection time (t 1). The NMR spectra ob-\ntained from these simulations are shown in Fig. 4(a), to-\ngether with the lineshapes for thermal state (Eq. 2) and\nideal end polarized state (Eq. 3). A few observations are\nworth pointing out. The NMR linewidth shows a pro-\ngressive narrowing as t 1is increased starting from t 1=\n0µs (corresponding to the thermal state). The linewidth\nis narrowest for t 1= 30.6µs and then starts to increase\ngradually but now with an anti-phase (dispersive) com-\nponent. A simple explanation of these features can be\nobtained by considering the signal as arising from a com-\npetition between the signal of the end-chain spins and\nthe signal from the spins in the bulk. The spins at the\nchain extremities are dipolarly coupled to only one spin,\nhence we expect a splitting of the NMR line into a dou-\nblet. The spins in the chain, instead, are each coupled\nto two nearest neighbors and hence produce a broader\nNMR line, split into a triplet. As the ﬁrst contribution\nincreases with increasing t 1time, the linewidth narrows.\nAt even longer times, (t 1beyond 30.6 µs) two-spin cor-\nrelations are created, that give rise to a dispersive spec-\ntrum under the subsequent dipolar Hamiltonian evolu-\ntion. These anti-phase terms keep increasing for longer\nt1times. We observe that none of the simulated lines ex-\nactly replicates the expected lineshape for the ideal end\npolarized state. This is most likely due to less than 100%\nﬁdelity of end selection and initialization step. However,\n−80 −60 −40 −20 0 20 40 60 80 \nFrequency (kHz) \nFIG. 6: (Color online)19F NMR lineshape measured for ther-\nmal (black, dotted) and end polarized (red) initial state us ing\npulse sequence P2.for t1= 30.6µs, the width of the spectrum is very close\nto the ideal lineshape.\nThe simulated lineshapes are however in good agree-\nment with the corresponding experimentally measured\nlineshapes (Fig. 4(b)), apart from a slight asymmetry,\nwhich might have been introduced by a misalignment of\nthe crystal with respect to the magnetic ﬁeld. In particu-\nlar, the experimental spectra fort 1= 30µsand 35µshave\na width very close to the ideal end selected spectrum.\nIn order to draw a quantitative comparison between\nsimulated and measured NMR spectra, we ﬁt the line-\nshapes at diﬀerent t 1times with a model comprising\n3 Gaussian lines, at frequencies shifted by the nearest-\nneighbor dipolar coupling. As shown in Fig. ( 5), both\nsimulated as well as experimental lineshapes could be\nﬁtted reasonably well with this model. Since the outer\nlines in the Gaussian model arise only from the spins\ninside the chains, their amplitude is expected to show\na zero crossing at t 1where the chain extremities have\nmaximum contribution to the measured lineshape and\nthe bulk spins polarization has a very small contribu-\ntion. In our data, this is seen at 35 µs indicating that end\nselection has maximum ﬁdelity for this value of t 1. This\nis in close accordance with 30.3 µs obtained from an op-\ntimization of the DQ-Hamiltonian evolution ﬁtting and\nused for our measurements.\nIn the lineshape analysisdescribed above, all the NMR\nmeasurements were carried out by using a π/2 pulse and\nmeasuring the resultant FID. This scheme reads out the\ncollective magnetization from all the nuclei within the\nchains. It would be interesting to isolate the signal con-\ntribution from the nuclei located at the chain ends. This\nwas achieved by means of pulse sequence P2,\nP1—π/2 — (τ) —π/2 —P1,(P2)\nwhere the end selection step (sequence P1) is used to po-\nlarizeaswellasreadoutthenucleiatthechainends. The\nselection and readout steps are separated by a variable\ndelay (τ) which enables measurement of free induction\ndecay (evolution under the natural dipolar Hamiltonian)\nas a function of time. The Fourier transform of this FID\nresults in a NMR lineshape where only the nuclei at the\nchain ends contribute.\nIV. CONCLUSION\nIn conclusion, we have studied spin transport in lin-\near chain of nuclear spins and experimentally demon-\nstrated addressability of spins at the ends of the chain\nby means of NMR control schemes. We have shown that\neven though NMR implementation allows only collective\ncontrolandobservables,wecould achieveinitializationas\nwell as readout capabilities through a combination of co-\nherent and incoherent control. These techniques can be\nused to prepare state of relevance for quantum informa-\ntion transport as well as to monitor the dynamics of the\nend-chain spins as it evolves under the DQ Hamiltonian8\nobtained via collective manipulation of the natural dipo-\nlar interaction. We validated our method by comparing\nthe experimental results with analytical solutions based\non an idealized model, which applies to the timescales\nexplored in the experiment. The good agreement of the\ndatawiththeanalyticalformulaconﬁrmsthepreparation\nand readout of the desired state.\nIn addition, we probed the states and their evolution\nby means of multiple quantum coherence measurements,\nwhich reveal information about multi-spin correlations.\nAgain,agoodconformityoftheexperimentalresultswith\nthe theoretical model was observed. We further opti-\nmized the end-selection scheme by a detailed analysis of\ntheF19NMRlineshapesobtainedfromcollectivethermal\nmagnetization and the end-polarized state respectively.\nAlthough we cannot achieve universal control of the\nend-chain spins, the initialization and readout capabili-\ntiesdemonstratedinthisworkwillallowustoexperimen-\ntally characterize quantum transport in spin chains. It\nwill enable us, for example, to explore non-idealities that\nemerge e.g. at longer times from non-nearest neighbor\ncouplings as well as couplings to other chain, and from\nthe interaction of the chains with the environment. Ad-\nditionally these methods will allow further experimental\nstudies of control schemes that can enable perfect ﬁdelity\ntransfer.\nAcknowledgments\nThis work was partially funded by NSF under grant\nDMG-1005926. We are grateful to Prof. Katsuya\nTeshima (Shinshu University, Japan) for providing the\nFluorapatite crystal used in this work. It is a pleasure\nto thank LorenzaViola and ChandrasekharRamanathan\nfor fruitful discussions.\nAppendix A: Analytical solution for the evolution\nunder the DQ-Hamiltonian\nInformation transport in linear spin chains has been\noften studies in the literature as arising from the evo-\nlution under the isotropic XY Hamiltonian, HXY=/summationtext\nj<ℓbjℓ\n2(σx\njσx\nℓ+σy\njσy\nℓ). For mixed-state chains, we\nshowed that the DQ-Hamiltonian (Eq. 1) can as well\ndrive the transport, since it is connected to XY Hamilto-\nnian by a similarity transformation. We can thus simu-\nlate quantumtransportwith an Hamiltonianthat (unlike\ntheXYHamiltonian)canbeimplementedexperimentally\nusing dipolar Hamiltonian and collective RF pulses using\na standard sequence [ 36,45].\nThe evolution of 1D spin system under a DQ Hamilto-\nnianisexactlysolvableinthenearestneighbor(NN)limit\n(only nearest neighbor couplings are present and all are\nequal to b), by invoking a Jordan-Wigner mapping onto\na system of free fermions [ 27,36,42,45]. Various for-\nmulas describing diﬀerent initial states and observables\nhavebeen reported[ 20,30,34] and wereproduceherethe\nformulas we used to interpret our experimental results inthe main text.\nThe analytical solutions for the evolution of the ther-\nmal state and end polarized state, when measuring the\ncollective magnetization, are given by:\nSth(t) =N/summationdisplay\np=1Ap,p(2t). (A1)\nSend(t) =N/summationdisplay\np=1A2\n1,p(t). (A2)\nwith\nAj,q(t) =∞/summationdisplay\nm=0i2mν[iδJ2mν+δ(2bt)−iσJ2mν+σ(2bt)]\n+∞/summationdisplay\nm=1i2mν[i−δJ2mν−δ(2bt)−i−σJ2mν−σ(2bt)]\nTransport from one end of the chain to the other is de-\nscribed by\nSsre(t) =A2\n1,1(t)+A2\n1,N(t), (A3)\nwhich corresponds to the experimental situation where\nwe prepared the end-polarized state and measured the\nchain ends only. Finally, we can derive the expected sig-\nnal arising from the collective magnetization when the\ninitial state is the logical state δρyL:\nSyL(t) =A1,2(2t)+AN−1,N(2t).(A4)\nAppendix B: Multiple Quantum NMR Spectroscopy\nMultiple Quantum spectroscopy is a powerful tech-\nnique in Nuclear Magnetic Resonance. It has the ability\nto simplify complex spectra by revealing some of the for-\nbidden transitions. Additionally, creation and evolution\nof quantum coherences can be used to probe the dynam-\nics of a correlated many-spin system giving insight into\ndimensionality of spin system, distribution of coupling\nconstants and eﬀects of motions and quantum trans-\nport [36,49–54]. Multiple Quantum Coherences (MQC)\ncan be excited by driving the spin system by irradiation\nwith cycles of multiple pulse sequences consisting of RF\npulses and delays. For example, the so-called Double\nQuantum Hamiltonian (Eq. 1) creates even quantum co-\nherences from the longitudinal magnetization. Standard\nNMR techniques, however, measure only single quantum\ncoherences. Thus, in order to probe the spin dynamics,\nit is necessary to indirectly encode the signature of the\ndynamics into single quantum coherences which can be\ndirectly measured inductively. This is done by labeling\neach coherence order with a diﬀerent phase: If ρiis the\ninitial density matrix, the ﬁnal density matrix ρfis given\nby:\nρf=U†\nMQUφUMQρiU†\nMQU†\nφUMQ,9\nwhereUMQ= exp(−i¯HDQt), andUφ= exp(−iφσz/2)\nis a rotation about the z axis by an angle φ. Under this\nrotation, a coherence term of order nwill acquire a phase\nnφ. In order to extract the information about the dis-\ntribution of MQC, each measurement must be repeated\nwhileincrementing φfrom0to2πinstepsofδφ= 2π/2K\nwhereKis the highest order of MQC we wish to encode.\nAπ/2 pulse is used to read out the signal at the end of\nthe experiment. Finally, Fourier transform of the signal\nwith respect to φyields the coherence order intensity:\nJn(t) =K/summationdisplay\nk=1Sk\nz(t)e−iknδφ,\nwhereSk\nz(t) = Tr/braceleftBig\nρk\nf(t)σz/bracerightBig\nis the signal acquired in the\nkth measurement. The technique of MQC spectroscopy\noutlined above is particularly well suited to study infor-\nmation transport by means of DQ Hamiltonian, since\nthe coherence intensities show characteristics signatures\nof the occurred transport [ 20].\nIna1DsystemwithNNcoupling, onlyzeroanddouble\nquantum coherence orders are created by the DQ Hamil-\ntonian. Starting from the thermal initial state, the nor-\nmalized intensities of the zero and DQ coherences pre-\ndicted by the analytical model are given by:\nJth\n0(t) =1\nN/summationdisplay\nkcos2[4btcos(ψk)],\nJth\n2(t) =1\n2N/summationdisplay\nksin2[4btcos(ψk)],(B1)\nwhere as before Nis the number of spins in the chain\nandψk=kπ/(N+1).\nFor the end-select state, zero and double quantum co-\nherence intensities given by the analytical model are asfollows:\nJend\n0(t) =2\nN+1/summationdisplay\nksin2(ψk)cos2[4btcos(ψk)],\nJend\n2(t) =1\nN+1/summationdisplay\nksin2(ψk)sin2[4btcos(ψk)].(B2)\nIn both Eqs. 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The spins (or interactions) in the original spin structure\nbecome the interactions (or spins) in the resulting spin structure. We identify conditions\nwhich ensure that each new spin structure is stable, that is, its spin con\fguration minimises\nits internal energy. Then, making a correspondence between spin con\fgurations and binary\nsequences, we propose a model of information growth and evolution based on the line-\ngraph operator. Since this operator can generate frustrations in newly formed spin chains,\nin the proposed model such frustrations are immediately removed. Also, in some cases,\nthe previously frustrated chains are allowed to recombine into new stable chains. As a\nresult, we obtain a population of spin chains whose dynamics is studied using Monte\nCarlo simulations. Lastly, we discuss potential applications to areas of research such as\ncombinatorics and theoretical biology.\n\u0003marcojavarone@gmail.com\nyjosh.oconnor128@gmail.com\n1arXiv:2103.12034v1  [cond-mat.stat-mech]  22 Mar 20211. Introduction\nThe line-graph operator [1] \u0015takes a graph G(V;E) withVvertices and Eedges\nand sends it to a new graph G0(V0;E0) withV0vertices and E0edges. For every edge\ninGthere exists a vertex in G0and if two edges in Gmeet at a vertex then their\ncorresponding vertices in G0are joined by an edge. Note that the graphs GandG0\nare isomorphic if Gis a cyclic graph. In this paper we investigate the application of\nthe operator \u0015to one-dimensional spin models with periodic boundary conditions.\nSpin models are a core topic in statistical physics and they have been applied to\na wide variety of \felds such as neural networks, information theory, computational\nneuroscience and evolutionary game theory among many other areas [2{11]. A simple\nspin model with Nspins can be described by the following Hamiltonian\nH=\u00001\nNNX\nhi;jiJij\u001bi\u001bj (1)\nwhere the spins \u001biare each equal to\u00061. Note thathi;jidenotes a sum over adjacent\nspins andJijdescribes the interaction between any pair of adjacent spins \u001biand\u001bj.\nPositive interactions Jij>0 are de\fned ferromagnetic, whereas negative interactions\nJij<0 are de\fned antiferromagnetic. Spin models which include both types of in-\nteraction are called spin glasses [12{15] and they have found remarkable applications\nin \felds such as neural networks and deep learning, among many others [16{18].\nThese models reach equilibrium once their free-energy F=U\u0000TSis minimised,\nwhereU,TandSare the internal energy, temperature and entropy of the system,\nrespectively. The temperature plays a key role in deciding if a state of equilibrium\ncan be reached by maximising Sor minimising U[19]. At high temperatures the\nentropy is the leading term, whereas the internal energy is most relevant at low\ntemperatures. From now on, we consider the regime of low temperatures where the\n2internal energy is described by the Hamiltonian in (1). Di\u000berent spin models can have\ndi\u000berent energy landscapes. For example, there are two equilibrium con\fgurations at\nlow temperatures in the ferromagnetic Ising model corresponding to the two states\nwith every spin aligned in the same direction. On the other hand, spin glasses have a\nmuch richer energy landscape with many more minimum energy states. For instance,\nsuch richness can be useful for storing patterns of information [20]. Interestingly, in\na spin glass there may be frustrations which make it impossible to stabilise at a local\nminimum of internal energy.\nBy making a correspondence between positive spins and the symbol 0, and between\nnegative spins and the symbol 1, every spin con\fguration may be mapped to a binary\nsequence. Note that this is just a convention and we could have mapped the symbols\nthe other way around. With this mapping, we are able to assign information content\nto a spin structure. We also introduce a correspondence between spin con\fgurations\nand cyclic graphs which have a vertex 2-colouring. Therefore, a con\fguration of N\nspins corresponds to a periodic binary sequence with period N, or to a cyclic graph\nwithNvertices and a vertex 2-colouring. By modifying the line-graph operator\nto produce a vertex 2-colouring of G0from a vertex 2-colouring of G, we can use\nthese correspondences to observe how periodic binary sequences and spin chains with\nperiodic boundary conditions evolve under the repeated action of \u0015. It turns out that\nperiodic binary sequences whose length is a power of 2 have particularly interesting\ndynamics under \u0015. Line-graph dynamics and some of its properties are discussed\nin Section 2. Then, in Section 3, we encode this dynamics for each length Ninto\nthe characteristic graphs \u0000 n. As an application, we propose a model of information\ngrowth and evolution based on \u0015and simple mechanisms for handling frustrations\nin Section 4. In particular, the proposed model studies the population dynamics of\nspin chains that evolve under \u0015and we observe connections with the characteristic\n3graphs from Section 3. Remarkably, line-graph evolution becomes an ergodic process\nwhen the process begins with an ordered spin chain whose length is a power of 2. All\nthe results are presented in Section 5. Finally, our observations and future possible\ndirections are discussed in Section 6.\n2. The line-graph operator\nWe now describe the action of the line-graph operator [21, 22] on one-dimensional\nspin chains with periodic boundary conditions (other applications of this operator in\nthe context of statistical physics can be found in [23{25]). Since all of these chains\nhave the topology of cyclic graphs, the action of \u0015must preserve the number of\nspins and interactions. As a result, two relevant possibilities for the action of the\nline-graph operator are:\n(i) Transforming interactions into spins\n(ii) Transforming spins into interactions\nIn the \frst case, one new spin chain is produced. In the second case, two new spin\nchains are produced and they are related by spin permutation, that is, swapping the\npositive spins for negative spins and vice versa. The spin chains produced by (i)\nand (ii) will, in general, not be the same. These two transformations can be thought\nof as inverses of each other and, without loss of generality, we label (i) and (ii) by\n\u0015and\u0014, respectively. At this point, we invoke the correspondence from Section 1\nbetween spins and the symbols 0 and 1 which leads to a correspondence between spin\nchains and binary sequences. A periodic binary sequence a= (:::;a\u00001;a0;a1;:::)\nwith period nsatis\fesai2f0;1gandai=ai+nfor alli2Z, and we label it by the\nrepeating binary string a1\u0001\u0001\u0001an. We also consider a correspondence between periodic\n4binary sequences and cyclic graphs with vertex 2-colourings. For the above sequence,\nthe cyclic graph Gcorresponding to it has nverticesv1;:::;v nlabelled clockwise\nwhere the colour of each vertex viisaifor alli. Here, rotational symmetry is imposed\nto ensure that the \frst vertex label v1is arbitrary. In doing so, the symmetry group of\nthis system becomes the cyclic group Cngenerated by the rotation \u001a:ai7!a0\ni=ai+1.\nEquivalently, two sequences aanda0are rotationally equivalent if they are related\nbya0\ni=ai+sfor some integer s. For example, \u001a(0001) = 0010. Equivalence classes\nof periodic binary sequences under Cnare called binary necklaces [26]. In addition,\nwe denote by Othe trivial sequence which has all of its digits equal to zero.\nWe are now ready to describe in detail how the line-graph can be applied to 2-\ncolouring graphs. Due to the previous correspondences, all results obtained here can\nalso be applied to spin chains and binary sequences. The line-graph operator \u0015takes\na 2-colouring of Gand produces a 2-coloring of G0which is de\fned in the following\nway. Letv1;:::;v ndenote the vertices of Gwhose colours are a1;:::;a n, respectively,\nand label the edge which joins viandvjbyeij. The colour a0\nijof the vertex v0\nijinG0\ncorresponding to the edge eijinGis de\fned as\na0\nij\u0011ai+aj(mod 2) (2)\nAs a result, a0\nij= 0 when ai=ajanda0\nij= 1 when ai6=aj. Now let Gbe a\ncyclic graph with nvertices. We can label the edge joining viandvi+1byeifor\ni= 1;:::;n\u00001, and the edge joining vnandv1byen. Denote by v0\nithe vertex in G0\ncorresponding to the edge eiinG. The colouring a0\niof the vertex v0\niinG0is given by\na0\ni\u00118\n><\n>:ai+ai+1(mod 2) if i= 1;:::;n\u00001\nan+a1(mod 2) if i=n(3)\n5For example, \u00152(010110010) = \u0015(111010110) = 001111011. The following result\nallows us to study the behaviour of periodic spin chains transforming under \u0015.\nTheorem 2.1. Let\u0006be a periodic binary sequence of length n. Then\u0015n(\u0006) = 0 for\nall\u0006if and only if nis a power of 2.\nProof. Suppose that \u0006 = a1a2:::a n. The action of \u0015on the periodic sequence \u0006\nmay be represented using n-dimensional vectors over the \fnite \feld F2as follows:\n\u0003n\u0006 =0\nBBBBBBBB@1 1 0\u0001\u0001\u00010 0\n0 1 1\u0001\u0001\u00010 0\n0 0 1\u0001\u0001\u00010 0..................\n0 0 0\u0001\u0001\u00011 1\n1 0 0\u0001\u0001\u00010 11\nCCCCCCCCA0\nBBBBBBBB@a1\na2\na3...\nan\u00001\nan1\nCCCCCCCCA(4)\nwhere \u0003 ndescribes the action of \u0015on binary sequences with period n. Now consider\nRn=0\nBBBBBBBB@0 1 0\u0001\u0001\u00010 0\n0 0 1\u0001\u0001\u00010 0\n0 0 0\u0001\u0001\u00010 0...............\n0 0 0\u0001\u0001\u00010 1\n1 0 0\u0001\u0001\u00010 01\nCCCCCCCCA(5)\nwhich is the n\u0002nrotation matrix representing \u001a. This implies that \u0003 n=In+Rn\nwhereInis then\u0002nidentity matrix and we use Rnn=Into obtain\n\u0003nn= (In+Rn)n= 2In+n\u00001X\ni=1\u0012n\ni\u0013\nRni(6)\nThe \frst term is even and the remaining terms are also even if and only if nis a\npower of 2 as a consequence of Lucas' theorem [27].\nMany sequences of period nwill terminate at Oafter fewer than niterations of \u0015.\nFor example, the sequence 1001 is 4 digits long but it will terminate after just 3\n6iterations since \u00153(1001) =\u00152(1010) =\u0015(1111) = 0000. To make contact with the\nnotation for spin chains, we can replace 0 and 1 with +1 and \u00001, respectively, which\nallows us to write the spin chain \u0006 = 1001 above as \u0006 = ( \u00001;+1;+1;\u00001). Applying\n\u0015to this three times, we \fnd \u00153(\u00001;+1;+1;\u00001) = (+1;+1;+1;+1), as expected.\nEquation (2) tells us how to colour the vertices of G0. It is also a manifestation of\nJ0\nij=\u001bi\u0001\u001bj (7)\nwhereJ0\nijare the interactions between spins \u001b0\niand\u001b0\njof the new spin structure.\nNote that the \fnal stable con\fguration is the trivial sequence O=:::000:::using\nequation (2), but this is equivalent to O= (:::;+1;+1;+1;:::) using equation (7).\n3. Characteristic graphs and equivalence classes\nIn this section, we encode the dynamics of periodic binary sequences of period n\nunder\u0015into characteristic graphs which we label \u0000 n. For instance, when n= 4, the\nequivalence classes under the symmetry group Cnaref0000g,f1111g,f0101;1010g,\nf0011;0110;1100;1001g,f0001;0010;0100;1000gandf0111;1110;1101;1011g. Then,\nwe use representatives from each class to label them: [0000], [0101], and so on.\nMoreover, in general, if any two sequences a1\u0001\u0001\u0001anandb1\u0001\u0001\u0001bnbelong to the same\nequivalence class, their images under \u0015also belong to the same equivalence class.\nFor example, 0100 and 0010 both belong to [0001] and we see that \u0015(0100) = 1100\nand\u0015(0010) = 0110 both belong to [0011]. The line-graph dynamics for n= 4 is\nshown in Figure 1.\n7FIG. 1. Line-graph dynamics for n= 4.\nIn this case, the graph is cycle-free and, as a result, every periodic binary sequence\nof period 4 terminates at some `eigensequence' xof\u0015which is de\fned as a sequence\nsuch that\u0015([x]) = [x]. Furthermore, this graph is connected so there is exactly one\neigensequence for n= 4. Other characteristic graphs, for \u0015are shown in Figure 2,\nwhere we observe the emergence of disconnected components.\nFIG. 2. Line-graph dynamics for n= 1,n= 2 andn= 3.\nFor the sake of simplicity, these digraphs can be reduced to the characteristic graphs\n\u0000nwhere each vertex is an equivalence class. The \frst eight are shown in Figure 3.\n8FIG. 3. Characteristic graphs for n= 1;:::; 8 withCnsymmetry.\nIn the previous \fgure, white vertices represent the trivial sequence and white vertices\nwith a dot in the middle represent non-trivial eigensequences of \u0015such as 011011.\nIt is now useful to introduce spin permutation symmetry which swaps the symbols\n0 and 1 in a binary sequence. Equivalently, it swaps positive and negative spins in\na spin chain. This symmetry is characterised by the symmetric group S2generated\nby the operator \u0019. For example, \u0019(0111001) = 1000110. If we group sequences into\nequivalence classes under both rotational and spin permutation symmetry, then the\nsymmetry group is Cn\u0002S2and we \fnd a di\u000berent collection of characteristic graphs\nwhich are shown in Figure 4.\n9FIG. 4. From left to right: line-graph dynamics with no symmetry (original), S2symmetry,\nCnsymmetry and Cn\u0002S2symmetry. From top to bottom: sequences with n= 2;:::; 8.\n10A more detailed discussion on this topic can be found in [28]. To conclude the\nsection, we remark that other symmetry groups might be identi\fed. For example,\nthe dihedral group Dngenerated by \u001aand the re\rection operator \u001c:ai7!an\u0000i\u0011a\u0000i\nwhose action is \u001c(a1\u0001\u0001\u0001an) =an\u0001\u0001\u0001a1.\n4. Population dynamics of spin chains\nIn this section we study the population dynamics [29] of spin chains under repeated\napplication of the operator \u0014. Recall that \u0014generates two new spin chains so if we\napply\u0014to the spin chain \u0006 = (+1 ;\u00001;\u00001;+1), then we obtain two new spin chains\ngiven by \u0006 1= (+1;+1;\u00001;+1) and \u0006 2= (\u00001;\u00001;+1;\u00001). To verify this, note that\n\u0015(\u00061) =\u0015(\u00062) = \u0006. A particularly useful quantity for studying the spin chains that\ncompose our population is the average magnetisation\nhMi=1\nNNX\ni=1\u001bi (8)\nwhich allows us to assess whether or not a con\fguration is ordered. In particular,\nordered states have hMi=\u00061 and disordered states have hMi\u00180. The sequence\nof spin chains obtained by repeatedly applying \u0014can be studied by computing the\nmagnetisation at each iteration as shown in Figure 5.\n11FIG. 5. On the left: a binary image (white pixels represent 0 and black pixels represent 1)\ndescribing the line-graph dynamics of a randomly chosen spin chain with length N= 210,\nand the magnetisation at each evolutionary step is shown below. On the right: a similar\nimage for a randomly chosen spin chain with length N= 383 along with its associated\nmagnetisation. The process for N= 383 was terminated after 3 Niterations. Each inset is\nan enlargement of the right edge of the related binary image and we observe that the spin\nchain approaches an ordered state for N= 210but not for N= 383.\nIn this \fgure, we see the evolution of spin chains of length N= 210andN= 383.\nThe length of the \frst chain is a power of 2 and it terminates after at most N\niterations, as expected from Theorem 2.1. However, for the same reason, the second\nchain does not terminate into a \fnal stable con\fguration, so we stop the process\n12after 3Niterations. The magnetisation \ructuates close to zero during the entire\nsimulation for the second chain, but it sharply converges towards +1 near the end\nfor the \frst chain. This interesting behaviour, expected from Theorem 2.1, can be\nobserved in the two black-and-white images in Figure 5. For instance, notice that\nthe right-most vector in the \frst image is completely white, but this convergence\ndoes not appear in the second image. It is also worth highlighting that this sort of\norder-disorder phase transition has an underlying dynamics which is very di\u000berent\nfrom those occurring in other classical spin models (e.g. [9, 10]). Usually, either spins\nor interactions are kept quenched , while in this model the line-graph operator varies\nspins and interactions with each iteration.\nNow we can proceed to describe the population dynamics of spin chains under re-\npeated action of \u0014. This model may represent a system of information growth and\nevolution. In particular, information is encoded into spin chains which evolve using\n\u0014. The mechanism to form a population is simple: we begin with an arbitrarily cho-\nsen spin chain (for simplicity, the initial spin chain could have only positive spins or\nonly negative spins). Then we apply \u0014to each new spin chain and we only allow new\ncon\fgurations into the population if they are not duplicates. Moreover, only stable\ncon\fgurations (i.e. those without frustrations) are allowed to generate an o\u000bspring.\nIf we start with a spin chain of length N, then the upper bound for the size of the\npopulation is Nmax= 2NHowever, from Sections 2 and 3 we know that the only\nsequences which can reach this upper bound are those whose length is a power of\n2. Once all accessible con\fgurations are generated by \u0014for a given N, frustrated\ncon\fgurations will appear in the system. However, as previously mentioned, the\nline-graph operator is not applied to them. To deal with this limitation we introduce\nthe following mechanism: frustrated con\fgurations are broken by removing the in-\nteraction responsible for the frustration. As a result, the topology of each frustrated\n13spin chain becomes linear. They are now unable to generate o\u000bspring since this\nwould not preserve the number of spins and the number of interactions under \u0014. In\nsome cases, these broken chains undergo a process where pairs of them recombine\ninto new stable chains which continue to evolve. A pair of broken chains will always\nrecombine when the sum of their lengths is a power of 2. On the other hand, every\nother pair of broken chains will recombine with a probability denoted pr. We have\nsummarised the evolution and recombination processes in Figure 6.\n14FIG. 6. This is a pictorial description of line-graph evolution and recombination. The red\ndotted line represents the interaction to be removed and the black dotted line represents\nthe interaction which is added during recombination. Positive spins and negative spins are\ncontained inside white and black vertices, respectively. The operators \u0015and\u0014are shown\nin blue and green, respectively.\n155. Results\nWe implement Monte Carlo simulations to study the proposed model. Each simula-\ntion begins with a single periodic spin chain whose length Nranges from 1 to 8 and\nthis is allowed to run for 21 evolution steps (or time steps). We compute three rel-\nevant quantities while averaging over 1000 attempts: the number of con\fgurations,\nthe number of frustrated con\fgurations, and the average length of spin chains. Note\nthat the need to implement Monte Carlo simulations is a result of the stochastic\nbehaviour induced by the recombination mechanism associated to a probability pr.\nFigure 7 shows how the number of con\fgurations changes during this evolution for\ndi\u000berent values of Nwhilepr= 0 is \fxed. This value of prallows recombination\nto occur only when the length of the chains is a power of 2. Recall that this pro-\ncess is ergodic when Nis a power of 2 and, as a result, spin chains with length\nN2f1;2;4;8gare able to generate all possible con\fgurations of that size before the\nrecombination phase which allows them to grow.\nFIG. 7. Number of con\fgurations from an initial spin chain with length N2f1;:::; 8g.\nThe inset in plot a)shows two curves \ftting the curve for N= 8. The red circles in plot\nb)highlight where the recombination phases occur for N= 1 andN= 8.\n16In Figure 7, the inset in plot (a)highlights the e\u000bect of recombination for N= 8 by\nshowing two functions which \ft the curve before and after the recombination phase.\nAfter recombination, the growth in the number of con\fgurations slows down up to\napproximately 50% and this e\u000bect is weaker for smaller values of N. Plot (b)in\nthe same \fgure emphasises the recombination phases for N= 1 andN= 8. Unless\notherwise speci\fed, Nrefers to the length of the initial spin chain in each simulation.\nWe now consider the cases where N2f3;5;6;7gby relaxing the constraint on pr.\nThe result obtained for pr2f0:0;0:1;:::; 1:0gare shown in Figure 8.\nFIG. 8. Number of con\fgurations emerging over 21 evolution steps from initial populations\ncomposed a single chain of length N2f3;5;6;7g. As indicated in the legend, there is a\ncurve for each value of pr.\nIn this \fgure, we observe that a higher value of prleads to a larger population, as\n17expected. In light of this, in Figure 9, we compare at di\u000berent evolution steps the\nnumber of con\fgurations obtained for di\u000berent Nwhile varying prfrom 0 to 1.\nFIG. 9. Number of con\fgurations for di\u000berent values of pr. Each plot refers to a speci\fc\nevolution step ESand each line refers to a di\u000berent value of N.\nIn Figure 10, we consider the average spin chain length hN(ES)ifor each value of\nN. We varyprfor the \frst four plots where N2f3;5;6;7g. The last plot considers\nall values of Nfrom 1 to 8 while keeping pr= 0.\n18FIG. 10. Average length hN(ES)iof spin chains during line-graph evolution. The \frst\nfour plots refer to N2f3;5;6;7gasprvaries, and the \fnal plot refers to N2f1;:::; 8g\nwithpr= 0 \fxed.\nResults from N= 1 toN= 8 at di\u000berent evolution steps are shown in Figure 11.\n19FIG. 11. Average length hN(ES)iof spin chains at di\u000berent steps during line-graph evo-\nlution while prvaries. As indicated in the legend, each line refers to the length Nof each\ninitial chain.\nFigure 11 shows an interesting phenomenon: when Nis not a power of 2, the average\nlength of spin chains in each population is greater than for populations where Nis\n20a power of 2. This phenomenon is related to a sort of probability threshold ^ prwhich\ncan be observed in the various plots of Figure 11. For instance, when ES= 5, the\nthresholds for N= 6 andN= 7 are ^pr= 0:4 and ^pr= 0:1, respectively, and when\nES= 21, the threshold for N= 5 andN= 6 are both ^ pr= 0:2. Lastly, in Figure 12,\nwe consider the number of frustrated con\fgurations emerging in each population.\n21FIG. 12. Number of frustrated con\fgurations. The \frst four plots refer to N2f3;5;6;7g\nasprvaries, and the \fnal plot refers to N2f1;:::; 8gwithpr= 0 \fxed.\nThe e\u000bect of recombination emerges as a sequence of steps in each curve in some of\nthe above \fgures, for example Figure 7 and Figure 12.\n226. Conclusion\nThis work investigates the action of the line-graph operator \u0015applied to one-\ndimensional spin models and also studies some of the properties of this operator. We\nencode line-graph dynamics into characteristic graphs \u0000 nbased on di\u000berent sym-\nmetry groups, for example Cn. In addition, we propose a model for investigating\nthe dynamics of populations of spin chains which can be thought of as a system\nof information growth and evolution. The properties of \u0015are studied analytically,\nwhile the evolutionary model is analysed with numerical simulations. Interestingly,\nsome of the results from our simulations re\rect the previously studied properties of\n\u0015. This is an example of a common general relationship between the topology of\na system and dynamical processes within it. For example, this relationship is well\nknown in complex network theory [30{32]. The analysis of \u0014showed that, at some\npoint, frustrated con\fgurations will be generated. In order to deal with this, the\nproposed model includes a mechanism for removing frustrations which sometimes\nallows broken spin chains to recombine into new stable structures. From this, we are\nable to observe a system whose information, encoded into spin chains, continuously\ngrow and evolve. This dynamics resembles that of an automata [33], as suggested\nby Figure 5, whose properties could be further explored. We also highlight that \u0015\ninduces a sort of phase transition from disordered con\fgurations to ordered con\fgu-\nrations only when the length Nof each spin chain is a power of 2. This model may\nalso be of interest to those working at the boundary between information theory\nand physics [34{37] as well as for those working in theoretical biology [38{43]. In\nparticular, this model contains spin chains whose evolution alternates between two\nprocesses which resemble asexual and sexual reproduction. The line-graph operator\ncan be seen as an asexual process, whereas the recombination mechanism resembles\n23a sexual process. Remarkably, some evidence found in nature [44] seems to con\frm\nprevious ideas about alternating sexual-asexual reproduction [45, 46] The probabil-\nity of recombination prcan be seen as a sort of Darwinian \ftness which measures\nthe chance for each initial spin chain to have a rich population of descendants. In\nthis context, we could identify a genealogy tree for every spin con\fguration under\nthe line-graph operator. We envision a potential application where patterns of in-\nformation can be analysed with the aim of computing their `most recent common\nancestor'. For example, this could be used to compare signals or images.\nThis investigation considers line-graph dynamics for spin chains with the following\nconditions: they are one-dimensional, they have periodic boundary conditions, and\nthere are two possible spins: \"and#. We deem that these aspects deserve further\nattention and, therefore, we plan to investigate line-graph dynamics for spin chains in\nhigher dimensions, with di\u000berent topologies (see also [47]), and with more than two\nspins. To conclude, we hope that our results and observations stimulate novel ideas\nin a variety of areas, such as combinatorics, theoretical biology, complex systems and\nother cross-disciplinary domains.\nAcknowledgments\nThe authors are grateful to Francesco Caravelli for his useful comments and obser-\nvations.\n[1] Reddy, P. Kota, S., Permi, K.S., Prashanth, B.: A note on line graphs Mathematical\nCombinatorics 1 119{122 (2011)\n24[2] Galam, S.: Sociophysics: a review of Galam models. International Journal of Modern\nPhysics C 19-03 409{440 (2008)\n[3] Galam, S., Walliserb, B.: Ising model versus normal form game. Physica A: Statistical\nMechanics and its Applications 389-3 481{489 (2010)\n[4] Castellano, C. and Fortunato, S. and Loreto, V.: Statistical physics of social dynamics.\nRev. Mod. Phys. 81-2 591{646 (2009)\n[5] Agliari, E., Barra, A., Sollich, P., Zdeborov\u0013 a, L.: Machine learning and statistical\nphysics: preface. Journal Of Physics A-Mathematical And Theoretical 53-50 500401\n(2020)\n[6] de Oliveira, B.F., ans Szolnoki, A.: Social dilemmas in o\u000b-lattice populations. Chaos,\nSolitons & Fractals 144110743 (2021)\n[7] Duh, M., Gosak, M., Perc, M.: Public goods games on random hyperbolic graphs\nwith mixing. 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Lett. 114\n028103 (2015)\n[21] Harary, F., Norman, R.Z.: Some properties of line digraphs Rendiconti del Circolo\nMatematico di Palermo 9161{168 (1960)\n[22] Beineke, L.W.: Derived graphs of digraphs Sachs, H.; Voss, H.-J.; Walter, H.-J.\n(eds.), Beitrage zur Graphentheorie, Leipzig: Teubner 17-33 (1968)\n[23] Caravelli, F., Markopoulou, F.: Properties of Quantum Graphity at Low Temperature\nPhys.Rev.D 84024002 (2011)\n[24] Chiu, S.C., Ma, D.-S., Song, Z.-D., Bernevig, A., Houck, A.A.: Fragile topology in\nline-graph lattices with two, three, or four gapped \rat bands Physical Review Research\n2043414 (2020)\n[25] Caravelli, F., Saccone, M., Nisoli, C.: On the Degeneracy of Spin Ice Graphs, and its\nEstimate via the Bethe Permanent arXiv:2101.12280 (2021)\n26[26] Mestrovic, R.: Di\u000berent classes of binary necklaces and a combinatorial method for\ntheir enumerations. arXiv:1804.00992 (2018)\n[27] Lucas, E.: Th\u0013 eorie des Fonctions Num\u0013 eriques Simplement P\u0013 eriodiques, Amer. 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From matter to life: information and\ncausality 283033312 (2017)\n[36] Landauer, R.: Information is physical. Physics Today 4423{29 (1991)\n[37] Zurek, W.H.: Complexity, entropy and the physics of information. CRC Press (2018)\n[38] Douglas L.T.: A formal test of the theory of universal common ancestry Nature 465\n219{222 (2010)\n[39] Walker, S.I.: The Natural History of Information. Arti\fcial Life Conference Proceed-\nings329-9 (2020)\n27[40] Walker, S.I.: The new physics needed to probe the origins of life. Nature 569 7754\n(2019)\n[41] Marshall, S.M., Moore,D ., Murray, A.RG., Walker, S.I., Cronin, L.: Quantifying the\npathways to life using assembly spaces. arXiv:1907.04649 (2019)\n[42] Sole, R., Sardanyes, J., Elena, S.F.: Phase Transitions in Virology Preprints\n2020020261 2020\n[43] Nigel, G., Woese, C.: Life is physics: evolution as a collective phenomenon far from\nequilibrium Rev. Condens. Matter Phys. 2375{399 (2011)\n[44] Charlesworth, B.: The cost of meiosis with alternation of sexual and asexual genera-\ntions Journal of Theoretical Biology 87-3 517{528 (1980)\n[45] Green, R.F., Noakes, D.L.G.: Is a little bit of sex as good as a lot? Journal of\ntheoretical biology 17487{96 (1995)\n[46] Neiman, M., Sharbel, T.F., Schwander, T.: Genetic causes of transitions from sexual\nreproduction to asexuality in plants and animals. Evol. Biol. 271346{1359 (2014)\n[47] Chiu, S.C., Ma, D.-S., Song, Z.-D., Bernevig, A., Houck, A.A.: Fragile Topology\nin Line-Graph Lattices with 2, 3, or 4 Gapped Flat Bands Supplementary Material\nPhysical Review Research 2043414 (2020)\n28"    },    {        "title": "0811.4731v2.Coherence_of_single_spins_coupled_to_a_nuclear_spin_bath_of_varying_density.pdf",        "content": "arXiv:0811.4731v2  [quant-ph]  5 May 2009APS/123-QED\nCoherence of single spins coupled to a nuclear spin bath of va rying density\nN. Mizuochi1, P. Neumann2, F. Rempp2, J. Beck2, V. Jacques2, P. Siyushev2, K.\nNakamura3, D. Twitchen4, H. Watanabe5, S. Yamasaki6, F. Jelezko2, J. Wrachtrup21\n11Graduate School of Lib., Inf. & Media Studies, University of Tsukuba, 305-8550 and PRESTO, JST, Japan\n23. Physikalisches Institut, Universit¨ at Stuttgart, Pfaﬀe nwaldring 57, D-70550 Stuttgart, Germany\n3Tokyo Gas Co., Ltd., 3-13-1 Minamisenju, Tokyo, 116-0003, J apan\n4Element Six Ltd, King’s Ride Park, Ascot, Berkshire SL5 8BP, UK\n5Diamond Research Center, AIST, Tsukuba Central 2, Tsukuba, 305-8568, Japan\n6Nanotechnology Research Institute AIST, Tsukuba Central 2 , Tsukuba, 305-8568, Japan\n(Received May 28, 2018)\nThe dynamics of single electron and nuclear spins in a diamon d lattice with diﬀerent13C nuclear\nspin concentration is investigated. It is shown that cohere nt control of up to three individual nuclei\nin a dense nuclear spin cluster is feasible. The free inducti on decays of nuclear spin Bell states and\nsingle nuclear coherences among13C nuclear spins are compared and analyzed. Reduction of a fre e\ninduction decay time T∗\n2and a coherence time T2upon increase of nuclear spin concentration has\nbeen found. For diamond material with depleted concentrati on of nuclear spin, T∗\n2as long as 30 µs\nandT2of up to 1.8 ms for the electron spin has been observed. The13C concentration dependence\nofT∗\n2is explained by Fermi contact and dipolar interactions with nuclei in the lattice. It has been\nfound that T2decreases approximately as 1/ n, where nis13C concentration, as expected for an\nelectron spin interacting with a nuclear spin bath.\nPACS numbers:\nDefect centers in diamond have attracted considerable\ninterest recently owing to their application for quantum\ninformation processing, communication and metrology.\n[1, 2, 3, 4, 5, 6, 7] Especially the nitrogen-vacancy (NV)\ncenter, with its strong and spin dependant optical transi-\ntionsallowsforsinglespinreadoutandexquisitecoherent\ncontrol which is crucial for quantum information applica-\ntions. [1, 2, 3, 4, 5] Owingto the highDebye temperature\nof diamond and weak coupling to acoustic phonons NV\nelectron spins show long coherence time. It was e.g. pro-\nposed to build small quantum registers by exploiting the\ninteraction between the electron spin and a small num-\nber of nuclear spins in the immediate vicinity. Five-qubit\nwould be suﬃcient to perform all functions necessary for\na node in a defect center based quantum repeater node.\n[4, 5] Up to now coherent control, swapping of coherence\nand even entanglement between up to two nuclei and the\nelectron spin was demonstrated. [3] To increase the size\nof the quantum register, more nuclei need to be coupled\nto the electron spin. The approach taken here is to in-\ncrease the concentration of paramagnetic13C nuclei in\nthe lattice. We systematicallydemonstrate coherentcon-\ntrol of up to three nuclear spins being coupled to an NV\ncenter electron spin in13C isotopically enriched crystals,\nnotwithstanding the fact that the electron decoherence\ntimeT2linearly scales with the13C concentration. Fur-\nthermore, our experiments provide experimental insight\ninto long studied problem of single central spin coupled\nto aparamagneticenvironment.[8, 9, 10] Owingto possi-\nbility to address individual electron spins in matrix with\nadjustable nuclear spin content we show the transition\nfrom diluted to dense spin bath (the situation relevant\nfor spins in GaAs quantum dots).The quantum system used in the present work is the\nnegatively charged NV center in diamond, which com-\nprises a substitutional nitrogen atom with an adjacent\nvacancy. (Fig. 1(h)) The electron ground state of it is a\nspintriplet. UponopticalexcitationtheNVcentershows\nstrong ﬂuorescence allowing it to be observed on an in-\ndividual basis by confocal microscopy. The ﬂuorescence\nintensity of the defect is spin-dependent owing to spin\nselective relaxation via singlet state, which allows opti-\ncal read out of the single electron spin resonance (ESR)\n[11] and an eﬃcient electron and nuclear initialization\nat room temperature. Microwaves (MW) and radiofre-\nquency (RF) ﬁelds are used for coherent manipulation\nof single electron and nuclear spins using conventional\nESR/NMRtechniques. Allmeasurementiscarriedoutat\n∼20◦C. Diamond enables for the unique opportunity to\ncontrol the concentration of paramagnetic nuclear spins.\nThe most abundant12C has zero nuclear spin. The con-\ncentration of13C nuclear spins ( I=1/2) can be adjusted\nto the suitable value by controlling the isotopic content\nof13C in the growth medium. Two types of synthetic\ndiamonds were used in this study. Crystals with 0.35%,\n1.1%, 8.4%, 20.7%13C concentrations were synthesized\nby a MW plasma-assisted homoepitaxial chemical va-\npor deposition technique (CVD) [12, 13] with13CH4or\n12CH4gases. The 0.03%13C and∼100%13C enriched\ndiamondsweresynthesizedbyhighpressurehightemper-\nature method (HPHT) using getters preventing incorpo-\nration of nitrogen into lattice. In all samples the concen-\ntration of paramagnetic impurity measured by ESR was\nunder detection limit (below 1 ppb) except 0.03% and\n100%13C diamond where the concentration of nitrogen\nwas at 1 ppm level. [14]2\n(e) \n(f) \n(g) \n(h) \nFIG. 1: (Color online) ESR spectra of single NVs with (a)\nzero, (b) one, (c) two, (d) three13C in the 1st shell. Blue\nsolid lines are simulation spectra. ESR spectra with (e) one\nand (f) two13C in the 3rd shell, and (g) one13C in the 1st\nshell and one13C in the 3rd shell. Blue solid lines are ﬁtting\ncurves with Gaussian shape. (h) Atomic structure of NV\ncenter. The numbers 1, 2, and 3 mark C in the 1st shell.\nNV centers with diﬀerent numbers of13C atoms in the\nimmediate vicinity of the electron spin have been inves-\ntigated in 8.4%13C diamond. Fig. 1(a-d) shows single\nESR spectra indicating the hyperﬁne coupling (HFC) of\nthe electron spin to zero, one, two, and three nuclear\nspins in the ﬁrst shell (ﬁg. 1(h)). A magnetic ﬁeld of\n∼83 Gauss was oriented along the NV axis ([111]-axis).\nTo characterize the spin quantum states associated with\nthe transitions in Fig. 1(b-d), the spectra were simulated\nby exact diagonalization of the spin Hamiltonian\nH=geβe˜SB+˜SDS+(˜SAIi−gnβnIiB) (1)\nHere the electron spin S= 1 and13C nuclear spins in\n1st shell are taken into account. βeis the Bohr and\nβnthe nuclear magneton, respectively. Reported val-\nues for zero ﬁeld splitting (ZFS) parameter ( |D|= 2.87\nGHz),[15]isotropicelectronandnuclearZeemang-values\n(ge=2.0028, gn=1.40483), and HFC parameters of A/bardbl=\n205 MHz and A⊥= 123 MHz [16] with angle of 106◦be-\ntween principal axes of ZFS and HFC, yield precise ﬁts\nof the experimental spectra. The small splittings in the\ncentral signals of the spectra in Fig. 1(c,d) are explained\nby a 2nd order perturbation approach. [3] The smaller\namplitudes in higher frequency are due to absorption of\nMW by wire on the sample.\nIn the ESR spectra of8.4%13C diamond, basicallytwo\ntypes of couplings are immediately visible (Fig. 1(b-g)),thosearound130MHzoriginatingfromﬁrstshell13Cand\nthose around 14 MHz. In C3vsymmetry, the number of\nequivalent atoms in close shells around NV is 3 or 6. In\nrecent theoretical study, [17] the 14 MHz splitting are\nassigned to13C at 3 and 6 equivalent sites in the 3rd\nshell (see Video in [18].) From measuring more than 250\nindividual centers and comparing the probability to ﬁnd\nthe 14 MHz splitting with the one predicted from theory\nwe assign this splitting to nuclei in the 3. shell. [18]\nIndividualnucleiinthespinclusteraroundtheelectron\nare addressed via their particular NMR frequency. Given\nthe increase in spectral density apparent from Fig. 1 one\nmight wonder in how far individual nuclei remain ad-\ndressable. However, coherent control even in dense spin\nclusters remains feasible as demonstrated in Fig. 2. Even\nin cases where there are three13C in the 1st shell i.e. in\ntotal four qubits, Rabi nutations of single nuclear spins\ncan be driven by an additional RF as shown in Fig. 2(c).\nThat is because even multiple RF transition frequencies\noriginating from nuclei at equivalent positions split due\nto higher order HFC contributions as shown above. The\nselectivityisnotlimitedtothe relativelylargesplitting in\nthe1stshellbutcanbeappliedto13Cnuclearspinsinthe\n3rd shell. A spectrum of single NV which has one13C in\nthe1stshellandone13Cin3rdshellisshowninFig.1(g).\nWe labeled the four nuclear spin states as |00/angbracketright,|01/angbracketright,|10/angbracketright,\nand|11/angbracketrightas shown in Fig. 2(a). Rabi oscillations between\n|00/angbracketrightand|01/angbracketrightcould be observed as shown in Fig. 2(d). To\nobtain a similar Rabi frequency for13C nuclear spins in\nthe 3rd shell as for those in the 1st shell, about 1 ×102\ntimes higher RF power was necessary. This can be in-\nterpreted mainly by hyperﬁne enhancement , [19] which\npredicts that Rabi frequency is almost proportional to\nHFC and the square root of the RF power. A 10 times\nsmallerHFC in the 3rd shell supports this interpretation.\nA ﬁgure of merit which characterizes the quality of co-\nherent control was derived by swapping quantum states\namong individual nuclei. It was estimated by transfer-\nring polarization back to a detectable electron spin state\nthat 82±5 % of polarization was transferred from |00/angbracketrightto\n|01/angbracketrightfor13C nuclear spins in the 3rd shell.\nWe are now in a position to engineer simple quantum\nstates in the spin cluster around the electrons spin. Bell\nstates Φ±= 1/√\n2(|00/angbracketright ±|11/angbracketright) and Ψ±= 1/√\n2(|01/angbracketright ±\n|10/angbracketright)canbegeneratedfromthefoureﬀectivenuclearspin\nstatesinFig.2(a). Inthepresentcasetheywereprepared\nfrom two13C spins at the 1st shell. The procedure fol-\nlows previous studies [3, 20] and is schematically shown\nfor Φ−in Fig. 2(k). After its generation, Φ−shows a\nfree induction decay (FID) which is made visible with an\nentanglement detectorsequence (Fig. 2(k)). In 8.4%13C\ndiamond, the free induction decay times T∗\n2of Ψ−(T∗\n2(Ψ))\nand Φ−(T∗\n2(Φ)) were estimated to be 22 .0±3.0µs and\n13.3±1.1µs, (see Fig. 2(f,g)) respectively, by ﬁtting with\nexp[���(t/T∗\n2)2] cos(∆ωt), [8] where ∆ ωis the detuned3\n(c) \nµ(b) \n(d) (i)1  C 13 \n(a) (ii)2  C 13 (iii)3  C 13 \nµΦΨ\nτ\n(e) \nFIG. 2: (Color online) (a) Energy levels for nuclear spins in\nMS=-1. Rabi oscillation of single13C in the 1st shell of neigh-\nbors around theNVwith (b) two13Cin the1st shell, (c)three\n13C in the 1st shell. (d) Rabi oscillation of single13C in the\n3rd shell with one13C in the 1st shell and one13C in the 3rd\nshell. The pulse sequence is π(MW)-Rabi(RF)- π(MW). [1]\nThe ESR transitions of the MW πpulse are those at lowest\nfrequency in Fig. 1 (c,d,g) and are indicated by blue arrows\nin (a). The NMR transitions of the RF pulse are indicated\nby orange arrows in (a). The recording the data in (b-d) re-\nquired about 20 minutes of averaging. (e) ENDOR spectrum\nof13C at 1st shell in (b) with the pulse sequence of π(MW)-\nπ(RF)-π(MW). FID of (f) Ψ−, (g) Φ−, (h) nuclear coherence\nbetween |00/angbracketright↔|10/angbracketrightand (j)|00/angbracketright↔|01/angbracketright. (k) Pulse sequence for\nΦ−generation and detection between two nuclear spins. E\nand N 1,2marks the electron and the two nuclear spins, re-\nspectively. Spin selective pulses are represented by squar es,\noperating on a target qubit. Vertical lines represent logic al\nconnections. The control qubit state |1/angbracketrightand the state |0/angbracketrightare\ndisplayed as ﬁlled ( •) and open ( ◦) circles. For example, ( ◦)\nindicates that the pulse is applied to the target qubit if the\nquantum state of the controlling qubit is |0/angbracketright.\nfrequency of FID. As expected from the view point of\ndecoherence free subspaces, [21] a longer T∗\n2(Ψ)compared\nto that of T∗\n2(Φ)is observed.\nThe diﬀerence among T∗\n2(Ψ)andT∗\n2(Φ)is best analyzed\nwhen compared with T∗\n2of a nuclear quantum coher-\nence among states |00/angbracketright↔|10/angbracketrightand|00/angbracketright↔|01/angbracketright. Those\ncoherences are labeled as single quantum coherences\nSQ1 and SQ2, respectively. Their T∗\n2are measured to\nbeT∗\n2(SQ1)=41.1±3.1µs andT∗\n2(SQ2)=15.8±1.4µs, re-\nspectively (Fig. 2(h,j)). The diﬀerence of T∗\n2(SQ1)and\nT∗\n2(SQ2)might be caused by a spatially inhomogeneous\nmagnetic noise around the defect caused by an inho-\nmogeneous distribution of13C around the two13C in\nthe 1st shell. Each spin-spin interaction between nu-\nclear spin ksurrounding the two13C in the 1st shell\nwith quantum numbers mI1andmI2cause oscillation\ngiven by/summationtext\nkexp[−i(∆ω1mI1+ ∆ω2mI2)t]. [22] Here(b) \n8.4% \n(d) (e) \n(c) \n1.1% (a) \n20.7% \nFIG. 3: (Color online) ESRspectraof single NVin (a) 20.7 %,\n(b) 8.4 %13C diamond with ﬁtted Gaussian lines (blue). (c)\nFourier transformed spectrum of FID of 1.1%13C diamond\nshown in (d) on expanded frequency axis. The splitting is due\nto HFC of distant13C nuclear spins. The hyperﬁne splitting\nto N is not visible in this spectrum due to polarization of\nthe N nuclear spin. [28] (e) Dependence of Inhomogeneous\nlinewidth on13C concentration. The error bars indicates the\ndistributions measured.\n∆ω1and ∆ω2are spin-spin interaction frequencies of\nthe two13C in the 1st shell due to surrounding13C\nnuclear spins. This implies that T∗\n2(Ψ)andT∗\n2(Φ)can\nbe approximated by 1 /T∗\n2(Ψ)=|1/T∗\n2(SQ1)−1/T∗\n2(SQ2)|\nand 1/T∗\n2(Φ)=1/T∗\n2(SQ1)+1/T∗\n2(SQ2), respectively. Insert-\ning the measured values for 1 /T∗\n2(SQ1,2), the results are\nT∗\n2 calc.(Ψ)=25.8+5.7\n−4.3µs andT∗\n2 calc.(Φ)=11.4+1.0\n−0.9µs, respec-\ntively, in good correspondence with measured values.\nBesides the eﬀect of the nuclear spin bath on individ-\nual13C spins the static interaction between the single\nNV electron spin and its environment for diﬀerent13C\nconcentrations was investigated by measuring T∗\n2i.e. the\ninhomogeneous ESR linewidth. It was observed that T∗\n2\nincreases i.e. the linewidth narrows, with decreasing13C\nconcentrationasshowninFig.3. In0.03%13Cdiamond,\nan extremely long T∗\n2of 30µs was found (see Fig. 3). In\nthe low13C concentration region ( ≦1.1%), the linewidth\nW(full width at half maximum) is derived from T∗\n2by\nW= 2√\nln2/πT∗\n2. Thecorresponding18kHzlinewidthis\nthe narrowesteverobservedfor an electronspin in a solid\nmaterial. In the high13C concentration region ( >1.1%),\nthe linewidth is derived from ﬁtting the ESR line of a\nsingle NV with a Gaussian lineshape. Average values are\nplotted as squares in Fig. 3(e).\nA likely cause for the inhomogeneous ESR linewidth is\nHFC to13C nuclear spins. In e.g. Si, the dependence of\nthe inhomogeneous linewidth of P donors in29Si is well\nﬁtted by the isotropic HFC ( al) due to Fermi contact\ninteraction with29Si nuclear spins with a concentration\n(f),W= 2√\n2ln2[f/summationtext\nl(al/2)2]1/2.[23, 24] The sum runs\nover all nuclear spin sites l. In Fig. 3(e), the solid line for\nhigh13C concentrations is calculated by summing only4\nover all the 9 sites in the 3rd shell with al= 14 MHz\n(see above for assignment of sites and HFC constants).\nIt should be noted that contributions from13C in the 1st\nshell were not considered in the linewidth calculations\nbecause they contribute to an observable splitting but\nnot to the linewidth. As seen from Fig. 3(e), it ﬁts the\nexperimental results well for high13C concentration.\nFor lower13C concentration, experimental data devi-\nate from this behavior. This is due to the fact that the\nprobability that any13C is located close to the NV cen-\nter is getting small upon reduction of13C concentration.\nFurthermore, the unpaired electron spin density rapidly\ndecreases with distance from the three dangling bonds\naroundthe vacancy. Thisis knownfrom the HFC param-\neters [16, 17] which indicates that almost100% spin den-\nsity is localized on the C sites in the 1st and the 3rdshell.\nThat is why in this situation the most prominent con-\ntribution to the inhomogeneous linewidth is the weaker\ndipole-dipole interaction between electron spin and13C\nnuclear spin at distant sites. The lower line in Fig. 3(e)\nis the linewidth\nW=/radicalbig\n(µ0µeµngegn/4πh)2(3.195×1046n),(2)\ncalculated from the 2nd moment [25] with more than\n3,000 lattice sites for each13C concentration ( n). Con-\ntributions from13C in the 1st and 2nd shell are not con-\nsidered. As seen from Fig. 3(e), Wﬁts the experimental\nresults in the low13C concentration ( ≦1%) quite well.\nObviouslyat low13C concentrationthe linewidth is dom-\ninated by dipole-dipole interaction.\nThe dephasingtime T2ofthe electron spin is measured\nby two pulse Hahn echo decay curves (Fig. 4(a-c)). We\nanalyzed T2of the diamond made by CVD and excluded\nthe 0.03 % and 100 %13C diamond made by HPHT be-\ncause paramagnetic impurities could not be suppressed\nin HPHT. In the 1.1 %13C diamond, a T2of 0.65 ms was\nfound, which is the longest T2in diamond with natural\nabundance of13C measured so far and for the lower13C\nconcentration of 0.3% an even longer T2of 1.8 ms was\nmeasured. [26] T2is found to be inversely proportional\nto the13C concentration as plotted in Fig. 4(d).\nIn a theoretical analysis of T2by the disjoint cluster\napproach, [5] the relationship of T2∼(¯CAc)−1/2is pro-\nposed, where ¯Cis the averaged nuclear-nuclear dipolar\ninteractioninthebathand Acissomecharacteristicvalue\nfor the electron-nuclear dipolar interaction. Since both\ninteractions scale linearly in13C concentration ( n),T2\ndecreases approximately as 1/ nin this model. The ﬁtted\nline to the data shown in ﬁg. 4 (d) supports this inverse\nproportionalityandﬁts ourdata. Note that ourdata also\nﬁt the values calculated in [5] within 30% deviation.\nIn conclusion, coherent control of up to three individ-\nual nuclei in a densenuclearspin clusteris demonstrated.\nThe13C concentration dependence of T∗\n2andT2of elec-\ntron spin point towards13C nuclei as the main cause forµµ\nFIG. 4: (Color online) Echo decays of electron spin in (a)\n20.7 %, (b) 8.4 %, and (c) 1.1%13C diamond. The MW pulse\nsequence is π/2-τ-π-τ-π/2 where τis delay [8]. Red lines are\ncurves ﬁtted with exp[−(t/T2)3]. (d) Plot of T2over13C con-\ncentration n. The solid line is ﬁtted with a 1/ ndependence.\ndephasing in otherwise clean diamond. The correspon-\ndence with the theoretical line of T2[5] is very important\nto elucidate the dephasing mechanism and to make T2\nlonger for quantum information devices [4] and magne-\ntometry. [6, 7] Furthermore, the results show that the\nthreshold ( ∼104operation) for quantum error-correction\nschemes [27] can be exceeded even in13C enriched dia-\nmond at room temperature with typical single-qubit ﬂip\nof several ns.\nThis work is supported by the EU (QAP, EQUIND,\nNANO4DRUGS, NEDQIT), DFG (SFB/TR21,\nFOR730), JST-DFG program, KAKENHI (20760006)\nand the Landesstiftung BW. VJ acknowledges support\nby the Humboldt Stiftung. We thank Dr. H. Kanda for\nproviding 100%13C diamond.\n[1] F. Jelezko, et al., Phys. Rev. Lett. 93, 130501 (2004).\n[2] M. V. G. Dutt, et al., Science 316, 1312 (2007).\n[3] P. Neumann, et al., Science, 320, 1326 (2008).\n[4] (a) L. Jiang, et al., Phys. Rev. A, 76, 062323 (2007). (b)\nL. Jiang, et al., quant-ph/0703029\n[5] J. R. Maze, et al., Phys. Rev. B, 78, 094303 (2008).\n[6] G. Balasubramanian, et al., Nature, 455, 648 (2008).\n[7] J. R. Maze, et al., Nature, 455, 644 (2008).\n[8] L. Childress, et al., Science, 314, 281 (2006).\n[9] Takahashi et al., PRL 101 047601 (2008).\n[10] R. Hanson, et al, Science, 320, 352 (2008).\n[11] A. Gruber et al., Science 276, 2012 (1997).\n[12] J. Isberg, et al., Science, 297, 1670 (2002).\n[13] N. Mizuochi, et al., J. Appl. Phys. 101, 103501 (2007).\n[14] K. Nakamura, et al., Diamond Relat. Mat. 16, 1765\n(2007).\n[15] N. B. Manson et al., Phys. Rev. B 74, 104303 (2006).\n[16] J. H. N. Loubser, et al., Diamond Res. 11-14 (1977).\n[17] A. Gali, et al., Phys. Rev. B, 77, 155206 (2008).\n[18] Supplemental materials are temporarily available in\nhttp://www.slis.tsukuba.ac.jp/∼mizuochi/suppl/Sup.html\n[19] A. Schweiger et al., Principles of Pulsed EPR (Oxford5\nUniv. Press, 2001)\n[20] M. Mehring, et al., Phys. Rev. Lett., 90, 153001 (2003).\n[21] C. F. Roos, et al., Phys. Rev. Lett., 92, 220402 (2004).\n[22] R. Ernst, et al., Principles of NMR in one and two di-\nmensions (Oxford press, 1989), Chap. 5.\n[23] W. Kohn, Solid state physics (Academic Press, New\nYork, 1957), vol. 5, p. 257.\n[24] E. Abe et al., arXiv:cond-mat/0512404[25] J. H. Van Vleck, Phys. Rev. 74, 1168 (1948).\n[26] G. Balasubramanian, et al., Nature Materials, 8, 383\n(2009).\n[27] D. D. Awshalom, et al., Semiconductor Spintronics and\nQuantum Computation (Spring-Verlag, Berlin, 2002).\n[28] V. Jacques et al., Phys. Rev. Lett., 102, 057403 (2009)."    },    {        "title": "1502.05400v1.Quenching_of_dynamic_nuclear_polarization_by_spin_orbit_coupling_in_GaAs_quantum_dots.pdf",        "content": "arXiv:1502.05400v1  [cond-mat.mes-hall]  18 Feb 2015Quenching of dynamic nuclear polarization by spin-orbit co upling in GaAs quantum\ndots\nJohn M. Nichol,1Shannon P. Harvey,1Michael D. Shulman,1Arijeet Pal,1Vladimir\nUmansky,2Emmanuel I. Rashba,1Bertrand I. Halperin,1and Amir Yacoby1\n1Department of Physics, Harvard University, Cambridge, MA, 02138, USA\n2Braun Center for Submicron Research, Department of Condens ed Matter Physics,\nWeizmann Institute of Science, Rehovot 76100 Israel\nThe central-spin problem, in which an electron spin interac ts with a nuclear spin bath, is a widely\nstudied model of quantum decoherence [1]. Dynamic nuclear p olarization (DNP) occurs in central\nspin systems when electronic angular momentum is transferr ed to nuclear spins [2] and is exploited\nin spin-based quantum information processing for coherent electron and nuclear spin control [3].\nHowever, the mechanisms limiting DNP remain only partially understood [4]. Here, we show that\nspin-orbit coupling quenches DNP in a GaAs double quantum do t [5], even though spin-orbit cou-\npling in GaAs is weak. Using Landau-Zener sweeps, we measure the dependence of the electron\nspin-ﬂip probability on the strength and direction of in-pl ane magnetic ﬁeld, allowing us to distin-\nguish eﬀects of the spin-orbit and hyperﬁne interactions. T o conﬁrm our interpretation, we measure\nhigh-bandwidth correlations in the electron spin-ﬂip prob ability and attain results consistent with\na signiﬁcant spin-orbit contribution. We observe that DNP i s quenched when the spin-orbit compo-\nnent exceeds the hyperﬁne, in agreement with a theoretical m odel. Our results shed new light on the\nsurprising competition between the spin-orbit and hyperﬁn e interactions in central-spin systems.\nDynamic nuclear polarization occurs in many con-\ndensed matter systems, and is used for sensitivity en-\nhancement in nuclear magnetic resonance [6] and for de-\ntecting and initializing solid-state nuclear spin qubits [7].\nDNP also occurs in two-dimensional electron systems [8]\nvia the contact hyperﬁne interaction. In both self-\nassembled [9–13] and gate-deﬁned quantum dots [3, 14–\n16], for example, DNP is exploited to create stabilized\nnuclearconﬁgurationsforimprovedquantuminformation\nprocessing. Closed-loop feedback [15] based on DNP, in\nparticular, is a key-component in one- and two-qubit op-\nerations in singlet-triplet qubits [3, 17, 18].\nDespite the importance of DNP, it remains unclear\nwhat factors limit DNP eﬃciency in semiconductor spin\nqubits [4]. In particular, the relationship between the\nspin-orbit and hyperﬁne interactions [19–21] has been\noverlooked in previous experimental studies of DNP in\nquantum dots. In this work we show that spin-orbit\ncoupling competes with the hyperﬁne interaction and\nultimately quenches DNP in a GaAs double quantum\ndot [5, 17], even though the spin orbit length is much\nlarger than the interdot spacing. We use Landau-Zener\n(LZ) sweeps to characterizethe static and dynamic prop-\nerties of ∆ ST(t), the splitting between the singlet Sand\nms= 1 triplet T+, and the observed suppression of DNP\nagrees quantitatively with a new theoretical model.\nFigure 1(a) shows the double quantum dot used in\nthis work [5, 17]. The detuning, ǫ, between the dots\ndetermines the ground-state charge conﬁguration, which\nis either (1,1) [one electron in each dot], or (0,2) [both\nelectrons in the right dot] as shown in Fig. 1(b). To mea-\nsure ∆ ST(t), the electrons are initialized in |(0,2)S∝angbracketright,ǫis\nswept through the S−T+avoided crossing at ǫ=ǫST,\nandtheresultingspinstateismeasured[Fig.2(a)]. Intheabsenceof noise, slowsweeps causetransitions with near-\nunity probability. For large magnetic ﬁelds, however, we\nﬁnd maximum transition probabilities of approximately\n0.5. This reduction is a result of rapid ﬂuctuations in\nthe sweep rate arising from charge noise (see Supple-\nmentary Information). Even in the presence of noise,\nhowever, the average LZ probability ∝angbracketleftPLZ(t)∝angbracketrightcan be ap-\nproximated for fast sweeps as2π/angbracketleft|∆ST(t)|2/angbracketright\n/planckover2pi1β(see Supple-\nmentary Information). Here ∝angbracketleft···∝angbracketrightindicates an average\nover the hyperﬁne distribution and charge ﬂuctuations,\nandβ=d(ES−ET+)/dtis the sweep rate, with ES\nandET+the energies of the SandT+levels. To ac-\ncurately measure σST≡/radicalbig\n∝angbracketleft|∆ST(t)|2∝angbracketright, we therefore ﬁt\n∝angbracketleftPLZ∝angbracketrightvsβ−1to a straight line for values of βsuch that\n0<∝angbracketleftPLZ∝angbracketright<0.1. [Fig. 2 (a)].\nWe ﬁrst measure σSTvsφatB= 0.5 T [Fig. 2(b)],\nwhereφis the angle between the magnetic ﬁeld Band\nthe z axis [Fig. 1(a)]. σSToscillates between its extreme\nvalues at 0◦and 90◦with a periodicity of 180◦. Fixing\nφ= 0◦and varying B, we ﬁnd that σSTdecreasesweakly\nwith with B, but when φ= 90◦,σSTincreases steeply\nwithB, reaching values greater than 10 times that for\nφ= 0◦, as shown in Fig. 2(c).\nWe interpret these results by assuming that both\nthe hyperﬁne and spin-orbit interactions contribute to\n∆ST(t) and by considering the charge conﬁguration of\nthe singlet state at ǫST[Figs. 1(b) and (c)]. The matrix\nelement between SandT+can be written as ∆ ST(t) =\n∆HF(t)+∆SO. ∆HF(t) =g∗µBδB⊥(t) is the hyperﬁne\ncontribution, which arises from the diﬀerence in perpen-\ndicular (relative to B) hyperﬁne ﬁeld, δB⊥(t), between\nthe two dots [22]. (In the following, we set g∗µB= 1.)\n∆HF(t), which is a complex number, couples |(1,1)S∝angbracketright\nto|(1,1)T+∝angbracketrightwhen the two dots are symmetric. ∆ SO2\nε250 nm \nenergy (0,2) S \n(0,2) S (1,1) T -\n(1,1) S \n(1,1) T 0\n(1,1) T +\nS T+\n(1,1) S ∆ST (t) |g*| µBB\ndetuning  (ε)(a)\n(c)(b)\nφ\nB\nxz\n(1,1) T +(1,1) S \n(0,2) S Hyperfine\nSpin orbit0εST \nB ΩSO B ΩSO φ=0° φ=90°\nFIG. 1. Experimental setup. (a) Scanning electron micro-\ngraph of the double quantum dot. A voltage diﬀerence be-\ntween the gates adjusts the detuning ǫbetween the potential\nwells, and a nearby quantum dot on the left senses the charge\nstate of the double dot. The gate on the right couples the\ndouble dot to an adjacent double dot, which is unused in this\nwork. The angle between Band the zaxis isφ. (b) En-\nergy level diagram showing the two-electron spin states and\nzoom-in of the S−T+avoided crossing. (c) The hyperﬁne\ninteraction couples |(1,1)S/angbracketrightand|(1,1)T+/angbracketrightwhen the two dots\nare symmetric, regardless of the orientation of B, and the\nspin-orbit interaction couples |(0,2)S/angbracketrightand|(1,1)T+/angbracketrightwhenB\nhas a component perpendicular to ΩSO= ΩSOˆz, the eﬀective\nspin-orbit ﬁeld experienced by the electrons during tunnel ing.\nis the spin-orbit contribution, which arises from an ef-\nfective magnetic ﬁeld ΩSO= ΩSOˆzexperienced by the\nelectron during tunneling [19]. Only the component of\nΩSO⊥Bcauses an electron spin ﬂip. ∆ SOtherefore\ncouples|(0,2)S∝angbracketrightto|(1,1)T+∝angbracketrightwhenφ∝negationslash= 0◦, and Ω SO\nis proportional to the double-dot tunnel coupling [19],\nwhich is 23.1 µeV here. At ǫST, the singlet state |S∝angbracketrightis a\nhybridized mixture: |S∝angbracketright= cosθ|(1,1)S∝angbracketright+ sinθ|(0,2)S∝angbracketright,\nwhere the singlet mixing angle θapproaches π/2 asB\nincreases (see Supplementary Information). Taking both\nθandφinto account, we write [19]\n∆ST(t) = ∆HF(t)+∆SO\n=δB⊥(t)cosθ+ΩSOsinφsinθ.(1)\nThe data in Fig. 2(b) therefore reﬂect the dependence\nof ∆ST(t) onφin equation (1). The data in Fig. 2(c) re-\nﬂect the dependence of ∆ ST(t) onθ. AsBincreases,\nθalso increases, and |S∝angbracketrightbecomes more |(0,2)S∝angbracketright-like,\ncausing ∆ HF(t) to decrease. When φ= 0◦, ∆SO= 0\nfor allB, but when φ= 90◦, ∆SO= ΩSOsinθ, andφ=0° φ=90°(a)\n(c) (b)time load sweep meas. \n(0,2) \n(1,1) εPLZ \n φ (degrees) σST /h (MHz) \nσST /h (MHz) (h/β) (x10-3  µs/GHz) (h β)-1  (x10 -4  µs/GHz) \nεST \nB=0.5 TB=1 T\nφ=90°\nB (T) 0 0.5 1020 40 60 80 100 \n-180 -90 090 180010 20 30 40 50 60 70 0 2 4 6 800.20.40.60.81\n0 1 2 300.050.1PLZ \nFIG. 2. Measurements of σST. (a) Data for a series of LZ\nsweeps with varying rates, showing reduction in maximum\nprobability due to charge noise. The horizontal axis is pro-\nportional to the sweep time. Upper inset: Data and linear\nﬁt for fast sweeps such that 0 </angbracketleftPLZ/angbracketright<0.1. Lower Inset:\nIn a LZ sweep, a |(0,2)S/angbracketrightstate is prepared, and ǫis swept\nthrough ǫST(dashed line) with varyingrates. Here h= 2π/planckover2pi1is\nPlanck’s constant. (b) σSTvsφ(dots) and simulation (solid\nline). (c) σSTvsBforφ= 0◦andφ= 90◦(dots) and ﬁts to\nequation (1) (solid lines). Error bars are ﬁt errors.\nσSTincreases with B. Fitting the data in Fig. 2(c) al-\nlows a direct measurement of the spin-orbit and hyper-\nﬁne couplings (see Supplementary Information). We ﬁnd/radicalbig\n∝angbracketleft|δB2\n⊥(t)|∝angbracketright= 34±1 neV and Ω SO= 461±10 neV, cor-\nresponding to a spin-orbit length λSO≈13µm [19], in\ngoodagreementwith previousestimates in GaAs[23–25].\nWe further verify that ∆ ST(t) contains a signiﬁ-\ncant spin-orbit contribution by measuring the dynam-\nical properties of PLZ(t). A key diﬀerence between\nthe spin-orbit and hyperﬁne components is that ∆ SO\nis static, while ∆ HF(t) varies in time because it arises\nfrom the transverse Overhauser ﬁeld, which can be con-\nsidered a precessing nuclear polarization in the semi-\nclassical limit [22]. To distinguish the components of\n∆ST(t) through their time-dependence, we develop a\nhigh-bandwidth technique to measure the power spec-\ntrum ofPLZ(t).\nInstead of measuring the two-electron spin state after\na single sweep, ǫis swept twice through ǫSTwith a pause\nof length τbetween sweeps [Fig. 3(a)] (See Supplemen-\ntary Information). Assuming that St¨ uckelberg oscilla-\ntions rapidly dephase during τ[18, 23], and after sub-\ntracting a background and neglecting electron spin re-3\n(a) \n(b) \n(c)\n(d) time (0,2) load \nτmeas. \n(1,1) εRPP (τ)\nf71 Ga -f 69 Ga f69 Ga -f 75 As f71 Ga -f 75 As \nf75 As f69 Ga f71 Ga sweep sweep SP(ω)   φ (degrees) \nφ=0°φ=25°φ=80°εST \n0 10 20 30 40 50 60 70 80 -0.1-0.0500.050.1\nτ ( µs) \n0510 15 \n020 40 60 80 \nω/2 π (MHz) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 B=0.1T , φ=0°\nFIG. 3. Correlations and power spectrum of PLZ(t). (a)\nPulse sequence to measure RPP(τ) using two LZ sweeps. (b)\nRPP(τ)forφ= 0◦andB= 0.1T.Thedataextendto τ= 200\nµs, but for clarity are only shown to 75 µs here. (c) SP(ω)\nvsφobtained by Fourier-transforming RPP(τ). Atφ= 0◦,\nthe diﬀerences between the nuclear Larmor frequencies are\nevident, but for |φ|>0◦, the absolute Larmor frequencies ap-\npear, consistent with a spin-orbit contribution to σST. The\nreduction in frequency with φis likely due to the placement of\nthe device slightly oﬀ-center in our magnet (see Supplemen-\ntary Information). (d) Line cuts of SP(ω) atφ= 0◦, 25◦, and\n80◦.\nlaxation, the time-averaged triplet return probability is\nproportional to RPP(τ)≡ ∝angbracketleftPLZ(t)PLZ(t+τ)∝angbracketright, the au-\ntocorrelation of the LZ probability [Fig. 3(b)]. Taking a\nFourier-transform therefore gives SP(ω), the power spec-\ntrum of PLZ(t) [Figs. 3(c) and 3(d)]. For PLZ(t)≪1,\nPLZ(t)∝ |∆ST(t)|2, soSP(ω)∝S|∆ST|2(ω), the power\nspectrum of |∆ST(t)|2. This two-sweep technique allows\nus to measure the high-frequency components of SP(ω),\nbecause the maximum bandwdith is not limited by the\nquantum dot readout time.\nBecauseitarisesfromtheprecessingtransversenuclear\npolarization, ∆ HF(t) contains Fourier components at the\nLarmor frequencies of the69Ga,71Ga, and75As nuclei in\nthe heterostructure, i.e., ∆ HF(t) =/summationtext3\nα=1∆αe2πifαt+θα,\nwhereα= 1,2,or3indexesthethreenuclearspecies, and\ntheθαare the phases of the nuclear ﬁelds. Without spin-orbit interaction, |∆ST(t)|2=|/summationtext3\nα=1∆αe2πifαt+θα|2\ncontainsonlyFouriercomponentsatthediﬀerencesofthe\nnuclear Larmor frequencies. With a spin-orbit contribu-\ntion, however, |∆ST(t)|2=|∆SO+ ∆HF(t)|2contains\ncross-terms like ∆ SO∆αe2πifαt+θαthat give |∆ST(t)|2\nFourier components at the absolute Larmor frequencies.\nA signature of the spin-orbit interaction would therefore\nbe the presence of the absolute Larmor frequencies in\nSP(ω) forφ∝negationslash= 0◦[26].\nFigure 3(b) shows RPP(τ) measured with B= 0.1 T\nandφ= 0◦. Figure 3(c) shows SP(ω) for 0◦≤φ≤90◦.\nAtφ= 0◦, only the diﬀerences between the Larmor fre-\nquencies are evident, but as φincreases, the absolute nu-\nclear Larmor frequencies appear, as expected for a static\nspin-orbit contribution to ∆ ST(t). These results, includ-\ning the peak heights, which reﬂect isotopic abundances\nand relative hyperﬁne couplings, agree well with simula-\ntions (see Supplementary Information).\nHaving established the importance of spin-orbit cou-\npling at the S−T+crossing, we next investigate how\nthe spin-orbit interaction aﬀects DNP. Previous research\nhas shown that repeated LZ sweeps through ǫSTincrease\nboth the averageand diﬀerential nuclear longitudinal po-\nlarization in double quantum dots [3]. However, the rea-\nsons for left/right symmetry breaking, which is needed\nfor diﬀerential DNP (dDNP), and the factors limiting\nDNP eﬃciency in general are only partially understood.\nHere, we measure dDNP precisely by measuring δBz,\nthe diﬀerentialOverhauserﬁeld, usingrapidHamiltonian\nlearning strategies [27] before and after 100 LZ sweeps to\npump the nuclei with rates chosen such that ∝angbracketleftPLZ∝angbracketright= 0.4\n(see Supplementary Information) [Fig. 4(a)].\nFigure 4(b) plots the change in δBzper electron spin\nﬂip forB= 0.2 T and B= 0.8 T for varying φ. In\neach case, the dDNP decreases with |φ|. Because the\nspin-orbit interaction allows electron spin ﬂips without\ncorresponding nuclear spin ﬂops, dDNP is suppressed as\n|∆SO|=|ΩSOsinφsinθ|increases with |φ|. The reduc-\ntion in dDNP occurs more rapidly at 0.8 T because ∆ SO\nis larger at 0.8 T than at 0.2 T. We gain further insight\ninto this behavior by plotting the data against σHF/σST,\nwhereσHF≡/radicalbig\n∝angbracketleft|∆HF(t)|2∝angbracketright[Fig 4(c)]. Plotted in this\nway, the two data sets show nearly identical behavior,\nsuggesting that the size of the hyperﬁne interaction rela-\ntive to the total splitting primarily determines the DNP\neﬃciency.\nBased on theoretical results and experimental data, to\nbe presented elsewhere, we expect that the dDNP should\nbe proportional to the total DNP, with a constant of\nproportionality that depends on B, but not βorφ. We\nthereforeexplainourmeasurementsofdDNPusingathe-\noretcal model in which we have computed the average\nangular momentum ∝angbracketleftδm∝angbracketrighttransfered to the ensemble of4\n(b) (a) \n(c)-60 -50 -40 -30 -20 -10 0-500 0500 1000 1500 2000 2500 \n φ (degrees) \nσHF /σST dDNP (Hz per flip) \n0 0.2 0.4 0.6 0.8 100.5 1Normalized dDNP time (0,2) δBz x 120 Probe Pump Probe \nload load meas. meas. δBz x 120 \n(1,1) εload meas. sweep x 100 \nεST \nB=0.8T \nB=0.2T \nB=0.8T \nB=0.2T \nFIG. 4. DNP quenching by spin-orbit coupling. (a) Protocol\nto measure DNP. δBzis measured before and after 100 LZ\nsweeps by evolving the electrons around δBz. (b) dDNP vs\nφat ﬁxed /angbracketleftPLZ/angbracketright= 0.4 forB= 0.8 T and B= 0.2 T and\ntheoretical curves (solid lines). dDNP is suppressed for |φ|>\n0 because of spin-orbit coupling. (c) Data and theoretical\ncurves for ﬁxed /angbracketleftPLZ/angbracketrightcollapse when normalized and plotted\nvsσHF/σST. Vertical error bars are statistical uncertainties,\nand horizontal error bars are ﬁt errors.\nnuclear spins following a LZ sweep as:\n∝angbracketleftδm∝angbracketright ∝σ2\nHF/angbracketleftbiggP′\nLZ(∆ST)\n|∆ST|/angbracketrightbigg\n, (2)\nwhereP′\nLZ(∆ST) is the derivative of the LZ probabil-\nity with respect to the magnitude of the splitting. (See\nSupplementary Information for more details.) Neglect-\ning charge noise, we have the usual Landau-Zener for-\nmula [23]\nPLZ(∆ST) = 1−exp/parenleftbigg\n−2π|∆ST|2\n/planckover2pi1β/parenrightbigg\n,(3)\nand equation (2) reduces to\n∝angbracketleftδm∝angbracketright ∝σ2\nHF2π\n/planckover2pi1β∝angbracketleft1−PLZ∝angbracketright. (4)The data in Figs. 4(b) and (c) can therefore be under-\nstood in light of equation(4) because asthe splitting σST\nincreases with |φ|, the sweep rate βwas also increased to\nmaintain a constant ∝angbracketleftPLZ∝angbracketright. Because the hyperﬁne con-\ntribution σHFis independent of φ,∝angbracketleftδm∝angbracketrighttherefore de-\ncreases. The data collapse in Fig. 4(c) can also be under-\nstood from equation (4), assuming a constant splitting\nand ﬁxed probability. In this case, β∝ |∆ST|2, as fol-\nlows from equation (3), and hence ∝angbracketleftδm∝angbracketright ∝σ2\nHF/|∆ST|2.\nMeasurements with ﬁxed rate βalso exhibit a similar\nsuppression of dDNP (see Supplementary Information).\nIn this case ∝angbracketleftPLZ∝angbracketrightincreases with |φ|, because of the in-\ncreasing spin-orbit contribution to σST, and according to\nequation (4), ∝angbracketleftδm∝angbracketrighttherefore decreases.\nThetwotheoreticalcurvesin Figs.4(b) and(c) arecal-\nculated using equation (4) multiplied by ﬁtting constants\nC, which are diﬀerent for the two ﬁelds, and agree well\nwith the data. As discussed in the Supplementary In-\nformation, we do not expect charge noise to modify the\nagreement between theory and data in Figs. 4(b) and\n(c) beyond the experimental accuracy. Interestingly, the\npeak dDNP is less at B= 0.2 T than at B= 0.8 T,\nperhaps because the electron-nuclear coupling becomes\nincreasingly asymmetric with respect to the center of the\nquantum dots at higher ﬁelds [28]. Finally, the peak\ndDNP value also approximately agrees with a simple cal-\nculation (see Supplementary Information) based on mea-\nsured properties of the double dot.\nIn summary, we have used LZ sweeps to measure the\nS−T+splitting in a GaAs double quantum dot. We ﬁnd\nthat the spin-orbit coupling dominates the hyperﬁne in-\nteractionandquenchesDNPforawiderangeofmagnetic\nﬁeld strengths. A misalignment of BtoΩSOby only 5◦\natB= 1 T can reduce the DNP rate by a factor of two,\nand DNP is completely suppressed for a misalignment\nof 15◦. The techniques developed here are directly ap-\nplicable to other quantum systems such as InAs or InSb\nnanowires and SiGe quantum wells, where the spin-orbit\nand hyperﬁne interactions compete. On a fundamental\nlevel, our ﬁndings suggest avenues of exploration for im-\nprovedS−T+qubit operation [23] and underscore the\nimportance of the spin-orbit interaction in the study of\nnuclear dark states [29, 30] and other mechanisms that\nlimit DNP eﬃciency in central-spin systems.\nThis researchwasfunded by the United States Depart-\nment of Defense, the Oﬃce of the Director of National\nIntelligence,IntelligenceAdvancedResearchProjectsAc-\ntivity, and the Army Research Oﬃce grant W911NF-11-\n1-0068. S.P.H was supported by the Department of De-\nfense through the National Defense Science Engineering\nGraduate Fellowship Program. 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Rashba,\nPhysical Review Letters 109, 236803 (2012).arXiv:1502.05400v1  [cond-mat.mes-hall]  18 Feb 2015Supplementary Information for\nQuenching of dynamic nuclear polarization by spin-orbit co upling\nin GaAs quantum dots\nJohn M. Nichol,1Shannon P. Harvey,1Michael D. Shulman,1Arijeet Pal,1Vladimir\nUmansky,2Emmanuel I. Rashba,1Bertrand I. Halperin,1and Amir Yacoby1\n1Department of Physics, Harvard University, Cambridge, MA, 0 2138, USA\n2Braun Center for Submicron Research,\nDepartment of Condensed Matter Physics,\nWeizmann Institute of Science, Rehovot 76100 Israel\n1. MEASURING σST\nHere we describe the ﬁtting procedure to extract σST. The experimentally measured\nquantity is the average triplet occupation probability ∝an}bracketle{tPT∝an}bracketri}ht, which we interpret as the aver-\nage Landau-Zener (LZ) probability ∝an}bracketle{tPLZ∝an}bracketri}ht, at the end of a sweep. Here ∝an}bracketle{t···∝an}bracketri}htindicates an\naverage over the hyperﬁne distribution and charge ﬂuctuations f or the same nominal sweep\nparameters. We calibrate the rate β=d(ES−ET+)/dtusing the spin-funnel technique [1]\nand assume a linear change in the S−T+splitting near the avoided crossing.\n∆HF(t) varies in time because of the nuclear Larmor precession and statis tical ﬂuctua-\ntions in the magnitude of the nuclear polarizations. We argue that bo th types of hyperﬁne\nﬂuctuations occur on time scales much longer than LZ transitions an d can be treated as\nquasi-static. In typical experiments, the S−T+splitting is swept through approximately\n5 GHz in less than 1 µs. For splittings of order 10 MHz, the total time spent near the\navoided crossing is less than 10 ns, which is much faster than the nuc lear Larmor period at\n1 T, roughly 100 ns. Furthermore, during 1 µs, the nuclear polarization diﬀuses by approx-\nimately 7 kHz [2], which is 3 orders of magnitude smaller than σHF. We therefore assume\nthat the splitting is constant during a single sweep. Numerical simulat ions discussed below\nalso support the hypothesis that nuclear Larmor precession does not signiﬁcantly aﬀect ∝an}bracketle{tPT∝an}bracketri}ht\nfor the sweep rates used here [Fig. S1].\nIn the absence of hyperﬁne or charge ﬂuctuations, the probabilit y for a transition is given2\nby the LZ formula: PLZ(t) = 1−exp(−2π|∆ST(t)|2/(/planckover2pi1β)) [3]. Neglecting high-frequency\ncharge noise, the exact form of the LZ probability averaged over t he hyperﬁne distribution\ncan be computed. Let the total splitting be ∆ ST= ∆HF+ ∆SO. We take ∆ SOto be the\nconstant, real spin-orbit part and ∆ HFthe complex hyperﬁne contribution. Assuming that\nthe real andimaginary parts of ∆ HF(uandv, respectively) areGaussian-distributed around\nzero such that the root-mean-square hyperﬁne splitting is σHF, the probability distribution\nfor the splitting to have magnitude ∆ = |∆ST|is\np(∆) =1\nπσ2\nHF/integraldisplay∞\n−∞du/integraldisplay∞\n−∞dv e−u2+v2\nσ2\nHFδ/parenleftBig\n∆−/radicalbig\n(∆SO+u)2+v2/parenrightBig\n(S1)\n=2∆\nσ2\nHFe−∆2+∆2\nSO\nσ2\nHFI0(2∆∆SO/σ2\nHF), (S2)\nwhereI0isthezeroth-ordermodiﬁedBessel functionoftheﬁrstkind. Not ethatwhen∆ SO=\n0, equation (S2) reduces to the familiar distribution p(∆) =2∆\nσ2\nHFe−∆2/σ2\nHF[4]. Integrating\nthe LZ probability over this distribution yields the average LZ probab ility∝an}bracketle{tPLZ∝an}bracketri}ht:\n∝an}bracketle{tPLZ∝an}bracketri}ht=/integraldisplay∞\n0d∆/parenleftbigg\n1−exp/parenleftbigg\n−2π∆2\n/planckover2pi1β/parenrightbigg/parenrightbigg\np(∆) (S3)\n= 1−Qexp/parenleftbigg\n−2π∆2\nSO\n/planckover2pi1βQ/parenrightbigg\n, (S4)\nwith\nQ=1\n1+2πσ2\nHF\n/planckover2pi1β. (S5)\nNote that this result agrees with another derivation [5]. Note also th at to leading order in\nβ−1,∝an}bracketle{tPLZ∝an}bracketri}ht ≈2π(∆2\nSO+σ2\nHF)//planckover2pi1β.\nThe average triplet return probability ∝an}bracketle{tPT∝an}bracketri}htmay be modiﬁed due to eﬀects of charge\nnoise on the deﬁning gates or in the two-dimensional electron gas its elf. High-frequency\ncharge noise in double quantum dots has recently been identiﬁed as a major source of\ndecoherence [6]. In the current setting, corrections to ∝an}bracketle{tPT∝an}bracketri}htshould occur, because charge\nﬂuctuations lead to time-dependent variations in S−T+detuning ES−ET+, on top of the\nlinear time-dependence due to the prescribed sweep rate β. Additionally, charge ﬂuctuations\ncan add noise to the oﬀ-diagonal coupling ∆ ST(t) =δB⊥(t)cosθ+ΩSOsinφsinθ, because\nthe singlet mixing angle θ= tan−1/parenleftBig\nǫ+√\nǫ2+4t2\n2t/parenrightBig\ndepends on ǫ. (Heret= 23.1µeV is the3\n0 0.5 100.20.40.60.81B=0.1 T, φ=90° B=0.3 T, φ=90° B=0.5 T, φ=90° B=0.7 T, φ=90° B=0.9 T, φ=90°\nB=0.1 T, φ=0° B=0.3 T, φ=0° B=0.5 T, φ=0° B=0.7 T, φ=0° B=0.9 T, φ=0°0 0.05 0.100.20.40.60.81\n0 0.02 0.0400.20.40.60.81\n0 0.01 0.0200.20.40.60.81\n0 0.005 0.0100.20.40.60.81\n0 0.5 1 1.500.20.40.60.81\n0 0.5 1 1.500.20.40.60.81\n0 0.5 1 1.500.20.40.60.81\n0 1 200.20.40.60.81\n0 1 2 300.20.40.60.81(h/β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) \n(h/β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) \nP\nLZ P\nLZ P\nLZ \nData Analytic Simulation with \ncharge noise,\nhyperfine averaging,\nLarmor precessionSimulation with \nhyperfine averaging,\nLarmor precession\nFIG. S1. Comparison of LZ data and simulations. Each panel sh ows data and simulations for a\ndiﬀerent magnetic ﬁeld strength and orientation. Red curves are experimental data for a series of\nLZ sweeps with varying rates. Blue curves are simulated data including charge noise, hyperﬁne\naveraging, and nuclear Larmor precession for the calculate d value of the splitting corresponding\nto the red curves. Green curves are simulated data with hyper ﬁne averaging and Larmor pre-\ncession for the same value of the splitting as the blue curves . Black curves are calculated via\nequation (S4) using the same value of the splitting. In all pa nels, the y axis is ∝an}bracketle{tPLZ∝an}bracketri}ht, and the x\naxis ish/β(µs/GHz). Here h= 2π/planckover2pi1is Planck’s constant.\ndouble-dot tunnel coupling.) As discussed below, however, the nois e in ∆ STshould have\nmuch less eﬀect than the detuning noise for the magnetic ﬁelds stud ied here.\nWe observe that for high magnetic ﬁelds and slow sweeps, the maximu m LZ probability\nfalls to 0.5 as shown in Fig. S1. It was previously noted that strong de tuning noise can\nhave such an eﬀect [7]. To conﬁrm that charge noise causes the pro bability reduction, we\nhave performed Monte Carlo simulations of the Schr¨ odinger equat ion for symmetric double\ndots undergoing LZ sweeps, including the eﬀects of wide-band char ge noise, nuclear Larmor4\nprecession, and averaging over the hyperﬁne distribution. The re sults of the simulations\nand experimental data are shown in Fig. S1. We generate random ch arge noise with power\nspectrum 14 ×10−14V2\nHz/parenleftBig\n1Hz\nf/parenrightBig0.7\nforf <1 GHz, and 0 otherwise. We generate the Fourier\ntransform of the charge noise time record by picking the amplitude c orresponding to the\nchosen power spectrum and a random phase for each frequency fin the desired range. We\nthen perform an inverse Fourier transform to obtain the charge n oise time record. The\nspectrum we chose corresponds to a noise amplitude of 3 nV/√\nHz atf= 1 MHz, which is\napproximately the measured level of charge noise in the double dot u sed here. Note that we\nhave extrapolated the f−0.7frequency dependence that was previously measured to f= 1\nMHz in ref. [6] up to f= 1 GHz in these simulations. However, one expects the results\nto be most sensitive to noise in the range of 10-100 MHz, correspon ding to the size of the\nsplitting. The ǫ-dependent Hamiltonian used in these simulations was\nH(ǫ) =\nǫ\n2−B δB ⊥(t)cosθ+ΩSOsinφsinθ\nδB∗\n⊥(t)cosθ+ΩSOsinφsinθ −1\n2√\nǫ2+4t2\n(S6)\nin the{|T+∝an}bracketri}ht,|S∝an}bracketri}ht}basis. Linear ǫsweeps through the S−T+crossingǫST=B2−t2\nBwere used\nin the simulation to replicate the actual experiments. For each stre ngth and orientation of\nthemagneticﬁeld, θwascalculated at ǫSTusing themeasuredtunnel coupling, andtheﬁtted\nvalues of the spin-orbit and hyperﬁne couplings from the main text w ere used to compute\nthe splitting. We assumed a lever arm of 10 to convert the voltage no ise on the quantum\ndot gates to ǫnoise.\nThe simulated LZ curves with charge noise agree well with the data as shown in Fig. S1.\nThesamesimulationsincludingaveragingoverthehyperﬁnedistribut ionandnuclearLarmor\nprecession, but without chargenoise, show very littlereduction inp robabilitycompared with\nthe analytic result, equation (S4), supporting the hypothesis tha t charge noise is responsible\nfor most of the observed probability reduction. A key feature in th ese experiments is the\ndecreasing maximum probability with increasing magnetic ﬁeld. We can u nderstand that\nthis trend occurs because the eﬀect of charge noise on the Landa u Zener probability is\ncontrolled by the ﬂuctuation in the energy splitting δE(ǫ) produced by a given ﬂuctuation\nin the detuning ǫ, which is proportional todE(ǫ)\ndǫ|ǫ=ǫST. SinceE(ǫ) =ǫ\n2−B+1\n2√\nǫ2+4t2, the\nmagnitude ofdE(ǫ)\ndǫ|ǫ=ǫSTincreases sharply with increasing magnetic ﬁeld.5\nφ=0° φ=90°\n0.2 0.4 0.6 0.8 1-25-20-15-10-5 05\nField (T) Systematic Error (%) \nFIG.S2. Fittingerror. Wecomputetheﬁttingerrorbysimula ting∝an}bracketle{tPLZ∝an}bracketri}htforthecalculated splitting\nat each of the magnetic ﬁeld conﬁgurations in the presence of charge noise. The simulated ∝an}bracketle{tPLZ∝an}bracketri}htvs\nβ−1is ﬁtted to a straight line for 0 <∝an}bracketle{tPLZ∝an}bracketri}ht<0.1, and the ﬁtted value of the splitting is subtracted\nfrom the value chosen for the simulation. The diﬀerence is the n divided by the simulated value of\nthe splitting. Error bars are ﬁt errors.\nEven in the presence of noise, however, the average LZ probability in the limit of fast\nsweeps is still 2 π|∆ST(t)|2//planckover2pi1β, which is identical to the leading order behavior of the usual\nLZ formula, as shown in section 3.1 of ref. [7]. Replacing the LZ formula in equation (S4) by\nits leading order behavior, and performing the integration over the quasi-static distribution\ngives∝an}bracketle{tPLZ∝an}bracketri}ht ≈2π(∆2\nSO+σ2\nHF)//planckover2pi1β. Such a result can be understood because the eﬀect of\ndetuning noise is reduced on short time scales. Figure S1 demonstra tes this idea because the\nanalytic curves deviate signiﬁcantly from the data for ∝an}bracketle{tPLZ∝an}bracketri}ht/greaterorsimilar0.2, but for 0 <∝an}bracketle{tPLZ∝an}bracketri}ht<0.1,\nthe analytic results agrees well with the data. Based on additional s imulations, we estimate\nthe systematic error in the deduced value of σSTas obtained by ﬁtting measured values of\n∝an}bracketle{tPLZ∝an}bracketri}htto a straight line for values of βsuch that 0 <∝an}bracketle{tPLZ∝an}bracketri}ht<0.1 to be small for most of\nthe experimental conditions as shown in Fig. S2.\nWenotethatthecoupling∆ ST(t) =δB⊥(t)cosθ+ΩSOsinφsinθdependson ǫthroughthe\nsinglet mixing angle θ. This dependence means that during a LZ sweep, the coupling ∆ ST(t)\nvariesbothdueto thelinear ǫsweep aswell aschargenoise. Weestimate thatdE(ǫ)\ndǫ≥40dσST\ndǫ\nfor the ﬁelds studied here. We therefore expect detuning ﬂuctua tions to be the dominant\nnoise source. Furthermore, when |E(ǫ)|< σST,σSTchanges by only a few percent during\nthe sweep and is likely not a signiﬁcant source of error in the measure ment of ∆ ST(t).\nAdditionally, we note that the simulations in Fig. S1, which include ǫ-depending coupling,6\ndemonstrate that the ﬁtting procedure described above allows an accurate measurement\nofσST. Finally, we have also performed additional simulations, taking into ac count the\nmeasured values of E(ǫ), which deviate slightly from the values predicted by assuming a\nconstant tunnel coupling, and we observe no signiﬁcant change in o ur results.\n2. DIRECTION OF Ω SO\nThe double quantum dot axis is aligned within ≈5◦of either the [ ¯110] or [110] axes\nof the crystal, but we do not know which. In the later case, both th e Rashba and Dres-\nselhaus spin-orbit ﬁelds are aligned with the z axis, and their magnitud es add [8]. In the\nformer case, the Rashba and Dresselhaus contributions are also a ligned with the z axis, but\ntheir magnitudes subtract. The techniques used here could be emp loyed to distinguish the\nRashba and Dresselhaus spin-orbit contributions by measuring σSTwith double quantum\ndots fabricated on diﬀerent directions with respect to the crysta l axes.\n3. FITTING σSTVS B AND φ\nWe ﬁt the data in Fig. 2(c) in the main text to a function of the form σST=\n/radicalbig\n∆2\nSOsin2θsin2φ+σ2\nHFcos2θ, with ∆ SOandσHFas ﬁt parameters. The singlet mixing\nangleθis computed by assuming that the (1 ,1) and (0 ,2) singlet branches are a two-level\nsystem with constant tunnel coupling, as discussed above.\n∆SOis held at 0 when ﬁtting data for φ= 0◦to determine the hyperﬁne coupling. We\nalso exclude data points for B <0.2 T in the ﬁt, as the hyperﬁne contribution appears to\ndecrease at very low ﬁelds. We determine the spin orbit length using e quation (28) of Ref.\n[8], where the spin-orbit ﬁeld is computed as Ω SO=4t\n3λDQD\nλSO, whereλDQD≈200 nm is the\ninterdot spacing, and λSOis the spin-orbit length. The simulation in Fig. 2(b) in the main\ntext is generated using the same equation with the ﬁtted values of t he ∆SOandσHF.\n4. MEASURING RPP(τ)\nHere we derive the triplet return probability after two consecutive LZ sweeps with a\npause of length τin between. In experiments, both sweeps were in the same direction , and7\nǫwas held in the (0 ,2) region between sweeps, as shown in Fig. 3(a) in the main text.\nSuppose the ﬁrst LZ sweep takes place at time twith probability PLZ(t). The probability\nfor the two electrons to be in the T+state isPLZ(t), while the probability to be in the\nSstate is 1 −PLZ(t). Then, the detuning is quickly swept into the (0 ,2) region. Here,\nelectron spin dephasing occurs rapidly, and there is very little T+occupation in thermal\nequilibrium because the SandT+states are widely separated in energy. Thus, after a wait\nof length τ, but before the second sweep, the triplet population is PLZ(t)e−τ/T1, and the\nsinglet population is 1 −PLZ(t)e−τ/T1, whereT1is the electron relaxation time. After the\nsecond sweep, the triplet occupation probability is\nPT(t+τ) =/parenleftbig\n1−PLZ(t)e−τ/T1/parenrightbig\nPLZ(t+τ)+PLZ(t)e−τ/T1(1−PLZ(t+τ)) (S7)\n=−2PLZ(t)PLZ(t+τ)e−τ/T1+PLZ(t+τ)+PLZ(t)e−τ/T1. (S8)\nThe second and third terms in equation (S8) vary slowly with τ. These terms are found by\nﬁtting the measured triplet probability to an exponential with an oﬀs et and are subtracted.\nWhenT1≫τ, relaxationcanbeneglected, andthepredicted time-averaged sig nal is∝an}bracketle{tPT(t+\nτ)∝an}bracketri}ht ∝RPP(τ), whereRPP(τ)≡ ∝an}bracketle{tPLZ(t)PLZ(t+τ)∝an}bracketri}ht, theautocorrelationoftheLZprobability.\nWhenφ= 0◦,T1≫τmax= 200µs, where τmaxis the largest value of τmeasured. The\nshortest relaxation time T1≈100µs in these experiments time occurs when φ= 90◦, which\nis consistent with spin-orbit-induced relaxation [9].\nThe eﬀect of T1relaxation is to multiply the measured correlation by an exponentially-\ndecaying window, which reduces the spectral resolution of the Fou rier transform, but does\nnot shift the frequency of the observed peaks. We expect statis tical ﬂuctuations in the\namplitude of the hyperﬁne ﬁeld to aﬀect the spectrum in a similar way, although we expect\nthis eﬀect to be less than that of electron relaxation. The raw data , [Fig. 3(b) in the\nmain text] consisting of 667 points (each a result of two sweeps with a 40 % chance of a\nLZ transition) spaced by 300 ns, were zero-padded to a size of 169 1 points to smooth the\nspectrum, and a Gaussian window with time constant 150 µs was applied to reduce the\neﬀects of noise and ringing from zero-padding before Fourier tran sforming.\nThe magnetic resonance frequencies in Fig. 3(c) appear to decrea se withφ. The inhomo-\ngeneity of the x-coil in our vector magnet is 1.6 % at 0.6 cm oﬀset from the center. Thus,\nthe ﬁeld could easily be reduced by more than 3 % for a misplacement of the sample by 1\ncm from the magnet center. We have simulated the data in Fig. 3(c) in the main text based8\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6020 40 60 80 \n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6020 40 60 80   φ (degrees)   φ (degrees) ω/2 π (MHz) Experiment \nSimulation \nω/2 π (MHz) (a)\n(b)\nFIG. S3. Simulations of SP(ω). (a) Experimental data. (b) Theoretical simulation takin g into\naccount known sweep rates, nuclear magnetic resonance freq uencies, hyperﬁne couplings, and a\n4.4% reduction in ﬁeld in the x direction. The expected frequ encies at B= 0.1 T are f69Ga=\n1.0248 MHz, f71Ga= 1.302 MHz, and f75As= 0.7315 MHz.\non the measured hyperﬁne and spin-orbit couplings and the known s weep rates. Assuming a\n4.4 % reduction in the ﬁeld from the x-coil, we obtain good agreement b etween theory and\nexperiment [Figs. S3(a) and (b)].\nWearguedinthe maintext that onlythe diﬀerence frequencies shou ld appear inthe spec-\ntrumSP(ω)withoutspin-orbitcoupling byconsidering thetime-dependence of |∆ST(t)|2and\nbecauseSP(ω)∝S|∆ST|2(ω) whenPLZ(t)≪1. Since PLZ(t) contains only even powers of\n|∆ST(t)|,SP(ω) can generally be expressed in terms of diﬀerences of the resonan ce frequen-\ncies, but will not contain the absolute frequencies in the absence of spin-orbit coupling,\nregardless of the value of PLZ(t).9\n5. DERIVATION OF NUCLEAR POLARIZATION CHANGE ∝an}bracketle{tδm∝an}bracketri}ht\nHere we derive equations 2 and 4 in the main text. Let ∆ ST= ∆SO+∆HFwhere ∆ SO\nis real and\n∆HF=/summationdisplay\njλjI+\nj, (S9)\nwhereI+\njis the raising operator for the jthnuclear spin, and the λjare individual coupling\nconstants. We assume that there are many nuclear spins, so that each coupling constant is\nsmall. Also,\nσ2\nHF≡ ∝an}bracketle{t|∆2\nHF|∝an}bracketri}ht=2\n3I(I+1)/summationdisplay\njλ2\nj=5\n2/summationdisplay\njλ2\nj, (S10)\nwhereI=3\n2is the spin of the nuclei, and the angular brackets refer to an avera ge over the\ndistribution of nuclear spins.\nWe pick one of the nuclear spins, j, and we wish to compute ∝an}bracketle{tδmj∝an}bracketri}ht, the mean value of the\nchange in Iz\njafter one sweep. Let PLZ(∆ST) be the probability of an S−T+transition for\na ﬁxed value of ∆ HF. Clearly, PLZdepends on |∆ST|. We calculate δmjas follows. Write\n∆ST=a+beiθj, (S11)\nwhereaincludes the contributions of spin orbit and of all nuclei other than t he nucleus j,\nand the second term represents the contribution (of order λj) from nucleus j. According to\nequation (31) of Ref. [4], the value of δmjfor this conﬁguration should be given by\nδmj=1\n2π/contintegraldisplay\ndθjPLZ(∆ST)dϕ\ndθj, (S12)\nwhereϕ= arctan(Im(∆ ST)/Re(∆ST)) speciﬁes the orientation of ∆ STin the complex plane.\nWithout loss of generality, we may suppose that ais real. Then we have, ignoring terms\nthat are higher order in b/a,\ndϕ\ndθj=b\nacosθj (S13)\nPLZ(∆ST) =PLZ(a)+bP′\nLZ(a)cosθj (S14)\nδmj=b2\n2aP′\nLZ(a), (S15)\nwhereP′\nLZ(a) is the derivative of PLZ(a) with respect to a. Averaging over nuclear conﬁgu-\nrations, we obtain\n∝an}bracketle{tδmj∝an}bracketri}ht=∝an}bracketle{tb2∝an}bracketri}ht/angbracketleftbiggP′\nLZ(a)\n2a/angbracketrightbigg\n, (S16)10\nwith∝an}bracketle{tb2∝an}bracketri}ht= (5/2)λ2\nj. In the case of no charge noise, we have\nPLZ(∆ST) = 1−exp/parenleftbigg\n−2π|∆ST|2\n/planckover2pi1β/parenrightbigg\n, (S17)\nso\nP′\nLZ(a)\n2a=2π\n/planckover2pi1β(1−PLZ(a)) (S18)\nand\n∝an}bracketle{tδmj∝an}bracketri}ht=2π\n/planckover2pi1β∝an}bracketle{tb2∝an}bracketri}ht∝an}bracketle{t1−PLZ(a)∝an}bracketri}ht. (S19)\nFinally, we sum over all nuclear spins and make the replacement a≈ |∆ST|, obtaining\n∝an}bracketle{tδm∝an}bracketri}ht=2π\n/planckover2pi1βσ2\nHF∝an}bracketle{t1−PLZ(∆ST)∝an}bracketri}ht. (S20)\nThe collapse demonstrated in Fig. 4(c) in the main text can be unders tood from equa-\ntion (S20), assuming constant ∆ STand ﬁxed probability. In this case, β∝ |∆ST|2from\nequation (S17), and hence ∝an}bracketle{tδm∝an}bracketri}ht ∝σ2\nHF/|∆ST|2.\nIn the case of a ﬁxed splitting, equation (S20) reduces to\n∝an}bracketle{tδm∝an}bracketri}ht=2π\n/planckover2pi1βσ2\nHFexp/parenleftbigg\n−2π|∆ST|2\n/planckover2pi1β/parenrightbigg\n. (S21)\nIn equation (S21), ∝an}bracketle{tδm∝an}bracketri}ht →0 for both β→0 andβ→ ∞. In practice however, experiments\nnecessarily average over the hyperﬁne distribution. Thus, using e quation (S4) with ∆ SO= 0\nto compute ∝an}bracketle{t1−PLZ(∆ST)∝an}bracketri}ht, we have\n∝an}bracketle{tδm∝an}bracketri}ht=2π\n/planckover2pi1βσ2\nHFQ (S22)\n=2πσ2\nHF\n/planckover2pi1β\n1+2πσ2\nHF\n/planckover2pi1β. (S23)\nAccording to equation (S23), in the limit of slow sleeps, where β→0,∝an}bracketle{tδm∝an}bracketri}ht →1, and in the\nlimit of fast sweeps, where β→ ∞,∝an}bracketle{tδm∝an}bracketri}ht →0, as expected.\nThe theory curves in Figs. 4(c) and (d) in the main text were genera ted by computing\nequation (S20). For each ﬁeld angle φ, the parameters θ, ∆SO, andσHFwere calculated\nusing the ﬁtted values of the spin-orbit and hyperﬁne couplings as w ell as the measured\ntunnel coupling. Equation (S4) was then solved using the calculated parameters to ﬁnd\nthe rate βsuch that ∝an}bracketle{tPLZ∝an}bracketri}ht= 0.4. In order to compare with data on the dDNP rate, the11\nφ (degrees) φ (degrees) dDNP rate (Hz per flip) B=0.8T \nB=0.2T \nB=0.8T \nB=0.2T (a) \n(b) -60 -50 -40 -30 -20 -10 0-50005001000150020002500\n-60 -50 -40 -30 -20 -10 00.30.40.50.60.70.80.9PLZ \nFIG. S4. DNP quenching with ﬁxed sweep rate. (a) dDNP vs φatB= 0.2 T and B= 0.8 T. For\neach ﬁeld, the sweep rate βwas chosen to give ∝an}bracketle{tPLZ∝an}bracketri}ht= 0.4 atφ= 0◦and then was held constant\nforφ∝ne}ationslash= 0◦. (b) As |φ|increases, σSTincreases. As a result, ∝an}bracketle{tPLZ∝an}bracketri}htalso increases and DNP is\nsuppressed, according to equation (S20). Error bars are sta tistical uncertainties. Lines between\npoints serve as a guide to the eye.\ntheoretical curves for ∝an}bracketle{tδm∝an}bracketri}htwere multiplied by ﬁtting constants C, which are diﬀerent for the\ntwo curves. As explained in the main text, and further discussed be low, we expect the ratio\nbetween the dDNP rate and ∝an}bracketle{tδm∝an}bracketri}htto depend on the magnetic ﬁeld but to be independent of\nthe sweep rate.\nData taken at ﬁxed sweep rate βalso show a suppression of DNP, as shown in Fig. S4(a).\nIn this case, ∝an}bracketle{tPLZ∝an}bracketri}htincreases with |φ|because of spin-orbit coupling [Fig. S4(b)], and ∝an}bracketle{tPLZ∝an}bracketri}ht\ntherefore increases, causing ∝an}bracketle{tδm∝an}bracketri}htto decrease, according to equation (S20).\nToaddresstheeﬀectofchargenoiseondDNP,werecomputeequa tion(S20)inthelimitof\nstrongnoiseusingtheresultsofRef.[7],makingthereplacement P(a) =1\n2/parenleftBig\n1−exp/parenleftBig\n−4πa2\n/planckover2pi1β/parenrightBig/parenrightBig\nforPLZ(a) both in the derivation leading to equation (S20) and in equation (S4) for the\ncomputation of β. The expected dDNP in the presence of strong noise is shown in Fig. S 5,\nand it does not signiﬁcantly deviate from the case without noise, at le ast at the level of the12\nB=0.8T \nB=0.2T No charge noise \nCharge noise \n-60 -50 -40 -30 -20 -10 0-50005001000150020002500\nφ (degrees) dDNP rate (Hz per flip) \nFIG. S5. The eﬀect of charge noise on dDNP. The data and solid li nes are the same as in Fig. 4\nin the main text, and the dashed lines are the theoretical est imates for dDNP in the presence of\ncharge noise. The dashed and solid lines are normalized to th e same values at φ= 0◦. Error bars\nare statistical uncertainties.\nexperimental accuracy.\n6. MEASURING δBz\nWemeasure δBzby ﬁrst initializing the double dot inthe |(0,2)S∝an}bracketri}htstateand thenseparat-\ning the electrons by rapidly changing ǫto a large negative value [10]. When the electrons are\nseparated, the exchange energy isnegligible, andthe magneticﬁeld gradient δBzdrives oscil-\nlations between |S∝an}bracketri}htand|T0∝an}bracketri}ht. In our experiments, we measure the two-electron spin state for\n120 linearly increasing values of the separation time. The resulting sin gle-shot measurement\nrecord is thresholded, zero padded, and Fourier transformed. T he frequency corresponding\nto the peak in the resulting Fourier transform is chosen as the value ofδBz. This technique\nis related to a previously described rapid Hamiltonian estimation techn ique [2].\n7. EXPECTED DNP RATE\nIn this section we give a simple calculation to explain the value of the pea k (φ= 0◦)\ndDNP rate, as shown in Fig. 4 of the main text. Additional measureme nts were carried\nout to measure the pumping rate of the sum hyperﬁne ﬁeld, ( Br+Bl)/2, where BrandBl\ndenote the longitudinal hyperﬁne ﬁelds in the right and left dots. Th is rate was determined\nby measuring the location of ǫSTbefore and after a series of LZ sweeps to polarize the nuclei13\natB= 0.2 T. We observe that the sum ﬁeld is pumped roughly twice as eﬃciently as the\ndiﬀerence ﬁeld, δBz=Br−Bl. Setting ( ˙Br+˙Bl)/2 = 2(˙Br−˙Bl), where ˙Bl(r)indicates\nthe pumping rate of the left(right) dot, we have ˙Bl= (3/5)˙Br, meaning that the left dot is\npumped 3 /5 as often as the right dot. Under these conditions, the average g radient builds\nup at a rate (per electron spin ﬂip) of ( ˙Br−˙Bl)/(˙Br+˙Bl) that is only 1/4 the rate that\nwould occur if nuclear spin ﬂips occurred in only one dot.\nTo determine the expected change in δBz, we require the approximate number of spins\noverlapped by the electronic wave function in the double dot. We hav e measured the inho-\nmogeneous dephasing time of electronic oscillations around δBzand ﬁnd T∗\n2= 18 ns [10].\nThis dephasing time corresponds to a rms value of the gradient σδBz≡/radicalbig\n∝an}bracketle{t|δBz|2∝an}bracketri}ht=\nh//parenleftbig\n|g∗|µB√\n2πT∗\n2/parenrightbig\n= 2 mT, where his Planck’s constant. The total number of spins over-\nlapped by thewavefunction is N= (h1/σδBz)2≈3×106, whereh1= 4.0 T[11]. If all nuclear\nspins were fullypolarized, then thedots would experience a hyperﬁn e ﬁeldof h0= 5.3T [11],\nand if the nuclear spins in the two dots were fully polarized in opposite d irections, the gra-\ndient would be 2 h0. Therefore, the expected change in the gradient per electron sp in ﬂip,\ncorresponding to a change in nuclear angular momentum of /planckover2pi1, is2π\n/planckover2pi1×2|g∗|µBh0\n2I(N/2)= 12 kHz,\nwhereI= 3/2 is the nuclear spin. The average dDNP under actual conditions is 1/ 4 of this\nvalue, or 3 kHz, in reasonable agreement with our observations. In addition, we note the\nreasonable agreement between the measured valueof σδBz= 2 mT andtheroot-mean-square\nhyperﬁne gap/radicalbig\n∝an}bracketle{t|δB⊥(t)|2∝an}bracketri}ht ≈34 neV/ ( |g∗|µB) = 1.5 mT.\n[1] J. R. Petta, H. Lu, and A. C. Gossard, Science 327, 669 (2010).\n[2] M. D. Shulman, S. P. Harvey, J. M. Nichol, S. D. Bartlett, A . C. Doherty, V. Umansky, and\nA. Yacoby, Nature Communications 5, 5156 (2014).\n[3] S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1 (2010).\n[4] I. Neder, M. S. Rudner, and B. I. Halperin, Physical Revie w B89, 085403 (2014).\n[5] C. Dickel, S. Foletti, V. Umansky, and H. Bluhm, (2014), a rXiv:1412.4551.\n[6] O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm, V. Umansk y, and A. Yacoby,\nPhysical Review Letters 110, 146804 (2013).\n[7] Y. Kayanuma, Journal of the Physical Society of Japan 53, 108 (1984).14\n[8] D. Stepanenko, M. S. Rudner, B. I. Halperin, and D. Loss,\nPhysical Review B 85, 075416 (2012).\n[9] P. Scarlino, E. Kawakami, P. Stano, M. Shaﬁei, C. Reichl, W. Wegscheider, and L. M. K.\nVandersypen, (2014), arXiv:1409.1016.\n[10] J. R. Petta, A. C. Johnson, J. M. Taylor, E. Laird, A. Yaco by, M. D. Lukin, C. M. Marcus,\nM. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005).\n[11] J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. M arcus, and M. D. Lukin,\nPhysical Review B 76, 035315 (2007)."    },    {        "title": "2403.12701v1.Unraveling_the_dynamics_of_magnetization_in_topological_insulator_ferromagnet_heterostructures_via_spin_orbit_torque.pdf",        "content": "Unraveling the dynamics of magnetization in\ntopological insulator-ferromagnet\nheterostructures via spin-orbit torque\nTaekoo Oh∗and Naoto Nagaosa∗\nRIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nE-mail: taekoo.oh@riken.jp; nagaosa@riken.jp\nAbstract\nSpin-orbit coupling stands as a pivotal determinant in the realm of condensed mat-\nter physics. In recent, its profound influence on spin dynamics opens up a captivating\narena with promising applications. Notably, the topological insulator-ferromagnet het-\nerostructure has been recognized for inducing spin dynamics through applied current,\ndriven by spin-orbit torque. Building upon recent observations revealing spin flip sig-\nnals within this heterostructure, our study elucidates the conditions governing spin\nflips by studying the magnetization dynamics. We establish that the interplay between\nspin-anisotropy and spin-orbit torque plays a crucial role in shaping the physics of\nmagnetization dynamics within the heterostructure. Furthermore, we categorize var-\nious modes of magnetization dynamics, constructing a comprehensive phase diagram\nacross distinct energy scales, damping constants, and applied frequencies. This re-\nsearch not only offers insights into controlling spin direction but also charts a new\npathway to the practical application of spin-orbit coupled systems.\n1arXiv:2403.12701v1  [cond-mat.mes-hall]  19 Mar 2024Keywords\nSpintronics, Magnetization dynamics, Topological insulator, Spin-orbit torque.\nIntroduction\nSpin-orbit coupling (SOC), acknowledged as one of the fundamental interactions in ma-\nterials,1reveals a fertile landscape within condensed matter physics. A standout illustration\nof its impact is evident in the burgeoning field of spintronics, where SOC-driven spin dynam-\nics unveils a promising horizon for practical applications. Accordingly, recent attention has\nbeen drawn to topological insulator-ferromagnet (TI-FM) heterostructures, as evinced by\nnotable reports.2–7The surge in interest stems from their remarkable efficiency in catalyzing\nspin dynamics, primarily attributed to the spin-orbit torque (SOT) induced by an applied\ncurrent.8,9\nThese heterostructures exhibit intriguing phenomena that closely follow its intricate dy-\nnamics of spins. For instance, an emergent inductance from the motive force in spiral mag-\nnets10–12was extended to the TI-FM heterostructures.13–15Nonreciprocal transport phenom-\nena associated with the SOC was explored.16,17The significant increase of Curie temperature\nwas reported as well.18–21Notably, it has been acknowledged that the SOT at the TI-FM in-\nterface is robust enough to flip magnetization and induce the sign change of Hall Effect.22–26\nSuch properties hold promise for applications in spintronics devices, particularly in utilizing\ntopological SOTs for magnetic memories.27–32\nMotivated by such intriguing phenomena due to the SOC and SOT, this study explores\nmagnetization dynamics in TI-FM heterostructures with an applied current, aiming to un-\nveil the conditions for magnetization flip. Considering both spin-anisotropy and SOT, we\nestablish the equilibrium state of magnetization and develop a model describing its dynam-\nics under direct current (DC) or alternating current (AC). In the absence of damping, we\nidentify oscillating, faltering, and flipping modes for DC, with the latest inducing magne-\n2a\n𝑚⃗𝐼⃗𝜎TIFM̂𝑧'𝑥'𝑦\n𝑚'𝑦̂𝑧'𝑥𝜙𝜃b\n𝑥=𝜎/𝐾01𝜃=arcsin𝑥Figure 1: Magnetization dynamics of TI-FM heterostructure. (a) The schematics\nof the TI-FM heterostructure. (b) The equilibrium state of magnetization in the different\nrelative strengths of SOT x=σ/K.\ntization flip. The choice between modes is determined by the relative strengths of SOT to\nspin-anisotropy. With the introduction of damping, the occurrence of spin flip hinges on the\nduration elapsed until the crossover from flipping to oscillating modes takes place.\nFor AC, on the other hand, we discern three final states—adiabatic, resonating, and\nchaotic. We classify five distinct modes in the adiabatic state at low frequencies, which is\nderived from DC modes. By providing the phase diagrams of adiabatic modes, we illus-\ntrate that the modes are determined by SOT, spin-anisotropy, driving frequency, and initial\ndriving phase. Lastly, we explore their transition to resonating and chaotic states at higher\nfrequencies in the viewpoint of the Fourier transform, revealing that the periodic array of\npeaks gives rise to the chaotic state. By overhauling the dynamics, we provide insights into\nthe complex dynamics of magnetization in TI-FM heterostructures.\n3Results\nModel and its equilibrium\nThe physical configuration is illustrated in Fig. 1(a), where a current flows along the x-axis,\nthe itinerant electron spin aligns with the y-axis, and the ferromagnet’s magnetization lie\nalong the + z-axis at t= 0. The polar and azimuthal angles with respect to the y-axis are\ndenoted as θandϕ. We represent the magnetization as ⃗ m=m(sinθsinϕ,cosθ,sinθcosϕ)\nand the itinerant electron spin as ⃗ σ=σ(t)(0,1,0). For simplicity, we exclusively focus on\nthe dynamics of the single ⃗ m, and ignore that of ⃗ σ. This is justified by the stronger spin-\nmomentum locking of the TI surface state compared to anisotropy.3,15,33–36For instance, Ni\nhas the easy-axis anisotropy energy about 2 .7µeV/atom in the bulk37and 4 .0 meV /atom\nin the monolayer,38while the spin-orbit coupling of Bi 2Se3is estimated as ∼1.2 eV/atom.\nAccordingly, we set our unit of energy to be 1 .0µeV∼2.4 GHz and the unit of time to\nbe (2 .4 GHz)−1≈0.42 ns. The spin flip is then denoted as the switching of spin direction\nbetween + zand−zhalf space.\nTwo crucial potential energies emerge: VA=−K\n2m2\nz(K > 0), representing magnetic\nanisotropy in the ferromagnet, and VS=−γ ⃗ m·⃗ σsignifying the Rashba-field-like39,40(or\ndomain-damping-like)41SOT. The equilibrium state of magnetization is obtained, showcased\nin Fig. 1(b) for fixed ⃗ σ. Normalizing m=γ= 1, key energy scales become the anisotropy\nenergy Kand SOT σ. The dimensionless parameter x=σ\nKis defined. As xincreases from 0,\nthe equilibrium states initially align along ±z-axis and gradually rotate towards the y-axis.\nIn terms of angles, sin θ=x, ϕ=±π/2. Before reaching the y-axis, the equilibrium states\nare twofold degenerate. For x≥1, the degenerate equilibrium states converge at the y-axis.\nTo describe magnetization dynamics out of equilibrium, the Lagrangian density is con-\nsidered:\nL=LB−VA−VS,LB=m(1−cosθ)˙ϕ. (1)\n4Note that LBis the Berry phase term, making polar and azimuthal angles conjugate.42,43\nThus, as magnetization rotates from the z-axis to its equilibrium state, deviations from\ntheyz-plane induce complex motion. Introducing Gilbert damping through the Rayleigh\ndissipation function,44R=η\n2∂ ⃗ m\n∂t·∂ ⃗ m\n∂t, with the dimensionless damping constant η, yields the\nequations of motion:\n(1 +η2)˙θ=−K\n2sinθsin 2ϕ+ηK\n2sin 2θcos2ϕ−ησ(t) sinθ,and (2)\n(1 +η2)˙ϕ=−Kcosθcos2ϕ+σ(t)−ηK\n2sin 2ϕ. (3)\nThe initial conditions are set at θ=π/2 and ϕ= 0. This system of equations governs physics\nfor both DC and AC scenarios. The numerical computation is performed by Mathematica.\nIt should be noted that the field-like SOT σ(t) precesses the magnetization around y-\naxis while the damping-like SOT ησ(t) cants the magnetization toward y-axis. This is\nconsistent with the Landau-Lifshitz-Gilbert equation. It is noteworthy that the relation\nbetween the damping constant ηand the Gilbert damping constant αisα=ηγ, where γis\nthe gyromagnetic ratio.44We already set γ= 1, resulting in α=η. As a consequence, the\nfield-like SOT is responsible for the spin flip, whereas the damping-like SOT deters it.\nMagnetization dynamics under DC\nIn the exploration of magnetization dynamics under DC, we initiate with a straightforward\nscenario where α= 0 and σ(t) = 0 for t <0 and σ(t) =σfort≥0. Lacking damping, we\nanticipate permanent magnetization precession, described by the simplified equations:\n˙θ=−Ksinθsinϕcosϕ,˙ϕ=−Kcosθcos2ϕ+σ. (4)\nThese equations are intricate to solve analytically, but insight can be gained by examining\ntwo limiting cases, x≪1 and x≫1 with α= 0. In the first limit, we find oscillatory\n5behavior:\nθ=π\n2+x(cos(Kt)−1), ϕ=xsin(Kt). (5)\nBoth θandϕoscillate with amplitude xand frequency K, termed the oscillating mode . In\nthe second limit:\nθ=π\n2+1\n4x(cos(2 σt)−1), ϕ=σt. (6)\nHere, θoscillates with amplitude 1 /(4x) and frequency 2 σ, while ϕmonotonically increases.\nThis implies the spin precession about the y-axis and a spin flip occurring at period of\nπ/σ, termed the flipping mode . Comparing the frequency of θdynamics in each mode,\nwe speculate that the transition from the oscillating to the flipping modes might occur at\nx= 1/2 where K= 2σ.\nThe guess can be verified by numerical computations of frequency in units of Kfor\nσ(t) =σfort≥0 and α= 0 in Fig. 2(a). The frequency variation exhibits ωDC≈K= 1\nforx≪1 and ωDC≈2σforx≫1. The singularity at x= 1/2 indicates the expected\ntransition. Only at this transition point, the spins are faltering between zaxis and xy-plane\nin period, so we call this faltering mode. For DC, we focus on the oscillating and flipping\nmodes, as the faltering mode requires a fine tuning.\nWhen damping αis introduced under DC, the magnetization eventually attains an equi-\nlibrium state. For x≥1, the equilbrium state aligns with y-axis, so the spin flip is absent.\nHowever, since two equilibrium states exists for x <1, the selection between them occurs,\nwhich can flip the spin. The selection is determined by both xandα. Figure 2(b) displays\ntwo examples of dynamics leading to an even-flipped ( x= 0.6, α= 0.05) and odd-flipped\n(x= 0.6, α= 0.1) final state, respectively. The magnetization in the even-flipped state stays\nin +zhalf space, while that in the odd-flipped state stays in −zhalf space. In both exam-\nples, owing to the damping effect, the crossover from the flipping to the oscillating mode\n6a\nb𝑥=0.60𝜂=0.05𝑥=0.60𝜂=0.10𝑡 (0.42 ns)050𝜔!\"𝑥0.51.01.52.01234c\n0.00.51.00.010.05𝛼0.020.030.04𝑥No / Even FlipsOdd FlipsOscillatingFlipping\n25\nFaltering\nFlippingOscillating\nFlippingOscillatingFigure 2: The magnetization dynamics under DC. (a) Computed frequencies of os-\ncillating ( x < 0.5), faltering ( x= 0.5), and flipping modes ( x > 0.5) under DC without\ndamping in varying x=σ/K. (b) Dynamics of magnetization at x= 0.6, α= 0.05 (lower)\nandx= 0.6, α= 0.1 (upper) in time. Chromatically, the red indicates + z, the green indi-\ncates in-plane, and the blue indicates −zdirections of magnetization. The direction inward\nthe paper is the ydirection. (c) A 2D phase diagram of the final states in varying xandα.\nThe dark blue indicates the even-flipped final state, and the yellow indicates the odd-flipped\nfinal state.\noccurs after a specific time τc. In Fig. 2(b), for the former, after τc≈10.8 ns, while for the\nlatter, τc≈4.2 ns. Empirically, we find that τc∝e4.53xα−1for small α. [See Supporting\nInformation (SI).] These show that the variation in τccaused by the interplay of xandα\nserves as a determining factor for the final state.\nWe further delve into the relation of the final state with xandαby a 2D phase diagram\nin Fig. 2(c). When x <1/2, exclusively the even-flipped final state manifests, whereas for\nx≥1/2, both odd-flipped and even-flipped states emerge, forming a fan-like configuration. It\nshould be noted that the fan-like configuration is not clear in α <0.01 due to the resolution.\nThe exclusive appearance of even-flipped state for x <1/2 is due to the absence of spin flip\n7in the oscillating mode. However, in the case of x≥1/2, the spin flip can occur multiple\ntimes before the crossover from the flipping to the oscillating modes takes place. If the\ncrossover time exceeds half of the flipping mode period, the spin flips to −zhalf space. With\na longer crossover time surpassing a full flipping mode period, the spin flips twice, returning\nto +zhalf space. By extending the crossover time further, the spin flips repeatedly. Even\n(odd) numbers of spin flips lead to an even-flipped (odd-flipped) final state, giving rise to\nthe distinctive fan-like feature in x≥1/2. This pattern persists until x= 1, where the\nequilibrium state converges to y-axis.\nMagnetization dynamics under AC\nBased on above results, we here explore the magnetization dynamics under AC, which is given\nbyσ(t) =σsin(ωACt+δ) for t≥0 and σ(t) = 0 for t <0.ωACis the driving frequency, and\nδis the initial phase of AC. The initial state of the magnetization is again aligned with + z-\naxis. Owing to the damping α, the relative strength of SOT x, and driving frequency ωAC,\nthe initial state overcomes the irregular dynamics and transits to the final states after some\ntime. We identify three distinct final states in Fig. 3(a): adiabatic, resonating, and chaotic\nstates. Both adiabatic and resonating states are the steady states coming after the decay\nof initial state. At each time t,σ(t) determines an equilbrium state as shown in Fig. 1(b).\nThe adiabatic state denotes that the magnetization mostly adheres to the equilibrium state\nat each time. The resonating state denotes that the magnetization dynamics is periodic but\ndetach from the equilibrium state at each time. On the other hand, the chaotic state is\nevolved from the initial state, retaining its irregularity, in which the magnetization never has\na periodic motion.\nWe primarily observe the adiabatic states for low-frequency or strong damping regime.\nWe observe five distinct modes in the adiabatic states: I) an even-flipped oscillating mode,\nII) an odd-flipped oscillating mode, III) an even-flipped faltering mode, IV) an odd-flipped\nfaltering mode, and V) a periodically flipping mode. These modes are depicted in Fig. 3(b).\n8The even-flipped (odd-flipped) oscillating mode or Mode I (II) is the oscillation of magne-\ntization within + z(−z) half space. The even-flipped (odd-flipped) faltering mode or Mode\nIII (IV) is the repeated faltering of magnetization between xy-plane and + z(−z) half space.\nThe periodically flipping mode or Mode V is the repeated magnetization flip. Notably,\nModes I and II (or III and IV) are almost identical, but differ in the number of spin flips\nbefore decaying to the adiabatic state. Only Mode V shows the continuous spin flip in time\nin its adiabatic state. As one can expect from their names, the adiabatic modes originate\nfrom DC modes. Modes I and II come from oscillating mode, Modes III and IV come from\nthe faltering mode, and Mode V comes from the flipping mode.\nThe modes are chosen by the interplay of x,ωAC,δ, and α. Both xandωACunderscore\ntheir importance after decaying to adiabatic states while δandαplay a pivotal role during\nthe decay process. Primary investigation is performed by phase diagrams for the modes in\nxandωACatα= 0.03 with different δpresented in Figs. 3(c,d). For x <1, Modes I and\nII appear, while for x≥1, Modes III, IV, and V manifest. This transition occurs around\nx∼1 since x= 1 is where two equilibrium states meet at the y-axis. Specifically, for x <1,\nas the equilibrium position does not reach y-axis at any time, the magnetization oscillates\nwithin either + zor−zhalf space. For x≥1, however, as the equilibrium reaches y-axis,\nthe magnetization either falters on the xy-plane or flips its position between + zand−zhalf\nspaces periodically.\nωACalso plays a role in determining the modes. Primarily, ωACchooses the faltering and\nflipping modes when x≥1, since it determines the time duration τythat the equilibrium\nstate at each time stays at the y-axis. One can obtain the duration by finding the maximum\nτysatisfying σ(t)≥1 int∈[t0, t0+τy]. In the case of sinusoidal σ(t), this can be expressed\nasτy=1\nωAC(π−2ζ), where ζ= arcsin(1 /x)∈(0, π/2]. For τy, as the equilibrium state is at\nthey-axis, the magnetization modulates near the y-axis. After τy, the equilibrium position is\ndivided again and deviates away from the y-axis. Then, depending on its modulated position,\nthe magnetization chooses one of the equilibrium state, which leads to either faltering or\n9Mode IMode IIMode IIIMode IVMode Vbcd\n𝛿𝛿ef𝜔!\"\n𝜔!\"𝜋/200.020.040.060.080.10\n0.020.040.060.080.10\n𝜋/3𝜋/6𝜋/20𝜋/3𝜋/6\nNo FlipFlip onceNo FlipFlip twiceFlip once𝑥𝑥𝜔!\"𝜔!\"0120120.010.050.030.020.040.010.050.030.020.04\naFinal StateDriving FrequencyPeriodicityAdhere to equilibrium at tAdiabaticSmallOOResonatingIntermediateOXChaoticLargeXX\nFigure 3: The modes in the adiabatic state under AC. (a) The classification and\ncomparison of final states. The final states transit with the driving frequency of AC, from\nadiabatic to resonating, and to chaotic state in sequence. While adiabatic and resonating\nstates have periodicity, chaotic state does not. While adiabatic state adheres to the equi-\nlibrium state at each t, others do not. (b) Schematics of distinct AC adiabatic modes after\ndecay. (c-d) 2D phase diagrams of adiabatic modes in xandωACwith α= 0.03, at (c) δ= 0\nand (d) δ=π/2. (e-f) 2D phase diagrams of adiabatic modes in δandωACat (e) x= 0.6\nand (f) x= 0.8. The violet denotes Mode I, the red denotes Mode II, the blue denotes Mode\nIII, the yellow denotes Mode IV, and the green denotes Mode V.\n10flipping modes. Additionally, the window of xfor every mode at higher ωACis opened\nup wider than that at lower ωAC, showing the fan-like feature in Figs. 3(c,d). Unlike the\nfaltering mode under DC, the window of Modes III and IV, originating from the faltering\nmode, expands to the finite range of x. Lastly, it is noteworthy that the phase of ωAC= 0\nin Figure 3(d) is the same as the phase in α= 0.03 line of Fig. 2(c). This substantiates that\nadiabatic modes can be derived from DC modes.\nOn the other hand, δacts as a switch to turn on the spin flip during the decay process\nto adiabatic states. Specifically, comparing Figs. 3(b) to (c), Modes II and IV barely appear\nwhen δ= 0, which means that the spin flip is turned off near δ= 0. This happens because\nthe average of σ(t) during the decay time is small, so the spin cannot flip before decaying\nthe adiabatic state. This can be supported by Figs. 3(e-f). In Fig. 3(e), we present phase\ndiagrams for the modes in δandωACatx= 0.6. The transition from Modes I to II occurs\nonce during the increase of δ. This happens because the spin flip is turned on by finite\nδ. Increasing x, the spin flips more times just as in DC case, so the repeated transition\nbetween Modes I and II can also be observed in Fig. 3(f). We should note that the increase\nofαreduces the decay time. Above arguments holds also for the triangular wave instead of\nsinusoidal wave. [See SI.]\nWe move on to high- ωACand low- αregime to investigate ferromagnetic magnetoresonance\n(FMR). One could expect that the resonance frequency is closely related to the frequencies\nωDCin Fig. 2(a). In fact, when ωACapproaches ωDC, the oscillating amplitude becomes\nlarger, and eventually the adiabatic state changes to the resonating state and to the chaotic\nstate sequentially. By checking the final state after t∼1 ms, we present 2D phase diagrams\ninωACandαatx= 0.3 and 0 .7 in Fig. 4(a). Both diagrams show similar behavior. It\nbegins with the adiabatic state at low frequencies, confronts the transition to the resonating\nstate, and reaches the chaotic state at a certain range of ωAC. For x= 0.7, the window of\nchaotic states is opened up widely as SOT becomes large.\nWe further illuminate the transition of states by Fourier transform. The transition is\n11c\n𝜔log!\"𝜔log!\"𝜙𝜔𝜔iiiiiiiviviviv𝜔!\"𝜔#2𝜔#−𝜔!\"2𝜔!\"2𝜔!\"2𝜔!\"2𝜔!\"𝜔iv𝜔\n𝑥=0.7\n𝜔!\"\n10𝛼×104𝛼×104𝑥=0.3ba\niiiiiiiiiiiiiii𝜔#𝜔!\"𝜔#−𝜔!\"0.00.51.01.52.010-610-510-40.0010.0100.100iiiiviviviviiiiv\n0.00.51.01.52.010-610-510-40.0010.0100.100iviiiiiiiiiiiiiii\n~1/𝜔𝜔!\"=0.60𝜔!\"=0.53𝜔!\"=0.10iviv2𝜔!\"2𝜔!\"\nlog!\"𝜔0.01.02.00.01.02.00.01.0-1-6-2-3-4-5-1-6-2-3-4-5-3-7-4-5-6Adiabatic\nChaoticResonating\n2.0log!\"𝜙𝜔log!\"𝜙𝜔\nlog!\"𝜙𝜔log!\"𝜙𝜔log!\"𝜙𝜔Figure 4: The transition from adiabatic to resonating and chaotic states under\nAC. (a) 2D phase diagrams of the final states in ωACandα. The adiabatic state is in dark\nblue, the resonating state is in green, and the chaotic state is in dark red. The top panel\nis at x= 0.3 and the bottom panel is at x= 0.7. (b) The schematics of change in Fourier\ntransform in for each state. (c) The corresponding examples to each state of (b) at x= 0.3\nandα= 0.\nillustrated in Fig. 4(b), where the change in |ϕ(ω)|byωACis schematically presented. Here,\nϕ(ω) corresponds to the Fourier transform of azimuthal angle ϕ(t) from t= 0 to t= 0.21 ms\n(5×105unit times). Figure 4(c) showcases representative examples of Fourier transforms at\nx= 0.3 and α= 0. For adiabatic and resonating states, four distinct peaks are observed in\nthe top panels: i) a main peak at ω1derived from ωDC, ii) another main peak driven by AC\natω2=ωAC, iii) induced peaks from i) and ii) at ω3=ω1+n(ω1−ωAC), and iv) subpeaks at\nω4=|ω1,2,3±2nωAC|(n∈N). Notably, as the time interval is elongated from 0.21 ms, Peaks\ni and iii decay while Peaks ii and iv gain intensity. [See SI.] This means that Peaks i and\niii are related to the decay process while Peaks ii and iv are related to the stable state after\ndecaying. Increasing ωACto the higher frequency, the change only occurs to the distance\nbetween peaks, as the frequency of Peak i decreases and that of Peak ii increases. Thus,\nadiabatic and resonating states are indistinguishable solely by Fourier transform. This is\nsubstantiated by the phase diagrams in Fig. 4(a), where the transition from adiabatic and\nresonating states is not distinctly delineated. This behavior is consistent for all modes in\n12Fig. 3(b). Near the chaotic state, the peaks form array in a period of ωAC−ω1as shown\nin the middle panels. As ωACincreases more, chaos sets in, destroying all peaks and being\nϕ(ω)∝1/ωas shown in the bottom panels. This behavior does not change although the\nrange of time is elongated.\nDiscussion\nWe address here about the typical values of parameters in the realistic systems. The typical\nvalue of ferromagnetic anisotropy is 1 −10µeV/atom,26that of the current density in the\nexperiments is ∼107A·cm−2, and that of the itinerant spin polarization by Rashba-Edelstein\nEffect is estimated about 10−4ℏper unit cell.45The typical value of Gilbert damping constant\nis∼10−3−10−2,46and that of the unit of AC frequency is estimated about 1 −10 GHz.\nSo far, we discuss the magnetization dynamics at the interface of TI-FM heterostructure.\nUnder DC, we observe the oscillating and flipping modes without damping, which is deter-\nmined by the relative strength of anisotropy and SOT. With damping, the number of spin\nflips is determined by the duration of the crossover from flipping to the oscillating mode.\nUnder AC, we observe five distinct modes in the low frequency regime and their evolution\nto the resonating and chaotic states in the high frequency regime. Although we mainly\ndiscuss the TI-FM heterostructure due to its high efficiency, our work can be applied to the\ngeneral systems with strong Rashba spin-orbit coupling. This is because the assumption un-\nderlying in our work is only that the current carries finite spin due to the Rashba-Edelstein\nEffect. Our work offers insights into the spin control by spin-orbit coupling, underscoring\nthe practical aspects of the world of spintronics.\nAcknowledgement\nWe thank Wataru Koshibae for the fruitful discussions. This work was supported by JST,\nCREST Grant Number JPMJCR1874, Japan.\n13Supporting Information Available\nThis material is available free of charge via the Internet at https://pubs.acs.org/\n•The dependence of crossover time from flipping to oscillating mode on xandα, the\ndependence of decay time to adiabatic modes on α, the comparison between triangular and\nsinusoidal waves, and the time evolution of peaks in Fourier transform.\n14Supporting information for “Unraveling the\ndynamics of magnetization in topological\ninsulator-ferromagnet heterostructures via\nspin-orbit torque”\nTaekoo Oh∗and Naoto Nagaosa∗\nRIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nE-mail: taekoo.oh@riken.jp; nagaosa@riken.jp\nTable of contents\n•The dependence of crossover time of DC in xandα.\n•The dependence of decay time of AC in α.\n•The comparison of triangular and sinusoidal waves.\n•The time evolution of peaks in Fourier transform.\n1The dependence of crossover time of DC in xand α\nab\n𝛼𝛼𝛼\nFigure 1: τcinxand α.(a)τcinxwith various α. (b) τcinαwith various x.\nUnder DC, the system shows flipping, faltering, and oscillating modes without damping.\nWhen damping is introduced, the crossover from flipping or faltering mode to oscillating\nmode occurs. We acquire the crossover time τcempirically. Figure 1(a) shows the relation\nofxandτc, while Figure 1(b) exhibits the relation of αandτc. For small α, one could note\nthat the empirical relation of τctoxandαisτc∝e4.53xα−1.\nThe dependence of decay time of AC in α\nUnder AC and damping, the system decays into an resonating or an adiabatic state or evolves\ninto a chaotic state. The decay time is defined as the duration of time reaching to resonating\nor adiabatic states from the initial state. We empirically observe the decay time at different\nx,ωACandαin Fig. 2 One could note that the decay time is proportional to α−1.\n2Figure 2: The decay time in α.The decay time in αwith different xandωAC.\nThe comparison of triangular and sinusoidal waves\nWe mainly discuss the magnetization dynamics under sinusoidal waves of AC. One could\nreplace the sinusoidal waves with triangular waves. The triangular wave is given by\nσ(t) =|mod[2ωACσ\nπ(t−t0),4σ]−2σ| −σ. (1)\nThe amplitude of this function is σ, and the frequency of this function is ωAC. At t0=\nπ/(2ωAC), the function becomes sine-like, while at t0= 0, the function becomes cosine-like,\nas shown in Fig. 3(a).\nWe set ωAC= 2π/240≈0.026,α= 0.03, and compare the adiabatic modes for sinusoidal\nand triangular waves in Fig. 3(b). In the manuscript, we describe that Modes II and IV does\nnot appear under sine waves. The argument is also consistent with triangular waves, since\nthe phase diagram under sine-like triangular waves does not show Modes II and IV as well.\nThe difference between sinusoidal and triangular waves is that the window for each mode\nwidens up for triangular waves.\n3a𝜎(𝑡)\n𝜎(𝑡)𝑡/𝑇𝑡/𝑇b𝑥0.51.01.52.0𝑥0.51.01.52.0SineSine-like triangular\n𝑥0.51.01.52.0𝑥0.51.01.52.0CosineCosine-like triangularSine-like triangular\nCosine-like triangularIIIVIIIIVFigure 3: The magnetization dynamics under triangular waves. (a) (upper panel)\na sine-like triangular wave, (lower panel) a cosine-like triangular wave. (b) The adiabatic\nmode diagrams at ωAC= 2π/240 and α= 0.03 for sine (first), sine-like triangular (second),\ncosine (third), and cosine-like triangular (fourth) waves. The violet denotes Mode I, the red\ndenotes Mode II, the blue denotes Mode III, the yellow denotes Mode IV, and the green\ndenotes Mode V.\nThe time evolution of peaks in Fourier transform\nIn our manuscript, we describe four types of peaks observed in the Fourier transform of ϕ(t).\nThese include: i) the main peak originating from the DC modes denoted as ω1, ii) a peak\ncorresponding to the AC driven frequency ω2=ωAC, iii) induced peaks resulting from the\nfrequency difference between i) and ii), given by ω3=ω1+n(ω1−ω2), and iv) subpeaks\nrepresented by ω4=|ω1,2,3±2nωAC|.\nPeaks i) and iii) are associated with the decaying process, whereas peaks ii) and iv)\nrepresent the stable state after decay. Figure 4 illustrates this phenomenon. In Fig. 4(a),\nthe Fourier transform is performed from t= 0 to 1 .26µs, while in Fig. 4(b), it is performed\nfrom t= 0.12474 to 0 .126 ms. As time progresses, Peaks i and iii diminish due to damping,\nwhile Peaks ii and iv remain robust owing to the driven frequency.\n40.00.51.01.52.010-510-40.0010.010\n0.00.51.01.52.010-510-40.0010.010iiviviviviviiiiviviiiiiviviviab𝜙𝜔𝜙𝜔𝜔𝜔Figure 4: The time evolution of peaks in Fourier transform of ϕ(t) (a) The Fourier\ntransform of ϕ(t) int= 0 to 1 .26µs. (b) The Fourier transform of ϕ(t) near t= 0.12474 to\n0.126 ms.\n5References\n(1) Witczak-Krempa, W.; Chen, G.; Kim, Y. B.; Balents, L. Correlated quantum phe-\nnomena in the strong spin-orbit regime. Annu. Rev. Condens. Matter Phys. 2014 ,5,\n57–82.\n(2) Hasan, M. Z.; Kane, C. L. Colloquium: topological insulators. Reviews of modern\nphysics 2010 ,82, 3045.\n(3) Qi, X.-L.; Zhang, S.-C. Topological insulators and superconductors. Reviews of Modern\nPhysics 2011 ,83, 1057.\n(4) Pi, U. H.; Won Kim, K.; Bae, J. Y.; Lee, S. C.; Cho, Y. J.; Kim, K. S.; Seo, S. 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Journal of the\nPhysical Society of Japan 2014 ,83, 104711.\n(44) Gilbert, T. L. A phenomenological theory of damping in ferromagnetic materials. IEEE\ntransactions on magnetics 2004 ,40, 3443–3449.\n24(45) Chen, W. Edelstein and inverse Edelstein effects caused by the pristine surface states\nof topological insulators. Journal of Physics: Condensed Matter 2019 ,32, 035809.\n(46) Barati, E.; Cinal, M.; Edwards, D.; Umerski, A. Calculation of Gilbert damping in\nferromagnetic films. EPJ Web of Conferences. 2013; p 18003.\n25TOC Graphic\nTopological InsulatorFerromagnet⃗𝐼⃗𝜎𝑀\n26"    },    {        "title": "0907.0475v2.Theory_of_frequency_dependent_spin_current_noise_through_correlated_quantum_dots.pdf",        "content": "arXiv:0907.0475v2  [cond-mat.mes-hall]  18 Jun 2010Theory of frequency-dependent spin current noise through c orrelated quantum dots\nC. P. Moca1, I. Weymann2,3, G. Zar´ and1\n1Department of Theoretical Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary\n2Physics Department, Ludwig-Maximilians-Universit¨ at, T heresienstrasse 37, 80333 Munich, Germany\n3Department of Physics, Adam Mickiewicz University, 61-614 Pozna´ n, Poland\n(Dated: November 4, 2018)\nWe analyze the equilibrium and non-equilibrium frequency- dependent spin current noise and spin\nconductance through a quantum dot in the local moment regime . Spin current correlations are\nshown to behave markedly diﬀerently from charge correlatio ns: Equilibrium spin cross-correlations\nare suppressed at frequencies below the Kondo scale, and are characterized by a universal function\nthat we determine numerically for T= 0 temperature. For asymmetrical quantum dots dynamical\nspin accumulation resonance is found at the Kondo energy, ω∼TK. At higher temperatures\nsurprising low-frequency anomalies related to overall spi n conservation appear.\nPACS numbers: 72.25.-b, 73.63.Kv, 72.15.Qm, 72.70.+m\nIntroduction. Coherent detection and manipula-\ntion of spin currents in nanostructures has recently at-\ntracted wide attention due to possible promising appli-\ncations in future storage technologies and quantum com-\nputing [1, 2]. Many proposals have been made to build\nspin batteries to inject spin-polarized current, and then\nﬁlter, manipulate, and detect it [3]. Often one makes use\nof ferromagnetic electrodes in these circuits [4], while in\nother cases the application of an external magnetic ﬁeld\n[5] or the presence of a ferromagnetic resonance process\n[6] enables one to ﬁlter and detect spin currents. Quan-\ntum dots play a special and important role in this regard:\nIn these devices, the strong electron-electron interaction\nenablesoneto manipulatethespin ofasingleelectron[7],\nand such quantum-dot devices provide a possible route\nto quantum computing [8].\nHowever, to use spin circuits eﬃciently, it would be of\ncrucial importance to characterize the noise in them. In\naddition, the structure of the noise provides valuable in-\nformation on interactions and correlations. In fact, a lot\nof attention has been devoted to noise analysis in corre-\nlated mesoscopic circuits for this reason [9–11]. Due to\nprogress in experimental technology, it is now possible to\nmeasure acconductance properties as well as frequency-\ndependent noise in these circuits down to very low tem-\nperatures, and even in the Kondo regime [12–14]. Fur-\nthermore, with eﬃcient spin ﬁltering methods [3] mea-\nsuring spin-resolved current noise in such circuits is also\nwithin reach. Surprisingly, while a lot is known about\nthe properties of ordinary noise in the correlated regime,\nmuch less is known about the structure of spin current\nnoise. So far, only spin correlationsin the sequential tun-\nneling [15, 16] and perturbative regimes [17] have been\nanalyzed, and these works focused almost exclusively on\nshot noise.\nHere we carry out a detailed analysis of the full fre-\nquency spectrum of the spin-dependent current noise in\nthe Kondo regime. We show that equilibrium spin cur-\nrent correlations are characterized by two universal func-tions, which we determine numerically for T= 0 tem-\nperature using the method of numerical renormalization\ngroup(NRG) [18], and compute analyticallyfor largefre-\nquencies. At ﬁnite temperatures, we analyze spin corre-\nlations using a perturbative approach. We ﬁnd in all\nregimes that correlations between electrons of the same\nand opposite spins behave markedly diﬀerently. In the\nperturbative regime these remarkable diﬀerences emerge\natfrequenciesbelowtheKorringarelaxationrate: whilea\ndip appears in the frequency-dependent noise of opposite\nspin directions, a largepeak developsfor the parallelspin\ncomponents. These surprising features are all intimately\nrelated to spin conservation.\nModel. Focusing on the Kondo regime, we shall as-\nsume that there is a single spin S= 1/2 electron on the\nquantum dot, which couples to the electrons on the leads\nthrough the Kondo interaction [19],\nHint=/summationdisplay\nr,r′=L,R/summationdisplay\nσ,σ′j\n2vrvr′Sψ†\nrσσσσ′ψr′σ′.(1)\nHereσstands for the three Pauli matrices, the ﬁelds\nψrσ=/integraltextD\n−Dcrσ(ε)dεdestroy electrons of spin σin leads\nr∈ {L,R}, and their dynamics are governed by the non-\ninteracting Hamiltonian, H0=/summationtext\nrσ/integraltext\nε c†\nrσ(ε)crσ(ε)dε\n[25]. The coupling jin Eq. (1) is the usual dimension-\nless coupling, which incorporates already the density of\nstates in the leads, and is related to the Kondo tempera-\nture asTK≈De−1/j, withDthe cut-oﬀ energy appear-\ning in��rσ. The dimensionless hybridization parameters\nare given by vL= cos(φ/2) andvR= sin(φ/2), withφ\nparametrizing the asymmetry of the dot: φ=π/2 cor-\nresponds to a symmetrical quantum dot with maximum\ntransmittance.\nEquilibrium noise. In view of the special structure\nof Eq. (1), it is natural to introduce the ’even’ and ’odd’\nlinearcombinations, Ψ ≡cos(φ/2)ψL+sin(φ/2)ψR, and\n˜Ψ≡sin(φ/2)ψL−cos(φ/2)ψR. AlthoughonlyΨcouples\nto the spin in Hint, changing the chemical potential in\none of the leads couples the ﬁelds ˜Ψ and Ψ, and both2\ncontribute to the spin noise.\nTo compute the noise, we ﬁrst deﬁne the spin compo-\nnentσof the current in lead rthrough the equation of\nmotion,Jrσ≡e˙Nrσ=e i[Hint,Nrσ]. The correspond-\ning current is found to have two distinct (even and odd)\nparts,Jrσ=Irσ+˜Irσ, with\nIrσ=ej γri(F†\nσΨσ−Ψ†\nσFσ),\n˜Irσ=ej˜γri(F†\nσ˜Ψσ−˜Ψ†\nσFσ), (2)\nand the prefactors deﬁned as γL/R= [1±cos(φ)]/4 and\n˜γL/R=±sin(φ)/4. The operator Fσ= (SσΨ)σdenotes\nthe so-called composite fermion operator [20], and repre-\nsents the universal (Kondo) part of the dot-electron.\nThe operator identity, Ir↑+Ir↓= 0, and the sim-\nple even-odd decomposition of Jrσimply that, in equi-\nlibrium and in the absence of external magnetic ﬁeld,\nthe sixteen components of the symmetrized noise Sσσ′\nrr′≡\n1\n2/an}bracketle{t{Jrσ(t),Jr′σ′(0)}/an}bracketri}ht, depend on just two universal func-\ntions,sand ˜s. Maybethemostinterestingleft-rightnoise\ncomponent, Sσσ′\nLR, can be expressed, e.g., as\nSσσ′\nLR(ω) =−e2\n2πTKsin2(φ) (δσσ′˜s(ω)+σσ′s(ω)),\nwheree2/2π=e2/hdenotes the universal conductance\nunit, and the dimensionless functions sand ˜sdepend\nexclusivelyontheratios ω/TKandT/TK. Thefunction s\nis related to the ’even’ current component, and it governs\nthe correlations between spin up and spin down carriers,\nhowever, its contribution cancels in the charge noise and\ncharge conductance, which are exclusively determined by\nthe ’odd’ component of the current, incorporated in ˜ s.\nIn equilibrium, the ﬂuctuation-dissipation theo-\nrem relates Sσσ′\nrr′(ω) to the real part of the spin-\nconductance through the dot, Re Gσσ′\nrr′(ω) =\n−1\nωcoth(ω/2T)Sσσ′\nrr′(ω),which can therefore also be\nexpressed in terms of two dimensionless universal con-\nductance functions, g(ω,T) and ˜g(ω,T). The left-right\nconductance, e.g., reads\nReGσσ′\nLR(ω) =e2\n2πsin2(φ)/parenleftbig\nδσσ′˜g(ω,T)+σσ′g(ω,T)/parenrightbig\n.\nUsing tedious but straightforward manipulations, we\ncan express gand ˜gin terms of the spectral functions\n̺F(ω,T) and̺IσIσ′(ω,T) of the composite fermion and\nof the “current” operator Iσ≡i(F†\nσΨσ−Ψ†\nσFσ),\n˜g(ω,T) =1\n2ω/integraldisplay\ndω′̺F(ω′,T)\n̺F(0)[f(ω′−ω)−f(ω′+ω)],\ng(ω,T) =−1\n2ω̺F(0)̺I↑I↑(ω,T), (3)\nwithf(ω) denoting the Fermi function [21]. Since Fσ\nandIσare local operators, we can compute gand ˜g(and\nthussand ˜s) by using the powerful method of numerical\nrenormalization group (NRG) [18, 22].10 -3 10 -2 10 -1 10 010 110 210 3 10 -8 10 -6 10 -4 10 -2 10 0\n10 -3 10 -1 10 110 30.0 0.4 0.8 1.2 \n  \n  \nPSfrag replacements\n˜s, s\n˜g, g\nω/TKω/TK−s˜s\n˜g\n−g\nFIG. 1: (color online) Zero-temperature universal functio nss\nand ˜scomputed by NRG. Inset: universal spin conductance\nfunctions gand ˜g.\nT= 0, equilibrium results. TheT= 0 temperature\nuniversal functions ˜ s(ω/TK), ands(ω/TK) and the con-\nductance functions g(ω/TK) and ˜g(ω/TK) are displayed\nin Fig. 1. The high-frequency behavior of sand ˜scan be\ncaptured by doing perturbation theory in jand summing\nup the leading logarithmic corrections to give\n˜s(ω/TK)≈ −3\n2s(ω/TK)≈3π2\n16|ω|\nTK1\nln2(|ω|/TK)\nforω≫TK. Though they look similar at high frequen-\ncies,sand ˜sbehave markedly diﬀerently in the Fermi\nliquid regime, ω≪TK, where ˜s= ˜α|ω|/TK+..., while\ns=α(|ω|/TK)3+..., withαand ˜αuniversal constants\nof the order of unity. The ω3scaling ofsis related to\nspin conservation: In the absence of external spin relax-\nation mechanism, the total number of spin up electrons\ncan ﬂuctuate between two values, N↑andN↑+1. Since\nthe spin up electrons couple to the spin down electrons\nonly at a single point (the quantum dot), no steady spin\ncurrent can be generated for the spin down electrons by\ninjecting spin up electrons in one of the leads. Thus the\nspin conductance G↑↓\nrr′(ω) must vanish at ω= 0, and by\nanalyticity, G↑↓\nrr′∼ω2. In equilibrium, however, the spin\ncurrentnoiseis simplyrelatedto the spinconductanceby\nthe ﬂuctuation-dissipation theorem, implying a |ω|3scal-\ning ofsatT= 0. This argument carries over to ﬁnite\ntemperaturestoo, where it leads to an asymptotic behav-\nior,s∼Tω2in the absence of external spin relaxation.\nWe should emphasize that, in our calculations, spin re-\nlaxation is due to the interaction part of the Hamiltonian\nwhich, however, conservesthe total spin, and leadsto the\nvanishing of ↑↓spin noise component at ω= 0. Introduc-\ning some source of an external spin relaxation, however,\nleads to a violation of spin conservation, and amounts in\na ﬁniteS↑↓\nLR(ω= 0)/ne}ationslash= 0 [21].\nThe fundamental diﬀerence between ↑↑and↑↓cor-\nrelations shows up even more strikingly in the spin-\nconductance (see Fig. 1): While G↑↑\nLR(ω) is dominated\nby ˜g(ω) and behaves qualitatively the same way as the\nconductance through the dot, G↑↓\nLR(ω)∼g(ω) exhibits a3\n(0) (0)disc Sconn S +\n+ ==\n= +\n+ + + =Π ΠΣ\nΣσr,  J\n  Jr',   ' σ σr,  J  Jr',   ' σ\nΠ Π\nΠ Πr',   'σ r',   'σ r,σ r,σW W W W\nFIG. 2: Diagrams contributing to the noise. The reduced\ndensity matrix of the spin evolves along the upper and lower\nKeldysh contours. Triangles denote the bare current vertic es,\ndots indicate the exchange interaction. Arrows correspond\nto conduction electron propagators. To leading order, the\ncurrent-current correlation function is given by the conne cted\ndiagrams (top), Sconn(ω), and the “disconnected” diagrams\n(second line), Sdisc(ω), with Π the propagator describing the\nevolution of the reduced density matrix of the spin (third\nline). The dressed current vertex Wrσis given by diagrams\nsimilar to those of the self-energy, Σ (last line), with one o f\nthe dots replaced by a triangle.\nresonance at a frequency ω≈0.5TK[26]. This can be\nunderstood in a simple and intuitive way: The spin con-\nductance ↑↓is generated by ﬂips of the localized spin.\nForω > T K, the coupling to the conduction electrons\ngets stronger with decreasing ω, and increases the con-\nductance. At verysmallenergyscales, ω≪TK, however,\nthe impurity spin is quenched, and with the above mech-\nanism being absent, the ↑↓conductance must vanish.\nT/ne}ationslash= 0, perturbative regime. Computation of the\nﬁnite temperature noise requires care: Usual ﬁnite tem-\nperature NRG broadening procedures lead to an unphys-\nical ﬁnite linear coeﬃcient for s(ω), conﬂicting with our\nexact ﬁnite Tresult,s(ω)∼ω2. Therefore, for T/ne}ationslash= 0,\nother methods must be used. For T≫TK, we carried\nout a systematic expansion in jfor the time-dependence\nof the reduced density matrix of the spin and the spin\ncurrent noise using the formalism of Refs. [23, 24]. De-\ntails of this involved calculation shall be published else-\nwhere [21], here we just outline the main results.\nNaively, to calculate the noise in leading order, one\nwould just compute the ﬁrst (connected) noise diagram\nof Fig. 2,Sconn(ω). This diagram accounts for short\ntime current correlations mediated by electron-hole ex-\ncitations in the leads, and dominates indeed the noise\nat high and intermediate frequencies, ω/greaterorsimilarT. At small\nfrequencies, however, a resummation of the perturbation\nseries is necessary, because there the “disconnected” con-\ntribution,Sdisc(ω), turns out to be of the same order injasSconn(ω), and becomes also important: This con-\ntribution accounts for correlations between subsequent\nincoherent tunneling processes, generated by the impu-\nrity spin itself. These correlations are due to the mere\nfact that a spin ﬂip process where a conduction electron’s\nspin is ﬂipped from up do down, ↑→↓, must be followed\nby a process ↓→↑. To account for them, one needs to\nsolve a Dyson equation for the propagator Π of the re-\nduced density matrix of the spin, as sketched in Fig. 2.\nIn this approach, spin relaxation is characterized by the\nrelaxation rate, Γ( ω), appearing in the self-energy Σ of\nthe propagator Π [23],\nΓ(ω) =T/summationdisplay\nr,r′j2v2\nrv2\nr′ˆL/parenleftbiggω\nT,µr−µr′\nT/parenrightbigg\n,(4)\nwhere Re ˆL(x,y) =π y+π[xsh(x)−ysh(y)]/[ch(x)−\nch(y)], and Im ˆL(x,y) =1\nπ/integraltext\ndx′ReˆL(x′,y)/(x−x′). In\ntheω→0 limit, Γ(ω) can be identiﬁed as the Korringa\nrelaxationrate, EK≡Γ(0)/2,oftheimpurityspin,which\nfor a simple voltage-biased quantum dot reads\nEK=πj2T/bracketleftbigg1+cosφ\n2+1−cosφ\n2eV\n2TcotheV\n2T/bracketrightbigg\n.\nIn the voltage-biased case, we can express the left-right\ncomponent of the spin current noise in a compact form\nSσσ′\nLR(ω) =−e2\n2πRe/bracketleftBigg\nσσ′\n16Γ2(ω)−R2(ω)\n−iω+Γ(ω)/2(5)\n+3−σσ′\n32Tj2sin2(φ)/bracketleftBig\nˆL/parenleftbiggω\nT,V\nT/parenrightbigg\n+ˆL/parenleftbiggω\nT,−V\nT/parenrightbigg/bracketrightBig/bracketrightBigg\n,\nwithR(ω) =j2cos(φ)ˆL(ω/T,0). Thesymmetrizednoise\nis shown in Fig. 3: At high frequencies, ω≫EK, the\nnoise is dominated by the result of simple-minded per-\nturbation theory, corresponding to the second line of\nEq. (5). This part of the correlation function describes\nshort-time correlations within a single tunneling process,\ngenerated by the dynamics of electron-hole excitations in\nthe leads. However, at time scales t∼1/ω>1/EK, con-\nsecutive incoherent tunneling processes start to correlate\nby the constraint mentioned before. These correlations\nare captured by the ﬁrst term in Eq. (5), coming from\nthe “disconnected” part of the noise (see Fig. 2). As\na consequence, for ω < E K, a large dip appears in the\nnoisecomponent S↑↓\nLR, while a bump emergesin S↑↑\nLR. For\nzero-bias,V= 0, we ﬁnd that S↑↓\nLR(ω= 0,V= 0) = 0, in\nagreement with the ﬂuctuation-dissipation theorem and\nthe observation that the linear spin conductance between\nspin up and spin down electrons must vanish. This has\na simple physical explanation: A spin- ↑electron injected\nfromthe left can giveriseto aspin- ↓outgoingelectronon\ntherightwith acertainprobability(see Fig.3). However,\nbefore such a process occurs again, another spin ﬂip pro-\ncess must take place, where the dot spin is ﬂipped back.4\n0.01 0.1 1 10 0.00 0.02 0.04 0.06 0.08 \nPSfrag replacements\n−Sσσ′\nLR(Te2/2π)\nω/TS↑↑\nLR\nS↑↓\nLReV= 0eV= 5Tj= 0.07 ∼ω≃EK/T\nFIG. 3: (color online) Top: Sketch of consecutive spin ﬂip\nprocesses. Bottom: Equilibrium and non-equilibrium noise\nspectraSσσ′\nLR(ω)intheperturbativeregime, max {T,ω,E K}>\nTK, as computed from a diagrammatic approach. In the non-\nequilibrium case a simple current bias was assumed, Vσ\nL≡V\nandVσ\nR≡0, while j= 0.07 andφ=π/2 in both cases.\nIn equilibrium, this second process (on the average) re-\nmovesexactlythe same amount of ↓spin from the right\nlead as injected in the ﬁrst process. Therefore, no equi-\nlibrium↑↓dcspin conductance is possible.\nRemarkably, the above correlations only show up in\nthe spin current noise, and cancel out in the charge\ncurrent noise, SLR≡/summationtext\nσ,σ′Sσσ′\nLR. Being the result of\nrather classical correlations between subsequent incoher-\nent processes, these low frequency features can also be\ncaptured by a much simpler rate equation approach (see\nRefs. [17, 21]), which, however, is unable to account for\nthe high frequency part of the noise at ω>T.\nAlthough the above results are perturbative in j,\nthey carry over to the whole regime max {T,ω,E K}>\nTKwith the small modiﬁcation that jmust be re-\nplaced by the renormalized coupling j→j(T,ω,eV)≈\n1/ln(max{T,ω,E K}/TK).\nFermi liquid regime. In the Fermi liquid regime,\nω,T,eV ≪TK, one can compute the spin current corre-\nlations by describing the dot in terms of scattering states\nthat interact at the impurity site. This is a rather cum-\nbersome approach for ﬁnite frequencies. However, ob-\nserving that correlations between spin up and down elec-\ntrons are generated only through the residual electron-\nelectron interaction, simple phase space arguments im-\nmediately give that the T= 0 shot noise is just given\nbyS↑↓\nLR(V) = (e2/2π)γsin2(φ)(eV)3/T2\nK, while for\nequilibrium we recover the numerically observed result,\nS↑↓\nLR(ω) = (e2/2π)αsin2(φ)|ω|3/T2\nK, withγandαtwo\nuniversal numbers. The discussion of the ﬁnite tempera-ture and ﬁnite frequency noise and the precise determi-\nnation of these universal constants is very complicated,\nand shall be considered in a future publication.\nConclusions .Analyzing the full frequency depen-\ndence of the spin current noise through a quantum dot\nin the Kondo regime we found that ↑↓correlations are\nstrongly suppressed at frequencies below the Kondo tem-\nperature and below the Korringa relaxation rate as com-\npared to ↑↑correlations due to overall spin conserva-\ntion. In the ↑↓conductance a resonance is predicted\natω∼TK. Observing these striking features is within\nreach with present-day noise measurement techniques.\nWe would like to thank Laci Borda for useful sugges-\ntions. This research has been supported by Hungarian\ngrants OTKA Nos. NF061726, K73361, and Romanian\ngrant CNCSIS PN II ID-672/2009, and the EU GE-\nOMDISS project. I.W. acknowledges support from the\nFoundation for Polish Science and the Ministry of Sci-\nence and Higher Education through a research project in\nyears 2008-2010.\n[1]Semiconductor Spintronics and Quantum Computation ,\ned. by D.D. Awschalom, D. Loss, and N. Samarth\n(Springer, Berlin 2002).\n[2] I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n[3] S. M. Frolov et al., arXiv:0801.4021 (unpublished).\n[4] V. Sih et al., Nature Physics 1, 31 (2005).\n[5] R. M. Potok et al., Phys. Rev. Lett. 89, 266602 (2002).\n[6] S. K. Watson et al., Phys. Rev. Lett. 91, 258301 (2003).\n[7] S. Sasaki et al., Nature 405, 764 (2000).\n[8] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n[9] A. Kaminski, Yu. V. Nazarov, and L. I. Glazman, Phys.\nRev. B62, 8154 (2000).\n[10] Y. Meir and A. Golub, Phys. Rev. Lett. 88, 116802\n(2002).\n[11] M. Sindel et al., Phys. Rev. Lett. 94, 196602 (2005).\n[12] J. Gabelli and B. Reulet, Phys. Rev. Lett. 100, 026601\n(2008).\n[13] P.-M. Billangeon et al., Phys. Rev. Lett. 98, 126802\n(2007).\n[14] T. Delattre et al., Nature Phys. 5, 208 (2009).\n[15] O. Sauret and D. Feinberg, Phys. Rev. Lett. 92, 106601\n(2004).\n[16] A. Cottet, W. Belzig, and C. Bruder, Phys. Rev. Lett.\n92, 206801 (2004).\n[17] M. Kindermann, Phys. Rev. B 71, 165332 (2005).\n[18] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[19] L.I. Glazman and M. Pustilnik, in ”Nanophysics: Coher-\nence and Transport,” eds. H. Bouchiat et al. (Elsevier,\n2005), pp. 427-478.\n[20] T. A. Costi et al., Phys. Rev. Lett. 73, 1275 (1994).\n[21] I. Weymann, C. P. Moca and G. Zar´ and, (unpublished).\n[22] We used the open access Budapest NRG code,\nhttp://www.phy.bme.hu/ ∼dmnrg/.\n[23] A. Thielmann et al., Phys. Rev. Lett. 95, 146806 (2005).\n[24] M. Braun, J. K¨ onig and J. Martinek, Phys. Rev. B 74,5\n075328 (2006).\n[25] The annihilation operators crσ(ε) satisfy\n{c†\nrσ(ε),cr′σ′(ε′)}=δrr′δσσ′δ(ε−ε′).[26] In the numerical calculations, we deﬁne TKas the half-\nwidth of the composite fermion’s spectral function."    },    {        "title": "2206.13593v1.Bridging_atomistic_spin_dynamics_methods_and_phenomenological_models_of_single_pulse_ultrafast_switching_in_ferrimagnets.pdf",        "content": "Bridging atomistic spin dynamics methods and phenomenological models of single pulse ultrafast\nswitching in ferrimagnets\nFlorian Jakobs and Unai Atxitia\nDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit ¨at Berlin, 14195 Berlin, Germany\nWe bridge an essential knowledge gap on the understanding of all-optical ultrafast switching in ferrimagnets;\nnamely, the connection between atomistic spin dynamics methods and macroscopic phenomenological models.\nAll-optical switching of the magnetization occurs after the application of a single femtosecond laser pulse to\nspeciﬁc ferrimagnetic compounds. This strong excitation puts the involved degrees of freedom, electrons, lattice\nand spins out-of-equilibrium between each other. Atomistic spin models have quantitatively described all-\noptical switching in a wide range of experimental conditions, while having failed to provide a simple picture\nof the switching process. Phenomenological models are able to qualitatively describe the dynamics of the\nswitching process. However, a uniﬁed theoretical framework is missing that describes the element-speciﬁc spin\ndynamics as atomistic spin models with the simplicity of phenomenology. Here, we bridge this gap and present\nan element-speciﬁc macrospin dynamical model which fully agrees with atomistic spin dynamics simulations\nand symmetry considerations of the phenomenological models.\nI. INTRODUCTION\nSince its experimental discovery [1], the theoretical de-\nscription of laser induced all-optical switching (AOS) of the\nmagnetization in GdFeCo ferrimagnetic alloys has remained\na challenge. Despite intense experimental and theoretical re-\nsearch in the ﬁeld [1–12], an established and uniﬁed picture\nof the process is still missing. Experimental ﬁndings are\nmostly compared or interpreted in terms of atomistic spin\ndynamics simulations [13–17], multisublattice spin dynam-\nics based on symmetry arguments [5, 18, 19], and based on\nthe Landau-Lifshitz-Bloch equation [20–22]. The main goal\nof the present work is the revision, extension and merging of\nthese approaches into a uniﬁed model.\nAtomistic spin dynamics (ASD) models have been used be-\nfore to quantitatively describe ultrafast dynamics in 3 dtransi-\ntion metals [23, 24] and 4 frare-earth ferromagnets [25, 26].\nThey have also been used in GdFeCo, to describe the equilib-\nrium thermal properties [13], the thermal character of AOS\n[4], the so-called transient ferromagnetic-like state [3], the\ndemonstration of spin-current-mediated rapid magnon local-\nisation and coalescence [27] and the possibility of AOS using\npicosecond-long laser pulses [16]. Results from atomistic spin\nmodels also compare qualitatively well to an analytical the-\nory based on the excitation of spin-wave exchange modes [8],\nprovide insights for optimal electron, phonon and magnetic\ncharacteristics for low energy switching [28] and predict max-\nimum repetition rate using two consecutive laser pulses [29].\nMore sophisticated, orbital-resolved atomistic models provide\ninsights on the role of the intra-exchange coupling between\n4fand 5 delectrons in the dynamics of GdFeCo alloys[14].\nAtomistic models can naturally describe switching in Gd/Fe\nmultilayers composed of very thin layers [30, 31]. Recent ob-\nservations [32, 33] of single pulse switching in Mn 2RuxGa\nalloys are also well-described by ASD methods [34]. De-\nspite the demonstrated success in modeling AOS, ASD sim-\nulation results are cumbersome to interpret without an ana-\nlytical model that unveils the role of the different processes\nand interactions during the switching process. This potential\nsemi-analytical model has to capture most of the features ofthe ASD simulations.\nSemi-phenomenological models describing switching al-\nready exist. A macroscopic theory for the description of the\ndynamics and relaxation of the macroscopic (sublattice) mag-\nnetization of ferromagnets and antiferromagnets was devel-\noped originally by Baryakhtar [9, 35]. An extension of such\nphenomenology to ferrimagnets in the context of ultrafast spin\ndynamics was introduced in Ref. [5]. At the ultrafast scale,\nmagnetization dynamics are dominated by atomic scale spin\nexcitations, these spin dynamics are driven by dissipative pro-\ncesses which in ferrimagnets are two-fold, relativistic and ex-\nchange driven. Relativistic processes allow for exchange of\nangular momentum between the spins and lattice degree of\nfreedom due to the presence of spin-orbit interaction connect-\ning them. Exchange processes can arise due to transport of\nspin angular momentum – spin and magnon transport – which\nis the only mean to exchange angular momentum in ferromag-\nnets. In multisublattice magnets another, different pathway\nopens, namely, local exchange of angular momentum. To ac-\ncount for such local exchange processes in ferrimagnets, the\nequation of motion for the magnetization dynamics proposed\nby Landau and Lifshitz [36] is enhanced by an exchange re-\nlaxation term [5, 9, 19, 37]. Within this macroscopic model,\nthe exchange relaxation dominates the dynamics when the\nmagnetic sublattices are driven into mutual non-equilibrium.\nQualitative agreement to experiments in two-sublattice mag-\nnets has been demonstrated [19], such as AOS in ferrimag-\nnetic GdFeCo using fs laser pulses [5] and ps laser pulses\n[38], AOS in Heusler semimetals Mn 2RuxGa [39], or element-\nspeciﬁc demagnetization of ferromagnetic NiFe alloys [18].\nQuantitative comparison of this model to neither experiments\nnor ASD simulations have been conducted so far. While the\narguments behind such phenomenology are robust, the range\nof applicability and the validity of the model parameters could\nbe questioned. For instance, the parameters deﬁning the rel-\nativistic and exchange relaxation are assumed to be constant\nand of the same order. The magnetic free energy functional\nis calculated for near thermal equilibrium states. This implies\na relatively strong coupling to the heat-bath, while switching\nconditions are supposedly fulﬁlled when exchange relaxationarXiv:2206.13593v1  [cond-mat.mtrl-sci]  27 Jun 20222\nbetween sublattices dominates over the relaxation to the heat-\nbath.\nAn alternative macroscopic model directly derived from an\natomistic spin model has also been proposed. This model is\nbased in the Landau-Lifshitz-Bloch (LLB) equation of mo-\ntion [20, 40–43]. The LLB model for two-sublattice mag-\nnets [20, 42] has been used in the context of AOS in GdFeCo,\ne.g. the element-speciﬁc demagnetization rates compare well\nto experiment, and it predicts that near the magnetic phase\ntransition the otherwise slower Gd sublattice becomes faster\nthan Fe [22], as recently observed [44]. The LLB model has\nbeen demonstrated to provide accurate analytical expressions\nfor the temperature dependence of the relativistic relaxation\nparameter as well as for the non-equilibrium effective ﬁelds\nbelow and above the critical temperature [42]. Moreover, the\nLLB model also describes the transverse motion of the mag-\nnetization. This makes it the preferred model for computer\nsimulations of heat-assisted magnetic recording [45] and re-\nalistic description of all-optical switching [46], and ultrafast\nspintronics, such as domain wall motion [47, 48] or skyrmion\ncreation by ultrafast laser pulses [49]. So far the LLB model\nand Baryakhtar-like models have been considered as comple-\nmentary approaches. Here, we merge them into one uniﬁed\napproach.\nIn this work we address the issues discussed above by di-\nrectly comparing both phenomenological models to ASD sim-\nulations. We do so since ASD simulations have been al-\nready quantitatively compared to experiments in literature.\nWe ﬁnd that quantitative comparison between ASD and both\nphenomenological models is partially possible for laser exci-\ntation producing small deviation from equilibrium. However,\nthose models hardly reproduce magnetic switching using the\nsame parameter values describing the relaxation of small per-\nturbations. Here, based upon those phenomenological mod-\nels, we propose a macroscopic model that compares precisely\nto the magnetization dynamics calculated using ASD simula-\ntions, including element-speciﬁc magnetization relaxation and\nswitching. This model bridges atomistic spin dynamics based\nmodels and previously proposed phenomenological models.\nNotably, it provides a deeper understanding to the parameters\nentering the phenomenological models and sheds some light\ninto the process of ultrafast switching in ferrimagnets.\nThe work is broken down in the following way: in Sec. II,\nwe present the atomistic spin model for the calculation of the\nmagnetic equilibrium properties and non-equilibrium dynam-\nics. The equilibrium properties are compared to a mean ﬁeld\nmodel. We then provide atomistic calculations of the ultra-\nfast magnetization dynamics with input from the two temper-\nature model. These results are the basis for the comparison to\nthe phenomenological models presented in Sec. III. Firstly,\nwe present the Baryakhtar model and the Landau-Lifshitz-\nBloch model. Secondly, we compare the ultrafast magneti-\nzation dynamics calculated with those models to the atomistic\nspin dynamics results. Finally, in Sec. III C we present the\nuniﬁed phenomenological model, a hybrid model combining\nBaryakhtar and LLB models, and its comparison to atomistic\nspin dynamics.II. ATOMISTIC SPIN MODEL\nFerrimagnetic materials characterise by spontaneous mag-\nnetization as a resultant of two or more components of non-\nparallel magnetic moments [50]. Atomistic spin models based\non the Heisenberg Hamiltonian can be considered one of the\nsimplest microscopic models able to reproduce the equilib-\nrium properties of ferrimagnets. The spin system energy due\nto only the exchange interactions can be described by an ef-\nfective Heisenberg model:\nH=\u0000å\ni6=jJaSa;i\u0001Sa;j\u0000å\ni6=jJbSb;i\u0001Sb;j\u0000å\ni6=jJabSa;i\u0001Sb;j(1)\nwhere Ja(b)(ab)is the exchange constant between neighbor-\ning sites represented by two classical spin vectors SiandSj\n(jSj=1). Further, one can include magnetic anisotropy terms\nto Eq. (1) to set a preferential axis for the magnetization.\nHowever, since the anisotropy energy is relatively low it plays\na marginal role in the switching process. This makes for a\nsimpler Hamiltonian and a more direct comparison to the phe-\nnomenological models. To model a ferrimagnet, one needs to\nconsider two alternating sublattices of unequal and antiparal-\nlel moments, with three exchange coupling constants: ferro-\nmagnetic for each sublattice ( JaandJb) and a third for the an-\ntiferromagnetic interaction between them, Jab. For instance,\nGdFeCo alloys are composed of a transition metal FeCo and\na Gd rare-earth sublattices. We model the Fe and Co spins\nas only one magnetic sublattice, and we assume a common\natomic magnetic moment of mFeCo=1:94mB. In these alloys\nthe rare-earth impurities add localised 4 fspins to the sys-\ntem assumed to be, mGd=7:6mB. The amorphous nature of\nGdFeCo is modelled by using a simple cubic lattice model\nbut with random placements of Gd moments within the lattice\nto the desired concentration. The applicability of the Heisen-\nberg approximation relies on the stability of local moments\nunder rotation and at high temperature where Stoner excita-\ntions are generally weak [51]. It is assumed that the electronic\nproperties are temperature-independent in the range where the\nsystem is magnetically ordered.\nA. Atomistic spin dynamics\nEquilibrium and non-equilibrium element speciﬁc mag-\nnetic properties of a ferrimagnet are calculated using atomistic\nspin dynamics simulations which are based in the stochastic-\nLandau-Lifshitz-Gilbert equation (s-LLG) [52]\n(1+l2\ni)ms;i˙Si=\u0000gSi\u0002[Hi\u0000li(Si\u0002Hi)]; (2)\nwhere gis the gyromagnetic ratio, and liis the so-called\nphenomenological sublattice speciﬁc damping parameter. By\nincluding a Langevin thermostat the spin dynamics includ-\ning statistical – equilibrium and non-equilibrium thermo-\ndynamic properties can be obtained. An effective ﬁeld-\nlike stochastic term ziis added to the effective ﬁeld Hi=\nzi(t)\u0000¶H\n¶Si, with white noise properties [53]: hzi(t)i=\n0 andhzi(0)zj(t)i=2likBTms;idi jd(t)=g:The variance3\n−2−1.6−1.2−0.8−0.400.40.81.21.6\n0 100 200 300 400 500 600 700M(T)[µB]\ntemperature / KMnet\nFe\nGd\nFIG. 1. Equilibrium magnetization of a GdFeCo alloy for Gd concen-\ntration, xGd=25%. Element-speciﬁc normalized equilibrium mag-\nnetization and net equilibrium magnetization, M(T) =xGdmGdmGd\u0000\nxFemFemFe, where mGd(Fe)is the atomic magnetic moment of Gd(Fe).\nLines correspond to the mean-ﬁeld approximation with renormalized\nexchange parameters. Symbols correspond to atomistic spin dynam-\nics simulations.\nof the Langevin noise is chosen such that the ﬂuctuation-\ndissipation theorem is full ﬁlled.\nB. Mean-ﬁeld approximation\nExact analytical expressions for the M(T)curve are cum-\nbersome to derive due to the many body character of the prob-\nlem. Here we resort the mean ﬁeld approximation (MFA),\nalready used in previous works [8, 13, 54]. We note that to\nbe able to apply the MFA for the GdFeCo impurity model,\nand thus translation non-symmetric with respect to spin vari-\nables Si, we need to transform the Heisenberg Hamiltonian to\na symmetric one. We use the spin analogy of the virtual crys-\ntal approximation (VCA) to transform the disordered lattice\nHamiltonian Hto a symmetric VCA Hamiltonian HVCA.\nWithin the VCA we evaluate the effective sublattice exchange\nparameters, given by the sum of the exchange interactions of a\ngiven spin at a site riof sublattice iwith all other atoms of this\nsublattice. This involves weighting the exchange parameters\nby the relative composition, xi\u0011concentration species i[8],\nJi=å\nri;r0\niJ(ri;r0\ni)\u0011|{z}\nVCAxiJ(ri;r0\ni)intrasublattice (3)\nwhereas the intersublattice effective exchange reads\nJi j=å\nri;r0\nj=2AiJ(ri;r0\nj)\u0011|{z}\nVCAxiJ(ri;r0\nj)intersublattice (4)\nThus the VCA Hamiltonian reads\nHVCA=å\nj2AiJiSi\u0001Sj+å\nj=2AiJi jSi\u0001Sj (5)where Airepresent the magnetic sublattice of the spin Si. In\nthe exchange approximation we deﬁne the MFA ﬁeld as\nmaHMFA\na=zaJaama+zabJabmb (6)\nThe element-speciﬁc equilibrium magnetization is calculated\nvia the self-consistent solution of ma=L(b maHMFA\na)and\nmb=L(b mbHMFA\nb).zaandzabcorrespond to the number\nof ﬁrst nearest neighbours of type aandb, respectively. It\nis well-known that the MFA overestimates the value of the\ncritical temperature TC. However, a very good agreement be-\ntween ASD and MFA can be obtained by using a reduced\nvalue for the exchange parameters, even for multilattice mag-\nnets [54]. Figure 1 shows element-speciﬁc Ma=xamama(T)\nusing ASD simulations and renornalized MFA for xGd=25%.\nNet magnetization is also shown in Fig. 1, which is deﬁned\nasM(T) =xGdmGdmGd\u0000xFemFemFe. The agreement between\nASD and MFA is good enough for all the temperature regions.\nWe observe the presence of compensation temperature TMat\nroom temperature for xGd=25% at which the thermally av-\nerage magnetization of both sublattices are equal but oppo-\nsite, so that the magnetization of the system is equal to zero\nM(TM) =0. The mapping of the atomistic spin model and the\ncorresponding mean-ﬁeld approximation turns out to be nec-\nessary for a quantitative comparison to the phenomenological\nmodels, and thereby paramount for the uniﬁcation of both pic-\ntures.\nC. Two Temperature Model\nSingle pulse all-optical switching has been demonstrated to\nbe a thermal process in ferrimagnetic GdFeCo alloys [4] and\nin Mn 2RuxGa Heusler semi-metals [32]. Ultrafast heating by\noptical or electric means are sufﬁcient to achieve switching in\nspeciﬁc GdFeCo alloys [55]. Although the minimum achiev-\nable duration of the electric pulses are limited to picoseconds,\nthose are better suited for potential integration into applica-\ntions. Laser pulses can be as short as only a few femtoseconds,\nwhich permits to excite the electron system in timescales of\nthe order of the exchange interaction allowing for the inves-\ntigation of fundamental physics governing switching. In this\nwork, we center in excitation of the ferrimagnetic GdFeCo\nusing femtosecond laser pulses. When a metallic ferrimag-\nnetic thin ﬁlm is subjected to a near infrared laser pulse, only\nthe electrons are accessible by the photon electric ﬁeld. Ini-\ntially, the absorbed energy is barely transferred to the lattice\nand consequently the electron system heats up. The electron\nand phonon temperatures are decoupled for up to several pi-\ncoseconds until the electron-phonon interaction equilibrates\nthe two heat-baths. This phenomenology is well captured by\nthe so-called two-temperature model (2TM) [56, 57] which\ncan be written as two coupled differential equations:\nCel¶Tel\n¶t=\u0000gep\u0000\nTel\u0000Tph\u0001\n+Pl(t) (7)\nCph¶Tph\n¶t= +gep\u0000\nTel\u0000Tph\u0001\n: (8)4\nCel=gelTelwhere gel=6\u0002102J/m3K2, and Cph=3:8\u0002106\nJ/m3K represent the speciﬁc heat of the electron- and phonon\nsystem. The electron-phonon coupling is taken temperature\nindependent, Gep=7\u00021017J/m3K. Here, P(t)is a Gaussian\nshaped pulse with a duration of 55 fs. The exact values of the\nparameters entering the TTM in GdFeCo are still unknown.\nThe values we use here are close to the commonly used, e.g.\nRefs. [4, 8, 34].\nD. Ultrafast magnetization dynamics using ASD\nElement-speciﬁc magnetization dynamics induced by a\nfemtosecond laser pulse are calculated by combining the\natomistic s-LLG equation for the spin dynamics (Eq. (2)) and\nthe 2TM for the electron temperature (Eq. (7)). The electron\nsystem acts as heat-bath for the atomic spins. We consider a\nlattice with N=50\u000250\u000250 spins, and damping parameters,\nlGd=0:01=lFe. Figure (2) shows, for t<0, the dynam-\nics of the element-speciﬁc magnetization from an initial sat-\nurated state ( T=0 K), towards thermal equilibrium with the\nheat-bath which is set to T=300 K. The relaxation dynamics\nof Fe sublattice is faster than those of the Gd sublattice. This\ncomes out naturally as the element-speciﬁc dissipation of an-\ngular momentum scales as ˙ mz\u0018gl=ms, in Gd ( mGd=7:6mB)\nis slower than in Fe sublattice ( mGd=1:94mB). Once the mag-\nnetic system is in thermal equilibrium with the heat-bath, we\napply the laser pulse, t>0, which introduces energy into the\nelectron system and induces ultrafast magnetization dynam-\nics. To illustrate the switching and no switching dynamics we\nconsider two limiting cases, dynamics induced by low laser\npower, P0, and large laser power, 2 P0. The electron temper-\nature increases up and above the Curie temperature in time\nscales of a few hundreds of femtoseconds Fig. (2) (a). This re-\nﬂects in the magnetic system as a fast demagnetization of both\nFe and Gd sublattices. For relatively low laser power, P0, the\nmagnetization of both sublattices reduces while the electron\ntemperature remains relatively high. Once the electron tem-\nperature reduces and equalizes to the lattice temperature, the\nmagnetization recovers to the thermal state given by the heat-\nbath temperature, which is higher than initially ( T=300 K).\nThis is why the ﬁnal magnetization value is smaller than the\ninitial one. For higher laser powers, 2 P0, the magnetization\nof both sublattices reduces quickly. The Fe sublattice faster\nthan the Gd one. Once the magnetization of the Fe sublattice\nhits zero, instead of remaining demagnetized, the magnetiza-\ntion starts to develop toward the opposite direction, while the\nmagnetization of the Gd sublattice is still in the process of de-\nmagnetization. During a couple of picoseconds, both sublat-\ntice magnetization are aligned along the same direction, simi-\nlar to a ferromagnet. Consequently, this non-equilibrium state\nhas been named the transient ferromagnetic-like state [3]. One\ncan observe in Fig. (2) (b) that the demagnetization rates of\nboth sublattices slow down when the Fe magnetization crosses\nzero. This change reveals the set in of a process driving the\nmagnetization dynamics different to the one driving the initial\ndemagnetization. It has been argued that at this point direct\nexchange of angular momentum between sublattices domi-\n30060090012001500\n−1−0.75−0.5−0.2500.250.50.751\n-10 -5 0 5 10 15relaxation tothermal stateswitchingno switchingT[K]electronlatticemztime [ps]FeCoGd(a)(b)\nFIG. 2. (a) Electron and lattice temperature dynamics for two laser\npulse power values, P0and 2 P0. Both electron and lattice tempera-\nture are kept constant, T=300K, for t<0. At t=0 a laser pulse is\napplied and the dynamics of the electron and lattice temperature heat\nup. The dynamics of those temperatures are theoretically described\nby the two-temperature model. (b) Element-speciﬁc magnetization\ndynamics induced by the heat proﬁle at (a). The dynamics are calcu-\nlated using atomistic spin dynamics methods. For lower laser powers\nP0, the magnetization of both sublattices demagnetize rapidly and re-\nmagnetize towards the new equilibrium. For laser power 2 P0, the\nmagnetization of both sublattices demagnetizes and switches. After\nswitching they relax towards the thermal equilibrium state. GdFeCo\nalloys with xGd=25% are calculated.\nnates over processes of relativistic origin, which in turn dissi-\npate angular momentum into the heat-bath. Interestingly, soon\nafter switching, both sublattice magnetization rapidly relax to\nequilibrium indicating that relaxation into the heat-bath dom-\ninates the dynamics.\nIII. PHENOMENOLOGICAL MODELS\nDifferently to ASD simulations, phenomenological mod-\nels describe the element-speciﬁc magnetization dynamics by\nsolving two coupled equations of motion, one for each sub-\nlattice. In this work we aim at ﬁnding a phenomenological\nmodel that describes the same element-speciﬁc magnetization\ndynamics as those coming out from the ASD simulations (Fig.\n2). The starting point is the comparison of the ASD simula-\ntions to well-known phenomenological models. We show that\nthose models are unable to describe in a satisfactory way the\ndifferent element-speciﬁc magnetization dynamics studied in\nthe previous section and summarized in Fig. 2.5\nA. Baryakhtar model\nThe simplest model to describe element-speciﬁc magneti-\nzation dynamics and switching in ferrimagnets was proposed\nby Mentink and co-workers [5]. Longitudinal spin dynamics\nwas derived from Onsager’s relations\nma\ngadma\ndt=aB\namaHa+aB\ne(maHa\u0000mbHb) (9)\nmb\ngbdmb\ndt=aB\nbmbHb+aB\ne(mbHb\u0000maHa) (10)\nhere, aB\na;bstands for the relaxation parameter of relativistic\norigin, which dissipates angular momentum out of the spin\nsystem, and aB\nestands for the exchange relaxation parame-\nter and describes the rate of dissipation of angular momen-\ntum between sublattices. By construction exchange relaxation\nconserves the total angular momentum. We emphasize here\nthe difference in the notation between the atomic relaxation\nparameter, l, describing the dissipation of the atomic spins\nin ASD simulations and the macrospin relaxation parameter,\na, describing the dissipation of the whole magnetic sample.\nWithin this model, the values for aB\na;bandaB\neare unknown\nbut used as ﬁtting parameters when compared to experiments.\nThe internal effective ﬁeld Ha(b), acting on sublattice a(b)are\nderived from a non-equilibrium mean-ﬁeld approximation,\nmaHa=\u0000b\u00001L\u00001(ma)+maHMFA\na (11)\nwhere, L\u00001(x)is the inverse Langevin function, b=1=kBT,\nwhere Trepresents the temperature of the heat-bath to which\nthe spin system is coupled to. At equilibrium, the effective\nﬁeld is Ha=0, as ma=L(b maHMFA\na). The same arguments\napply for sublattice b. It turns out that by solving Eqs. (9)\nand (10) together with the 2TM, described in Eqs. (7) and\n(8), one obtains similar ultrafast magnetization dynamics as\nthose using ASD simulations (Fig. (2)). Element-speciﬁc\ndemagnetization [18] and switching dynamics [19] based on\nthis approach have been discussed thoughtfully before. On\nthose works, the values for the relaxation parameters, rela-\ntivistic and exchange, are taken constant and of the same or-\nder,aB\nFe\u0019aB\nGd\u0019aB\ne. We note that here aB\nadeﬁnes the rate\nof change of angular momentum ( mm=g). It differs from the\ndeﬁnition of intrinsic damping parameters in ASD, which are\nrelated to the rate of change of the magnetization ( m). Sim-\nilarly to ASD methods though, within the Baryakhtar model\nthe observed fast dynamics of the Fe sublattice is related to a\nsmaller value of atomic magnetic moment.\nThe switching process within the Baryakhtar-like model\nis explained in the following manner. Since the Fe sublat-\ntice reacts faster than Gd to heating it is expected to remain\ncloser to thermal equilibrium with the heat-bath. This trans-\nlates into a smaller non-equilibrium effective ﬁeld acting on\nFe than in Gd, HFe\u001cHGd, during the action of the laser pulse.\nFor strong enough pulses, the Fe magnetization rapidly re-\nduces, mFe\u00190, still HFeis small in comparison to HGd, in a\nway that the dynamics of Fe can be fairly approximated by\n˙mFe\u0019aB\neHGd. This drives the magnetization of Fe towards\n(a)\n(b)\n-1-0.500.51\n-10 -5 0 5 10laser powerP0laser power 2P0-1-0.500.51laser powerP0laser power 2P0mztime [ps]αBe/αBa=0αBe/αBa=0.3αBe/αBa=3mzFeGdFIG. 3. Element-speciﬁc magnetization dynamics of GdFeCo cal-\nculated using atomistic spin dynamics (symbols) and macroscopic\nBaryakhtar-like equation (solid lines) for two laser pulse power val-\nues, (a) P0and (b) 2 P0. Both electron and lattice temperature are\nkept constant, T=300 K, for t<0. At t=0 a laser pulse is ap-\nplied. In the Baryakhtar-like model the relativistic relaxation pa-\nrameters aBahave a value different to the Gilbert damping in ASD\nsimulations, (g=mFe)aB\nFe=0:005 and (g=mGd)aB\nGd=0:01. The ex-\nchange relaxation parameter is varied, aBe=aB\nFe=0;0:3 and 3. The\nrelaxation to thermal state ( t<0) is only well described for the Fe\nsublattice. (a) For P0, the laser induced dynamics is well described\nbyaBe=aB\nFe=0:1. (b) For 2 P0the demagnetization phase of both\nsublattices is relatively well described in comparison to ASD sim-\nulations. Switching is also possible, here one instance, for a value\naBe=aB\nFe=3.\nthe opposite direction. The ﬁeld, HGdis deﬁned by the en-\nergy of the system, HMFA\nGd(Eq. (6)) and aB\nefrom the cou-\npling between the Gd and the Fe sublattices. After switching,\nHFe\u0019HGdand relativistic relaxation processes dominate the\ndynamics and drive magnetization to complete the switching.\nThe question here is to what extent the non-equilibrium ﬁelds\nas given by Eq. (11) are accurate, and how are the relaxation\nparameters related to atomic damping parameters in ASD.\nSo far the connection between the relaxation parameters in\nthe ASD and Baryakhtar-like model is unknown. In ASD sim-\nulations shown in Fig. 2 we have used lFe=lGd=0:01 as\natomistic relaxation parameter. One would expect that the re-\nlaxation parameters in the atomistic and macroscopic models\nare related as la\u0019aB\na(ga=ma). In an attempt to ﬁnd this cor-\nrespondence, we directly compare results from ASD simula-\ntions and Baryakhtar-like models for different values of aB\na\nandaB\nein Eqs. (9) and (10). We numerically solve Eqs.\n(9),(10), and (11) coupled to the 2TM with exactly the same\nparameters as for the ASD simulations. After exploring the\nresults of the Baryakhtar model for a range of values for aB\na\nandae, we ﬁnd that for some values the agreement is good,\nas one observes in Fig. 3, however, it is not possible to ﬁnd a\ngood match for all scenarios.6\nIn order to illustrate this, we ﬁrst focus on the dynamics in-\nduced by the laser pulse with power P0(Fig. (3)(a)). We ﬁnd\na good match for the laser induced magnetization dynamics\n(t>0 for (g=mFe)aFe=0:005 and (g=mGd)aGd=0:01, and\nfor values of exchange relaxation of up to aB\ne=aB\nFe=0:3. For\nvalues aB\ne=aB\nFe<0:3, thermal relaxation ( t<0) of the Fe is\nalso well described, however the relaxation of the Gd sublat-\ntice is signiﬁcantly faster. For larger values of the exchange\nrelaxation aB\ne=aB\nFe=3, the dynamics of both sublatttices are\nsubstantially speed up and strongly disagree with ASD simu-\nlations.\nFor larger laser pulse power 2 P0the magnetization switches\nusing ASD simulations. We keep the same values for the re-\nlaxation parameters in Baryakhtar-like model as for P0, and\ncompare to the ASD simulations. For small values of aB\ne\n(Fig. (3)(b)), differently to the P0case (Fig. (3)(a)), the dy-\nnamics described by the Baryakhtar-like model is not only\nslower than those of ASD simulations but it hardly reproduces\nmagnetization switching. In order to reproduce switching, we\nneed to use larger values of the exchange relaxation parameter,\naB\ne=aB\nFe=3. These ﬁndings are in agreement with previous\nworks using Baryakhtar-like model where switching was re-\nproduced for comparable values of aB\ne. However, as we have\ndiscussed before, for those values of aB\ne, thermal relaxation\ndynamics ( t<0) is much faster than in ASD simulations.\nThis brings us to the question of how much understanding\nabout switching can we gain by using this bare Baryakhtar-\nlike model, are we missing something?\nB. The Landau-Lifshitz-Bloch model\nSince the Baryakhtar-like model is based on symmetry ar-\nguments, the macroscopic magnetization dynamics coming\nout from ASD simulations should also be described by that\nmodel with adequate expression for the relaxation parameters\nand non-equilibrium effective ﬁelds. The magnetization dy-\nnamics coming out from ASD simulations is well described\nby the LLB equation of motion.\ndma\ndt=Gk;a(ma\u0000m0;a); (12)\nwhere\nGk;a=2lag\nmakBT1\nxaL(xa)\nL0(xa); (13)\nwith xa=b maHMFA\na, where HMFA\na is given in Eq. (6), and\nm0;a=L(xa). The same equation applies to the second sublat-\nticeb. Here, the relaxation rate Gk;adepends non-linearly on\nthe non-equilibrium sublattice magnetization, ma(b), through\nthe parameter xa. We note that Eq. (12) can be expanded\naround equilibrium for small perturbations of the magnetiza-\ntion. By doing so, the relaxation rates and effective ﬁelds are\nexpressed in terms of equilibrium properties such as equilib-\nrium magnetization and zero-ﬁeld susceptibilities [20]. In the\npresent work, however, we use the version in Eq. (12). Direct\ncomparison between ASD simulations and the LLB model\n-1-0.500.51\n-10 -5 0 5 10laser powerP0laser power 2P0-1-0.500.51laser powerP0laser power 2P0mztime [ps]αe/αa=0αe/αa=0.1αe/αa=1mzFeGd(a)\n(b)FIG. 4. Element-speciﬁc magnetization dynamics of GdFeCo calcu-\nlated using atomistic spin dynamics (symbols) and macroscopic LLB\nequation (solid lines) for two laser pulse power values, (a) P0and (b)\n2P0. For t<0, electron and lattice temperature are T=300K, and at\nt=0 a laser pulse is applied. The exchange relaxation parameter is\nvaried, ae=aa=0;0:1 and 1, where aa=0:01, and a=FeCo or Gd.\nThe initial relaxation dynamics is well described by ae=aa=0. (a)\nFor laser power P0, the element-speciﬁc dynamics is well-described\nforae=aa=0:1. (a) For ae=aa=1, exchange relaxation dominates\nand the element-speciﬁc dynamics are similar. (b) For laser power\n2P0, the switching dynamics is not described by the LLB model.\nof element-speciﬁc magnetization dynamics is possible and\nwith relatively good agreement. Importantly, since the LLB\nmodel is derived directly from the ASD microscopic model,\nthe damping parameters, la(b)in Eqs. (13) and (2) stand for\nthe same physics, the rate of angular momentum dissipation of\nthe atomic spins. Differently to the Baraykhtar model where\naB\na(b)is taken as a ﬁtting parameter, within the LLB model the\nvalue of la(b)in Eq. (13) is the same as in the ASD simu-\nlations. A key difference between the Baryakhtar-like model\nand the LLB model is that in the latter an exchange relaxation\nterm is missing. In order to ﬁnd a meeting point between these\nphenomenological models, we rewrite Eq. (12) in terms of a\ndamping term multiplied by an effective ﬁeld,\ndma\ndt=2laL(xa)\nxag\nmama\u0000m0;a\nbL0(xa)=gaaHa; (14)\nwhere\naa=2laL(xa)\nxa: (15)\nDifferently to Baryakhtar-like model, in the LLB model, the\nrelaxation parameter strongly depends on temperature and\nnon-equilibrium sublattice magnetization through the thermal\nﬁeld, xa=b maHMFA\na. At the same time, the non-equilibrium\nﬁelds maHawithin the LLB and Baryakhtar-like models differ.7\nThe effective ﬁeld in the LLB model is deﬁned as\nmaHa=(ma\u0000m0;a)\nbL0(xa): (16)\nEquation (16) provides a microscopic description of the effec-\ntive ﬁeld driving the magnetization dynamics in ferrimagnets,\nbased on the Heisenberg spin model (Eq. (1)). Under the as-\nsumption of small perturbations around the equilibrium both,\nLLB and Baryakhtar-like effective ﬁelds, simplify to Landau-\nlike expressions [19]. Equation (14) describes with a very\ngood degree of accuracy the relaxation of the angular mo-\nmentum via dissipation to the heat-bath, which corresponds\nto the relativistic term in Eqs. (9) and (10). Previously, it has\nbeen found that ASD simulations compare well to Eq. (14) for\ncoupling parameters of around la\u00190:1\u00001 [20, 42]. These\nvalues can be considered to correspond to the intermediate-to-\nhigh coupling regime. Direct comparison between ASD sim-\nulations and experiments of single pulse switching in GdFeCo\nhas suggested values of lFe\u00190:06 and lGd\u00190:01 [16]. In\nthe context of the present work we ﬁnd that Eq. (14) describes\nrelatively well the thermal relaxation dynamics in direct com-\nparison to ASD simulations (Fig. (4)).\nIn order to account for the exchange relaxation in the LLB\nmodel, we follow the Baryakhtar-like model ((9) and (10)),\nand add an exchange relaxation term to Eq. (14),\ndma\ndt=gaaHa+gae\nma(maHa\u0000mbHb) (17)\nwhere aeis a phenomenological exchange relaxation param-\neter to be determined by comparison to ASD dynamics. The\ninclusion of the exchange relaxation (second term in r.h.s) in\nthe LLB improves the agreement to ASD simulations. With\nthis addition, the LLB model describes well thermal relax-\nation for small values of the ratio ae=aaas demonstrated in\nFig. 4. For large values of aethe LLB model is unable to\ndescribe thermal relaxation dynamics ( t<0 in Fig. 4(a) and\n(b)). For laser power P0(Fig. 4(a) ( t>0)) the magnetization\ndynamics is slightly slower using the LLB model than those\ngained by ASD simulations for ae=aa=0. For ae=aa=0:1,\nthe agreement is even better than without exchange relax-\nation. The agreement vanishes when the exchange relax-\nation is increased to ae=aa=1. Critically, when the laser\npower is increased from P0to 2P0, for which ASD simula-\ntions show ultrafast switching, the LLB model only shows\ndemagnetization-remagnetization of both sublattices. We ﬁnd\nsome agreement on the demagnetization time scales when a\nquite large exchange relaxation is used, ae=aa=1. These\ndynamics are similar to those observed using the Baryakhtar-\nlike model for intermediate values of the exchange relaxation\nparameter (Fig. (3)). It has been demonstrated previously\nthat by including the transverse components of the equation\nof motion, switching is possible via a precessional path when\na canting between the magnetization of each sublattice exists\n[21]. Here, we restrict to purely longitudinal switching within\nthe LLB model.\n02468 1 0-1-0.500.51−10−50 5 1 0laser powerP0laser power 2P0time [ps]mztime [ps]FeGd(a)(b)FIG. 5. Element-speciﬁc magnetization dynamics of GdFeCo calcu-\nlated using atomistic spin dynamics (symbols) and the uniﬁed phe-\nnomenological model derived here, following Eq. (17) (solid lines)\nfor two laser pulse power values, (a) P0and (b) 2 P0. Both electron\nand lattice temperature are kept constant, T=300 K, for t<0. At\nt=0 a laser pulse is applied, the same as in Figure (2). GdFeCo\nalloys with xGd=25% are calculated.\nC. Uniﬁed phenomenological model\nSo far we have constructed a phenomenological model\nbased on the LLB and Baryakhtar-like models, the dynam-\nics is given by Eq. (17), the effective ﬁeld by Eq. (16)\nand the relativistic relaxation parameter Eq. (15). We still\nneed an expression for the exchange relaxation parameter. We\nconstruct this expression starting with single species ferro-\nmagnets, where sublattices aandbrepresent the same spin\nlattice, hence exchange of angular momentum is non-local.\nTherefore, maHa\u0000mbHb=maHexa2\n0Dma, with a0represent-\ning the lattice constant. Hence, the rate of non-local angu-\nlar momentum transfer reads Gnon\u0000loc:\nex =aex(maHa\u0000mbHb) =\naa(A=Ma(T))Dma, where Ais the so-called micromagnetic\nexchange stiffness [58]. Ma(T) = (ma=ua)mais the magne-\ntization density at temperature T, where uais the unit cell\nvolume. Therefore, we ﬁnd that aex=aa=(zma). By con-\nsidering that the exchange relaxation rate should conserve the\nsymmetry under the exchange of lattice index, aex(M1;M2) =\naex(M2;M1), we ﬁnd that\naex=1\n2\u0012aa\nzabma+ab\nzbamb\u0013\n: (18)\nThis expression is the extension of the non-local exchange re-\nlaxation in ferromagnets to local exchange relaxation in ferri-\nmagnets. This explicit expression for the exchange relaxation\nparameter in Eq. (18) completes our uniﬁed model, which\nbridges the atomistic spin dynamics model and the Baryakhtar\nand LLB macroscopic models.\nWe ﬁnd that the agreement between our uniﬁed phe-\nnomenological model and ASD simulations is excellent, see\nFig. (5)(a) and (b). Figure 5(a) shows that for t<0, the sub-\nlattice magnetization relaxation towards thermal equilibrium\nvalue is described with a high level of accuracy by our model.\nFort>0 and a relatively low laser power P0, the agreement\nis also excellent for the demagnetization and remagnetization\ndynamics. Figure 5(b) shows the comparison between the uni-\nﬁed model and ASD simulations of the switching dynamics.\nWe conclude that Eq. (17) for the sublattice magnetization8\ndynamics together with the Eq. (16) for the effective ﬁeld\nand Eqs. (15) and (18) for the relaxation parameters, unify\nthe Barayakhtar and the LLB phenomenological models for\nsingle-pulse all-optical switching in ferrimagnets.\nIV . DISCUSSION AND CONCLUSION\nThe macroscopic model presented in this work solves some\nopen questions in the ﬁeld of ultrafast magnetization dynam-\nics in ferrimagnets. For example, it answers the question of\nthe range of applicability and the validity of the parameters of\nthe Barayakhtar and LLB phenomenological models. In the\none hand, within our model, the relativistic relaxation param-\neters ( aa) are element-speciﬁc and strongly depend on both\nthe temperature and the non-equilibrium sublattice magneti-\nzation. The temperature and magnetization dependence of\nthe relativistic relaxation parameters are well described by the\nLLB model. In the other hand, the exchange relaxation pa-\nrameter ( aex) is cast in terms of the element speciﬁc relativis-\ntic relaxation parameters and sublattice magnetization. We\nhave demonstrated that in order to reproduce the ASD sim-\nulations results, the relaxation parameters in the Barayakhtar\nmodel have to be both temperature and magnetization depen-\ndent. The explicit expression of the exchange relaxation pa-\nrameter is the main result of the present work since it allows\nus to unify the Barayakhtar and LLB models. While for the\nBarayakhtar model aeis unconnected to aa, within our pro-\nposed model they are proportional to each other, ae\u0018aa=ma.\nThis relation is the key to bridge both ASD simulations and\nBarayakhtar and LLB models together. Additionally, we have\nalso demonstrated the validity of the non-equilibrium effective\nﬁelds given in Eq. (16) as derived in the LLB model instead\nof the Barayakhtar model.\nSingle-pulse switching in ferrimagnets has been described\nbefore by the Baryakhtar model. A necessary condition for\nswitching is that the system transits from the relativistic relax-\nation regime to the so-called exchange-dominated relaxation\nregime. Although details of switching in such a regime have\nbeen already discussed in detail [5, 19], so far it has remained\nunknown how this transition could be described theoretically.\nOur model resolves this question. When the system is at equi-\nlibrium or weakly excited, the exchange-relaxation parameter\nfulﬁlls, ae\u001caa. For strong excitation, such that the mag-\nnetic order of one sublattice reduces signiﬁcantly, close to\nzero ma!0, the exchange relaxation will dominate the dy-namics since ae\u0018aa=ma\u001daa. From our model, one can\nderive universal criteria for switching in ferrimagnets, includ-\ning GdFeCo and Mn 2RuxGa [59].\nThe provided understanding is paramount for further re-\nsearch on material engineering, for example, to ﬁnd alter-\nnative material classes showing all-optical switching. No-\ntably, our model predicts that the exchange relaxation term\nis enhanced as the number of neighbours reduces. This de-\npendence suggests that magnetic systems of lower dimen-\nsion, e.g. 2D magnets [60], could show a faster, more ef-\nﬁcient switching than bulk materials. Further, the extension\nof our model to the micromagnetic level will allow to opti-\nmize switching conditions. The use of micromagnetic com-\nputational solvers permits for a realistic description of ultra-\nfast AOS processes in ferrimagnetic alloys, such as helicity-\nindependent and helicity-dependent AOS, where multidomain\nstates and thermal gradients play an important role in the pro-\ncess [46].\nTo summarize, in the present work we have presented a\nuniﬁed model for single-pulse all-optical switching in ferri-\nmagnets. Our model merges and improves previous semi-\nphenomenological models, the Landau-Lifshitz-Bloch model\nand Barayakhtar-like models. To verify the accuracy of the\nproposed model, we directly compare the laser induced mag-\nnetization dynamics to atomistic spin dynamics computer\nsimulations. Differently to previous models, our model has\nthe advantage that it can be directly compared to ASD simu-\nlations. Further, we have established the connection between\nASD and macroscopic equations of motion. Importantly, we\nprovide here the stepping stone for the construction of a mi-\ncromagnetic model valid for ferrimagnets including exchange\nrelaxation between sublattices. This is paramount for a ro-\nbust construction of a multiscale scheme of the switching pro-\ncess in which not only local magnetization dynamics is de-\nscribed but also magnetic domain nucleation and motion un-\nder strong non-equilibrium. 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San-\ntos, The magnetic genome of two-dimensional van der waals\nmaterials, ACS Nano 10.1021/acsnano.1c09150 (2022), pMID:\n35442017, https://doi.org/10.1021/acsnano.1c09150."    },    {        "title": "0806.3843v3.Signal_propagation_in_time_dependent_spin_transport.pdf",        "content": "arXiv:0806.3843v3  [cond-mat.other]  12 Aug 2008Signal propagation in time-dependent spin transport\nYao-Hui Zhu, Burkard Hillebrands, and Hans Christian Schneider∗\nPhysics Department and Research Center OPTIMAS,\nKaiserslautern University of Technology, 67653 Kaisersla utern, Germany\n(Dated: November 5, 2018)\nThis paper analyzes theoretically the signal propagation i n spin transport by modulating the\ncurrent passing through magnetic multilayers. Using a macr oscopic description of spin transport\nbased on the dynamical Boltzmann equation, we show that time -dependent spin transport possesses\na wave-like character that leads to modiﬁcations of pure spi n-diﬀusion dynamics. In particular, the\nwave-like characteristics allow one to extract a ﬁnite spin signal-propagation velocity.\nPACS numbers: 72.25.-b, 75.40.Gb, 75.47.-m, 85.75.-d\nI. INTRODUCTION\nTime-dependent spin transport in magnetic multilay-\ners with current perpendicular to the plane (CPP) is\nstudied because of its signiﬁcance in physics and promis-\ning applications in spintronics devices.1,2Most theoreti-\ncalinvestigationsarebasedonadiﬀusionequationforthe\nspin accumulation or magnetization.3,4,5,6These theories\nshow that if one drives a spin-polarized current through\nan interface from a magnetic to a non-magnetic metallic\nlayer, the spin propagates by “diﬀusing” into the non-\nmagnetic layers. If one considers time-dependent spin\ntransport, such as spin transfer torque switching,3,5al-\nternating current (AC)4, or magnetization switching,6\nwhere a time-dependent signal is encoded in the spin ori-\nentation, one faces a diﬃculty of the diﬀusion equation\nin that no propagation velocity for the spin signal in the\nnonmagnetic layer can be deﬁned. Or, stated diﬀerently,\nthe diﬀusion equation yields an inﬁnite propagation ve-\nlocity for the spin signal in the metal, because the signal\nwill appear everywhere as soon as the source is switched\non.3In this paper, we show how a physical propagation\nvelocity for spin signals in the CPP conﬁguration can be\ndetermined by deriving and analyzing macroscopic dy-\nnamical equations for spin transport.\nIt is an interesting connection that a problem anal-\nogous to that of an inﬁnite signal propagation velocity\nin the spin diﬀusion equation exists for the heat diﬀu-\nsion equation, which yields an inﬁnite heat-conduction\nvelocity. This diﬃculty was resolved by recognizing that\nthe theoretical description of heat transport needs to be\ngeneralized by substituting the Maxwell-Cattaneo equa-\ntion7,8for Fourier’s law. In this way, one obtains the\nphysical picture that heat conduction is characterized\nby a wave-diﬀusion duality. Formally, the heat diﬀusion\nequation needs to be replaced by an equation that is es-\nsentially a telegraph equation.9,10As we show in this pa-\nper, a similar modiﬁcation of the spin-diﬀusion equation\nis necessary in the case of spin transport.\nWe base our derivation of the macroscopic equations\nfor spin transport through multilayers on the theory de-\nveloped for steady-state spin transport across magnetic\nmultilayers by Valet and Fert.11Instead of using the\ntime-independent Boltzmann equation as in Ref. 11, wetreat time-dependent spin transport starting from the\ndynamical Boltzmann equation, which allows us to de-\nrive macroscopic equations and to generalize the spin-\ndiﬀusion equation.\nThis paper is organized as follows. The macroscopic\ndynamical equations are derived in the Sec. II of our pa-\nper. Since the central equations (18) and (19), can also\nbe cast in a form that resembles telegraph equations, we\ndiscuss qualitative aspects of dynamical spin transport\nin Sec. III using these telegraph equations. In Sec. IV,\nweanalyzetwoconcreteexamplesoftime-dependent spin\ntransportnumerically, andthe mainconclusionsaresum-\nmarized in Sec. V.\nII. TIME-DEPENDENT EQUATION SYSTEM\nIn this section, the model of Valet and Fert11for spin-\ndependent transport of conduction electrons through\nmetallic multilayers will be extended to take into account\nthe time-dependence of spin transport. The electron dis-\ntributionfunction fs(z,v,t) satisﬁesthe linearizedBoltz-\nmann equation\n∂fs(z,v,t)\n∂t+vz∂fs(z,v,t)\n∂z−eE(z,t)vz∂f0(v)\n∂ε\n=/integraldisplay\nd3v′δ[ε(v′)−ε(v)]Ps[z,ε(v)]/bracketleftbig\nfs(z,v′,t)−fs(z,v,t)/bracketrightbig\n+/integraldisplay\nd3v′δ[ε(v′)−ε(v)]Psf[z,ε(v)]/bracketleftbig\nf−s(z,v′,t)−fs(z,v,t)/bracketrightbig\n,\n(1)\nwhere−eandε(v) =mv2/2 denote, respectively, the\ncharge and kinetic energy of the electrons, and E(z,t) =\n−∂V(z,t)/∂zis the local electric ﬁeld.12Ps(z,ε) and\nPsf(z,ε) are the spin conserving and spin-ﬂip transition\nprobabilities, respectively. Following Ref. 11, we assume\nfs(z,v,t) to be the sum of the Fermi-Dirac distribution\nf0(v) and small perturbations:\nfs(z,v,t)=f0(v)+∂f0\n∂ε/braceleftbig/bracketleftbig\nµ0−µs(z,t)/bracketrightbig\n+gs(z,v,t)/bracerightbig\n,(2)\nwhereµ0=mv2\nF/2 andµs(z,t) are the equilibrium and\nnonequilibrium chemical potentials, respectively. Due to2\nthe cylindricalsymmetryofthe systemaroundthe zaxis,\ngs(z,v,t) can be expanded in Legendre polynomials of\ncosθ, whereθis the angle between vand thezaxis, as\ngs(z,v,t) =∞/summationdisplay\nn=1g(n)\ns(z,t)Pn(cosθ). (3)\nHere, the zero-order (isotropic) term is absent be-\ncause (∂f0/∂ε)gs(z,v,t) was deﬁned by Eq. (2) as the\nanisotropic partoftheelectrondistributionperturbation.\nUsing Eq. (3), we obtain\n∂gs(z,v,t)\n∂t+vz∂gs(z,v,t)\n∂z+/parenleftbigg1\nτs+1\nτsf/parenrightbigg\ngs(z,v,t)\n=∂µs(z,t)\n∂t+vz∂¯µs(z,t)\n∂z+¯µs(z,t)−¯µ−s(z,t)\nτsf,(4)\nwhere ¯µs(z,t) =µs(z,t)−eV(z,t) is the electrochemical\npotential for electrons with spin s. The derivation of this\nequation is detailed in Appendix A2. Note that vin\nEq. (4) has been restricted to the Fermi velocity vF, i.e.,\n|v|=vFandvz=vFcosθ.\nWith the relaxationtimes τsandτsf(see Eqs.(A9) and\n(A10)inAppendixA2), thelocalelectronmeanfreepath\nλs, diﬀusion constant Ds, and spin diﬀusion length lscan\nbe deﬁned, respectively, as λs=vFτ′\ns,Ds=vFλs/3, and\nls= (Dsτsf)1/2, where the momentum relaxation time τ′\ns\nis deﬁned by\n1/τ′\ns= 1/τs+1/τsf. (5)\nThe appropriate “average” spin-diﬀusion length lsfcan\nbe deﬁned as (1 /lsf)2= (1/l+)2+ (1/l−)2. Throughout\nthe paper, subscripts + and −stand for the absolute\nspin directions “up” and “down”, respectively, whereas\nsubscripts ↑and↓stand for the majority and minority\nspin directions, respectively.\nUsingthemethodofAppendixBinRef. 11, weexpress\nthe time-dependent current density for spin sas\nJs(z,t) =−e\nV/summationdisplay\nvfs(z,v,t)vz=κg(1)\ns(z,t),(6)\nwhereκ=σs/(eλs). Note that κis independent of s\nand of the material in the Valet-Fert model. The con-\nductivity σscan be written as σs=e2nsτ′\ns/m, where\nns= 4π(mvF/h)3/3 is the number of electrons with spin\ns. It is easy to see that σssatisﬁes Einstein’s relation\nσs=e2NsDs, where\nNs=1\n4π2/parenleftbig\n2m//planckover2pi12/parenrightbig3/2/radicalbig\nµ0, (7)\nis the density of states for spin sat the Fermi level µ0,\nandN+=N−.\nSubstituting Eqs. (3) and (6) into Eq. (4), we obtain\ne\nσs∂Js(z,t)\n∂z−1\nDs∂µs(z,t)\n∂t=¯µs(z,t)−¯µ−s(z,t)\nl2s,(8)\nJs(z,t) =σs\ne∂¯µs(z,t)\n∂z−τ′\ns∂Js(z,t)\n∂t.(9)In steady state, Js(z,t) and ¯µs(z,t) become time-\nindependent and then Eqs. (8) and (9) reduce to the\nEqs. (10) and (11) in Ref. 11, respectively.\nEquations (8) and (9) will be transformed to more\ndirectly usable forms next. Without loss of generality,\nthe magnetization of the ferromagnet is set to be “up”.\nThen, the majority (minority) spins, which are antipar-\nallel (parallel) to the local magnetization (electron mag-\nnetic moment is µ=−(e/m)s) and denoted by subscript\n↑(↓), point to the absolute spin direction “down” (“up”)\ndenoted by subscript −(+). In terms of Jm(z,t) =\nJ+(z,t)−J−(z,t) andµm(z,t) =µ+(z,t)−µ−(z,t),\nEqs. (8) and (9) can be written as\n∂Jm(z,t)\n∂z−eNs∂µm(z,t)\n∂t=eNsµm(z,t)\nT1,(10)\nJm(z,t) =eNs¯D∂µm(z,t)\n∂z−τ∂Jm(z,t)\n∂t−˜βJ(z,t),(11)\nwhere\nT1=τsf/2 (12)\ncan be regarded as the spin relaxation time.13The “av-\nerage” diﬀusion constant ¯Dis deﬁned as ¯D=c2τ, with\nthe wavefront velocity cdeﬁned by\nc2=v2\nF/3. (13)\nThe “average” momentum relaxation time τis\n1/τ= (1/τ′\n++1/τ′\n−)/2, (14)\nand we have the identity lsf=c√τT1. In Eq. (11),\n˜β= (τ′\n−−τ′\n+)/(τ′\n−+τ′\n+) equals βand 0 for the ferro-\nmagnetic and nonmagnetic layers, respectively. The bulk\nspin asymmetry coeﬃcient βin the ferromagnetic layer\nis deﬁned by ρ↑(↓)= 1/σ↑(↓)= 2ρ∗\nF[1−(+)β], where ρ∗\nF\nis the total resistivity of the ferromagnetic layer. In the\nnonmagnetic layer, we have ρ↑(↓)= 2ρ∗\nN, whereρ∗\nNis the\ntotal resistivity of the nonmagnetic layer.\nIn Eq. (11), J(z,t) stands for the total current density\nJ(z,t) =J+(z,t) +J−(z,t). By introducing µ(z,t) =\n[µ+(z,t)+µ−(z,t)]/2, we can also derive equations for\nthe charge dynamics\n∂J(z,t)\n∂z−2eNs∂µ(z,t)\n∂t= 0, (15)\nJ(z,t) = 2eNs¯D∂\n∂z[µ(z,t)−eV(z,t)]\n−τ∂J(z,t)\n∂t−˜βJm(z,t).(16)\nTo describe spin accumulation by spin density instead\nof the chemical potential, it is necessary to transform\nEqs.(10) and(11) usingthe followingidentity (Eq.(A15)\nin Appendix A3)\nnm(z,t) =−eNsµm(z,t), (17)3\nwherenm(z,t) =n+(z,t)−n−(z,t) is the spin density\nandns(z,t) the nonequilibrium charge density for spin s.\nUsing Eq. (17), we can rewrite Eqs. (10) and (11) as\n∂Jm(z,t)\n∂z+∂nm(z,t)\n∂t=−nm(z,t)\nT1,(18)\nJm(z,t) =−¯D∂nm(z,t)\n∂z−τ∂Jm(z,t)\n∂t−˜βJ(z,t).(19)\nTo proceed further, we need the following identity\n(Eq. (A16) in Appendix A3)\nn(z,t)−2n0\ns=−eNs/bracketleftbig\n2µ(z,t)−2µ0/bracketrightbig\n,(20)\nwheren(z,t) =n+(z,t)+n−(z,t) is total nonequilibrium\ncharge density and n0\nsthe equilibrium charge density for\nspins. Using Eq. (20), we can rewrite Eqs. (15) and (16)\nas\n∂J(z,t)\n∂z+∂n(z,t)\n∂t= 0, (21)\nJ(z,t) =−¯D∂n(z,t)\n∂z−2e2Ns¯D∂V(z,t)\n∂z\n−τ∂J(z,t)\n∂t−˜βJm(z,t).(22)\nIngeneral,Eqs.(18)and(19)shouldbesolvedtogether\nwith Eqs. (21), (22), and Poisson’s equation. However,\nin metals and degenerate semiconductors, the accumula-\ntion of charge occurs on a much smaller length scale and\nvaries much faster than that of spin.3,4,5,6Thus as an ap-\nproximation, it is assumed that the charge accumulation\ndescribed by n(z,t) can always reach its steady state in-\nstantaneously when spin transport is considered. This\nmeans that we always set ∂n(z,t)/∂t= 0, which leads\nto∂J(z,t)/∂z= 0 according to Eq. (21). Therefore, the\ncurrent density J(z,t) in Eqs. (11) and (19) becomes in-\ndependent of zand can be written as J(t) instead.\nIII. “TELEGRAPH” EQUATION\nIn order to see the physical signiﬁcance of the dynam-\nics described by Eqs. (18) and (19) and to compare it\nwith the spin diﬀusion equation used in Refs. 3,4,5,6, we\ncombine Eqs. (18) and (19) to yield the following equa-\ntions\n∂2nm(z,t)\n∂t2+(1\nτ+1\nT1)∂nm(z,t)\n∂t+nm(z,t)\nτT1\n=c2∂2nm(z,t)\n∂z2,(23)\n∂2Jm(z,t)\n∂t2+(1\nτ+1\nT1)∂Jm(z,t)\n∂t+Jm(z,t)\nτT1\n=c2∂2Jm(z,t)\n∂z2−˜β/bracketleftbigg1\nτ∂J(t)\n∂t+J(t)\nτT1/bracketrightbigg\n.(24)0246810 \n0.001 0.01 0.1 1 10 Normalized length \nωτ s'η=0.1 η=0.01 \nη=0.02 \nl\nd/λ\ns,z λ/λ\ns,z η=0.1, 0.02,\n    0.01 \nFIG. 1: Variation of ld/λs,zandλ/λs,zwithωτ′\nsfor three\ndiﬀerent values of η. The short-dashed, solid, and long-dashed\ncurves correspond to η= 0.1, 0.02, and 0 .01, respectively.\nBecause of the formal similarity of each of Eqs. (23) and\n(24) with the telegraph equation, we will refer to them\nas telegraph equations in the following.\nEach of the telegraph equations contains a second-\nordertimederivative,whichisabsentinthespindiﬀusion\nequation. This term originates from the time derivative\nof the spin current in Eq. (19), which is also absent in\nthe corresponding equation for the spin current in spin\ndiﬀusion theory; see, for instance, Eq. (8) of Ref. 3. This\nadditional term shows that it takes a ﬁnite time for the\nspin current to adjust to the gradientofthe spin accumu-\nlation.9,14The second-order time and space derivatives\nlead to a wave character of dynamical spin transport in\naddition to its diﬀusion character described by the ﬁrst-\norder time and second-order space derivatives. Thus,\nthese equations show that time-dependent spin transport\nshould be understood using a wave-diﬀusion duality pic-\nture. The occurrenceof spin accumulation waves enables\noneto determine awell-deﬁned propagationvelocity cfor\nthe signal in time-dependent spin transport. Although\nthe spin diﬀusion equation does not yield spin accumu-\nlation waves and thus a ﬁnite signal propagation veloc-\nity, it can be regarded as an approximation of the wave-\ndiﬀusion duality of the time-dependent spin transport in\nthe long-time limit.\nIn the following, the telegraph equation of the non-\nmagnetic layer will be analyzed in detail. Here, we have\nτ=τ′\nsand˜β= 0. Thus, Eqs. (23) and (24) have the\nsame structure and we discuss only Eq. (23) without loss\nof generality. We seek a damped and dispersive wave\nsolution to Eq. (23) of the form\nnm(z,t) =n0\nmexp[i(kz−ωt)]. (25)\nAtthisstage,wecanseteither ω=ωr+iωiork=kr+iki.\nThe complex ωandkwill yield damping factors in time4\nand space, respectively. Since we are more interested\nin the damping length (or the dynamical spin diﬀusion\nlength), we will follow the method of Ref. 15 and assume\nk=kr+iki. Substituting Eq. (25) into Eq. (23), we get\nthe dispersion relation\n−ω2−iαω+ξ=−c2k2, (26)\nwhereα= 1/τ′\ns+1/T1andξ= 1/(τ′\nsT1). Separating the\nreal and imaginary parts of Eq. (26), we obtain\nk2\nr,i=1\n2c2/bracketleftBig/radicalbig\n(ω2−ξ)2+α2ω2±(ω2−ξ)/bracketrightBig\n,(27)\nThe wavelength, deﬁned as 2 π/|kr|, can be written as\nλ\nλs,z= 2π√\n2/bracketleftBig/radicalbig\n(˜ω2−η)2+(η+1)2˜ω2+(˜ω2−η)/bracketrightBig−1\n2,\n(28)\nwhereλs,z=cτ′\nsis thezcomponent of the electron mean\nfree path. Moreover, we have introduced dimensionless\nquantities\n˜ω=ωτ′\nsandη=τ′\ns/T1. (29)\nThe damping length, deﬁned as ld= 1/|ki|, can be\nwritten as\nld\nλs,z=√\n2/bracketleftBig/radicalbig\n(˜ω2−η)2+(η+1)2˜ω2−(˜ω2−η)/bracketrightBig−1\n2.\n(30)\nNote that ldcan also be regarded as the dynamical spin\ndiﬀusion length. When ˜ ω→0 or∞, the damping length\nldwill approach lsfor 2lsf/radicalbig\nτ′sT1/(τ′\ns+T1), respectively.\nFigure 1 shows the variation of ld/λs,zandλ/λs,zwith\nωτ′\nsfor three diﬀerent values of η. Note that the curves\nofλ/λs,zfor diﬀerent ηare very close to each other in\nthe frequency range shown in the ﬁgure. The damping\nlengthlddecreases with frequency, which is analogous to\nthe skin eﬀect of the electromagnetic wave propagating\nin metal. The intersection of ld/λs,zandλ/λs,zindi-\ncates the critical angular frequency ωc, which separates\nthewave-likeregionfromthediﬀusiondominatedregime,\nbecause the wave character becomes signiﬁcant only if\nthe damping length exceeds the wavelength. Stated dif-\nferently, the wave character is signiﬁcant if the typical\ntime scale τsigof the time-dependent process is smaller\nthan the critical period Tc= 2π/ωc. On the contrary,\nthe diﬀusion character is dominant if τsig> Tc, and the\nspin-diﬀusion picture becomes a good approximation of\nthe wave-diﬀusion duality in the limit τsig≫Tc.\nAn explicit expression for the critical angular fre-\nquencyωcis obtained by combining λ=ldwith Eq. (27)\nωcτ′\ns=1\n2/bracketleftBig\nγ(1+η)+/radicalbig\nγ2(1+η)2+4η/bracketrightBig\n≈γ+(γ+1\nγ)η,(31)\nwhereγ=π−1/(4π)≈3.06. Then, we have ωcτ′\ns=\n3.06+3.4ηapproximately.0.4 0.5 0.6 0.7 0.8 0.9 1\n0.1 1 10 100 vp/c\nωτ s'ωcτs'\nFIG. 2: The variation of vp/cwithωτ′\nsfor three diﬀerent\nvalues of η. The short-dashed, solid, and long-dashed curves\ncorrespond to η= 0.1, 0.02, and 0 .01, respectively. The thick\nvertical dot-dashed line indicates the critical angular fr equen-\nciesωcτ′\nsfor the three diﬀerent values of η, which are very\nclose to each other according to Eq. (31).\nThe phase velocity, deﬁned as vp=ω/|kr|, of the spin\naccumulation wave can be written as\nvp\nc=√\n2\nη+1/bracketleftBig/radicalbig\n(˜ω2−η)2+(η+1)2˜ω2−(˜ω2−η)/bracketrightBig1\n2.\n(32)\nWhen ˜ω→0 or∞, the phase velocity vpwill ap-\nproach 2c/(η1/2+η−1/2) orc, respectively. When η= 1,\nthe phase velocity becomes equal to cfor all frequen-\ncies. Furthermore, the group velocity can be deﬁned as\nvg=dω/dk rand calculated from Eq. (26).\nFigure 2 shows the phase velocity vpas functions of\n˜ωforη= 0.1, 0.02, and 0 .01. The phase velocity is ap-\nproximately equal to the wavefront velocity cwhen the\nwave character is signiﬁcant ( ω > ω c). In this case, the\nphase velocity provides a good description of the wave-\nlike dynamics. On the contrary, when the wave char-\nacter is insigniﬁcant ( ω < ω c), the wave amplitude is\ndamped strongly and the phase velocity is not meaning-\nful any more. In this region, the propagation velocity is\nthe wavefront velocity c, albeit only on the length scale\nof a damping length.\nIn the special case where η= 1 (τ′\ns=T1), we have\n|kr|=w/cand|ki|= 1/lsf. This means that the spin\naccumulationwavebecomes anon-dispersivebut dissipa-\ntive wave with the constant phase (and group) velocity\ncand penetration depth lsf. However, this case is likely\nnot realized because T1is usually much larger than τ′\ns\nand Valet-Fert theory is justiﬁed to be valid only when\n(τ′\ns/2T1)1/2≪1.5\nTABLE I: Parameters for Cu and Co used in numerical cal-\nculation. The units of vF,ρ∗\nN(F), andlN(F)\nsf are nm/ps, Ω ·nm\nand nm, respectively. τandT1are given in ps.\nMaterial vFρ∗\nN(F)lN(F)\nsf˜β τ T 1\nCu 1570a6b450b0 0.07e3.5e\nCo 1570a86c60d0.5c0.005e0.9e\naFrom Ref. 16; Cu and Co are assumed to have a common Fermi\nvelocity in the Valet-Fert model.\nbFrom Ref. 17\ncFrom Ref. 11\ndFrom Ref. 18\neCalculated from Eqs. (12) and (14).\n-20 -10 010 20 30 40 50 \n-2000 -1600 -1200 -800 -400 0 400 Spin current density J m (nA nm -2 )\nz (nm) -20 -10 010 20 30 40 50 \n(a) t=1.75 T a\n(b) t=5.75 T a\nFIG. 3: Spin current density Jm(z,t) as a function of z. The\nsolid curves in (a) and (b) are Jm(z,t) att= 1.75Taand\nt= 5.75Ta(charge current J(t) =−J0) with AC drive, re-\nspectively. The dashed curves in (a) and (b) are the spin\ncurrent density Jm(z) resulting from the DC current density,\nJ=−J0.\nIV. NUMERICAL RESULTS\nIn this section, the general analysis of the telegraph\nequations for spin-transport is augmented by numerical\nsolutions for two illustrative examples of signal prop-\nagation using spin polarized currents through a ferro-\nmagnet/metal junction: (i) injection of an alternating\ncurrent, and (ii) instantaneous magnetization switching.\nThe results are obtained by numerically solving the sys-\ntem of Eqs. (10) and (11). Our numerical method is\noutlined in Appendix A4. These equations are equiva-\nlent to the telegraph equations Eqs. (23) and (24), which\nhave been discussed in the previous section, but are eas-\nier to solve. Alternatively, we could solve the equation00.05 0.1 0.15 0.2 \n-2000 -1600 -1200 -800 -400 0 400 \nz (nm) Spin accumulation µm (meV) t=5.75 T a\nFIG. 4: Spin accumulation µm(z,t) for the same parameters\nas in Fig. 3(b).\nsystem consisting ofEqs. (18) and (19), in which the spin\naccumulation is described by the spin density. However,\nit is more convenient to work with the electrochemical\npotential than the spin density when we deal with the\nboundary conditions.19\nWe choose a ferromagnet/metal junction consisting of\nCo and Cu as the material system in both of the sce-\nnarios. The interface of the junction is placed at z= 0\nand the Co (Cu) occupies the half-space z >0 (z <0).\nThe positive direction of the current is parallel to the\npositive direction of the zaxis. For simplicity, the inter-\nface resistance of the junction will be neglected. Then,\nthe electrochemical potential and the current density are\ncontinuous across the interface. Consequently, the spin\ntransport across two layers can be described by one com-\nmon equation system with diﬀerent material parameters\nfor the two layers.\nThe material parameters used in our numerical calcu-\nlation are shown in Table I. All other parameters can\nbe obtained from the values in Tab. I. In particular, the\nwavefrontvelocityiscalculatedtobe c= 910nm/ps from\nEq. (13). In the nonmagnetic layer, η=τ′\ns/T1= 0.02.\nThe wavelength λand damping length ldare shown as\nthe solid curves in Fig. 1. The critical period Tccan be\nestimated to be 2 τ′\ns≃0.14ps from Eq. (31), and the\nphase velocity is plotted in Fig. 2.\nA. AC current injection\nThealternatingchargecurrentdensitypassingthrough\nthe ferromagnet/metal junction is assumed to be of the\nformJ(t) =J0sin(ωt), where J0= 100 nA/nm2. Note\nthat the z-dependence of the charge current J(z,t) in\nEq. (11) is neglected for the investigation of the spin\ntransport as pointed out in Sec. II. Two typical fre-\nquencies are studied in the case of the AC drive: νa=6\n010 20 30 40 50 \n-2000 -1600 -1200 -800 -400 0 400 Spin current density J m (nA nm -2 )\nz (nm) \nFIG. 5: Spin current density Jm(z,t) as a function of z. The\nsolid curve is Jm(z,t) att= 1.75Tb(charge current J(t) =\n−J0) with AC drive. The dashed curve is the spin current\ndensityJm(z) for the case of a DC current density J=−J0.\n00.05 0.1 0.15 0.2 \n-2000 -1600 -1200 -800 -400 0 400 Spin accumulation µm (meV) \nz (nm) \nFIG. 6: Spin accumulation µm(z,t) for the same parameters\nas in Fig. 5.\nωa/(2π) = 8.33THz and νb=ωb/(2π) = 0.23THz,\nwhich are larger and smaller than the critical frequency\nνc=ωc/(2π) = 7.11THz of Cu, respectively. The corre-\nsponding periods of the two frequencies are Ta= 0.12ps\nandTb= 4.4ps, which satisfy Ta< Tc< Tb. The nu-\nmerical results for the two frequencies are discussed in\nthe following.\nHigh-frequency case ( ω=ωa> ωc):Figure 3 shows\nsnapshots of the spin current density Jm(z,t) at times\nt= 1.75Taandt= 5.75Ta. At both times, the charge\ncurrent density J(t) reaches its minimum J(t) =−J0.\nThe wavefront, i.e., the spin signal, can be seen clearly\nin Fig. 3(a), where the time tis so small that the wave-\nfronthasnotpropagatedbeyondthescaleofthedamping\nlengthld. In Fig. 3(b), the signal has propagated fur-ther, and due to the attenuation of the wave, the wave\nfront is less clearly visible. Nevertheless, the wavefront\nvelocityccan be determined numerically (or experimen-\ntally) by tracking the motion of the wavefront over a\nshort time interval after switching on the drive current.\nSince we are using a signal time scale shorter than the\ncritical time, we expect from the analysis in Sec. III (see\nalso Fig. 2) the phase velocity to be vp≃c=910nm/ps\nfrom Eq. (13), and a wavelength λ= 108nm. These ex-\npectations are borne out by the numerical results. The\ndynamical damping length ldcan also be extracted from\nthe numerical data, or from an experiment, by ﬁtting a\ndecay time to the envelope of the spin-current signal for\nlongertimes. Duetoinaccuraciesoftheﬁttingprocedure,\nthisquantityismorediﬃculttodeterminequantitatively,\nbut agrees well with the damping length ld= 126nm ex-\npected from Eq (30). An important qualitative conclu-\nsion can be drawn by comparing the decay of the dynam-\nical spin signal in Fig. 3(b) with the spin current density\nJm(z) that results from a DC current density J=−J0,\nwhich is also shown. Since our dynamical equations and\nthespin-diﬀusionequationhavethesamelong-timelimit,\ntheDC resultisidenticalwith steady-statespindiﬀusion.\nIt is apparent that the damping length ldbecomes much\nshorter than the spin diﬀusion length lsfof the steady-\nstate spin transport with DC bias. This is the “skin”\neﬀect, which is already present in the analytical results\nin Sec. III.\nFigure 4 shows the z-dependent spin accumulation µm\nfor the same parameters as in Fig. 3(b). The wavelength,\ndamping length, and phase velocity given by Fig. 4 are\nverysimilartothoseinFig.3(b). Note, however,thatthe\namplitude of the dynamical spin accumulation is much\nsmaller than the spin accumulation of the steady-state\nspin transport shown by the dashed curve. The rea-\nson is that the AC drive oscillates too fast so that the\nspin accumulation does not have enough time to reach\nits steady-state value.\nLow-frequency case ( ω=ωb< ωc):Before analyz-\ning the signal propagation velocity, we ﬁrst show how\nthe results change qualitatively compared with the high-\nfrequency case. In Figure 5, the spin current Jmis plot-\nted as a function of zdriven by an AC current with\nfrequency ωb, which is smaller than the critical angular\nfrequency ωc. The period Tbof the AC drive is 4 .4ps,\nwhich is much larger than Tain Fig. 3. The solid curve\nisJm(z,t) att= 1.75Tb(charge current J=−J0) with\nAC drive. The dashed curve is again Jm(z) driven by\na DC current density J=−J0. For this driving fre-\nquency, the wave character is insigniﬁcant, because the\nwavelength λ= 1856 nm becomes much larger than the\ndamping length ld= 268 nm, and thus the wave ampli-\ntude is damped to zero within just one wavelength. From\na practical point of view, the wavelength and the phase\nvelocityvp= 0.47close their meaning in this case. Com-\nparison between the solid and dashed curves shows that\nthe damping length for Tbbecomes longer than that for\nTain Fig. 3, which is a consequence of the “skin” eﬀect.7\n-50 -40 -30 -20 -10 0\n-1000 -800 -600 -400 -200 0 200 400 \nz (nm) Spin current density J m(z,t) (nA nm -2 )\nFIG. 7: Spin current density Jm(z,t) as a function of z.\nThe solid, dot-dashed, and dashed curves are Jm(z,t) at\nt=Tb/16, Tb/8, Tb/4, respectively, with AC drive.\n-0.1 -0.08 -0.06 -0.04 -0.02 0\n-1000 -800 -600 -400 -200 0 200 400 \nz (nm) Spin accumulation µm(z,t) (meV) \nFIG. 8: Spin accumulation µm(z,t) for the same parameters\nas in Fig. 7.\nFigure 6 shows the spin accumulation µmas a func-\ntion ofz. The parameters used are the same as those in\nFig. 5. The features of the spin accumulation are again\nreminiscent of the spin current in Fig. 5. Note that the\nspin accumulation has become larger compared with the\nAC drive with period Tain Fig. 4. This is reasonable\nbecause the AC drive oscillates more slowly than that in\nFig. 4, so that the spin accumulation has more time to\napproach its steady-state value.\nAt the time the snapshots in Figs. 5 and 6 are taken,\nno “wave front” of the spin current, or signal, can be\ndistinguished. To determine the propagation velocity, we\nshowJm(z,t) andµm(z,t) att=Tb/16,Tb/8, andTb/4\nin Figs. 7 and 8, respectively. By tracking the motion of\nthewavefrontwithtime, wecanestimatethepropagation\nvelocity ofthe signal. The result is in agreementwith thetb-40 -20 020 40 \n-2000 -1600 -1200 -800 -400 0 400 \nz (nm) -40 -20 020 40 \n-40 -20 020 40 Spin current density J m(z,t) (nA nm -2 )\n(c) t = 3.5 ps (b) t = 0.5 ps (a) t = 0 ps \nFIG. 9: Spin current density Jm(z,t) as a function of z.\nThe solid curves in Figs. (a), (b) and (c) are Jm(z,t) at\nt= 0,0.5,3.5 ps, respectively. The dashed curves in (b)\nand (c) are Jm(z,t) att= 0 ps plotted again as a reference.\n“wave front” velocity c, which according to our analysis\nof the telegraph equation (23) is still the propagation\nvelocity.\nTime-dependent spin transport in the low-frequency\ncase can be described approximately by the conventional\nspin diﬀusion equation. However, it is impossible to es-\ntimate the signal propagation velocity from conventional\nspindiﬀusiontheorybecausethereisnowavefrontinthat\ncase and the signal appears in inﬁnity once the charge\ncurrentJ(t) is switched on.3,20\nB. Magnetization switching\nThe instantaneous switching of the magnetization in\nthe ferromagnet, through which the current passes into\nthe nonmagnetic metal, provides perhaps the conceptu-\nally cleanest picture of a spin-switching process. For\na numerical study of this process, we consider again a\nferromagnet/metal junction consisting of Co and Cu.\nWe assume that the system is in a steady state in the\npresence of the DC drive with a charge current density\nJ0= 100 nA/nm2before the magnetization of the fer-\nromagnetic layer is switched from “up” to “down” at\nt= 0. We model the switching as an idealized instan-\ntaneous process, and only consider the evolution of the\nspin current density and spin accumulation afterwards.\nThe spin-up electrons become the majority, and the spin-\ndown electrons become the minority after the instanta-\nneous switching. The conductivities of the majority and8\ntb-0.2 -0.1 00.1 \n-2000 -1600 -1200 -800 -400 0 400 \nz (nm) -0.2 -0.1 0-0.2 -0.1 0Spin accumulation µm(z,t) (meV) \n(c) t = 3.5 ps (b) t = 0.5 ps (a) t = 0 ps \nFIG. 10: Spin accumulation µm(z,t) for the same parameters\nas in Fig. 9.\nminority channels are also exchanged by the switching.\nAlthough the evolution of Jm(z,t) andµm(z,t) does not\ntake the waveform used in Sec. III, it can be decomposed\ninto diﬀerent frequencies by Fourier transformation, so\nthat the analysis of the telegraph equation still applies.\nFigures9 and10showthe dynamicsofthe spin current\nand spin accumulation. Starting from the steady-state\nvalue shown in parts (a), the magnetization is switched\ninstantaneously at t= 0. Figs. 9(b) and Fig. 10(b) show\nsnapshots 0.5ps after the switch, when a pronounced\nkink has developed. This kink indicates the leftmost po-\nsition to which the magnetization-switching signal has\npropagated after 0.5ps. The kink is noticeable only if\nthe time tis so small that it does not propagate be-\nyond the length scale of the spin-diﬀusion length lsf, over\nwhich the steady-state signal decays. Thus the signal-\npropagation velocity can be estimated roughly by track-\ning the motion of the kink with time at the early stage\nof the switching. The result is very close to the wave-\nfront velocity c= 910 nm/ps calculated from the ana-\nlytical result, Eq. (13). Moreover, Fig. 9(c) shows that\nthe spin current density reaches the steady state with\n“down”magnetizationon thetime scaleofthe spin relax-\nation time T1. SinceT1≫τ′\ns, we can consider t=T1as\nthe long-time limit. This behavior is consistent with the\nresult calculated from the diﬀusion equation in Ref. 3,\nso that again the diﬀusion character of spin transport\nemerges as an approximation of the wave-diﬀusion char-\nacter in the long-time limit.V. SUMMARY\nWe studied signal propagation in time-dependent spin\ntransport through magnetic multilayers using an exten-\nsion of the Valet-Fert theory to time-dependent phenom-\nena. We established that time-dependent spin transport\nhas a wave character in addition to its diﬀusive char-\nacter, which enabled us to determine the ﬁnite propa-\ngation velocity of signals in spin transport, such as AC\nspin injection and magnetization switching. The propa-\ngation velocity is the wavefront velocity c=vF/√\n3. The\nwave character is signiﬁcant if the signal time scale τsig\nis smaller than a critical time Tc. When the wave char-\nacter is signiﬁcant ( τsig< Tc), the time-dependent spin\ntransport should be modeled by the dynamical equations\nintroduced in this paper, or, equivalently, the telegraph\nequations. However, pure diﬀusive spin transport can be\nregarded as an approximation of the wave-diﬀusion du-\nality for slow switching times ( τsig≫Tc). In this limit,\nthe spin diﬀusion equationcan be used to study the time-\ndependence ofspin transportapproximately, but it incor-\nrectly yields an inﬁnite signal-propagation velocity.\nAcknowledgments\nWe acknowledge ﬁnancial support from the state of\nRheinland-Pfalz through the MATCOR program, and a\nCPU-time grantfrom the Johnvon Neumann Institut for\nComputing (NIC) at the Forschungszentrum J¨ ulich.\nAPPENDIX A: IDENTITIES AND DERIVATIONS\n1. Useful identities\nSeveral useful identities will be established by the help\nof Eq. (3). Multiplying sin θand integrating over θfrom\n0 toπon both sides of Eq. (3), we have\n/integraldisplayπ\n0dθsinθgs(z,v,t)=∞/summationdisplay\nn=1g(n)\ns(z,t)/integraldisplayπ\n0dθsinθPn(cosθ).\n(A1)\nThe right-hand-side (RHS) of Eq. (A1) can be further\nwritten as\nRHS=∞/summationdisplay\nn=1g(n)\ns(z,t)/integraldisplay1\n−1duP0(u)Pn(u).(A2)\nUsing the orthogonality relation between Legendre poly-\nnomials\n/integraldisplay1\n−1duPn′(u)Pn(u) =2\n2n+1δn,n′,(A3)\nwhereδn,n′is the usual Kronecker symbol, we obtain\nfrom Eqs. (A1) and (A2)\n/integraldisplayπ\n0dθsinθgs(z,v,t) = 0. (A4)9\nEq. (A4) further leads to\n/summationdisplay\nvgs(z,v,t) =Vm3\nh3/integraldisplay\nd3vgs(z,v,t)\n=Vm3\nh3/integraldisplay2π\n0dϕ/integraldisplayπ\n0dθsinθ/integraldisplay∞\n0dvv2gs(z,v,t) = 0.(A5)\n2. Derivation of Eq. (4)\nFollowing Ref. 11, we substitute Eq. (2) into Eq. (1)\nand use the following identity\n∂f0\n∂ε=1\nmv∂f0\n∂v=−δ(v−vF)\nmvF. (A6)Then we can write the RHS (the collision terms) of\nEq. (2) as\n∂fs(z,v,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ncollision=−∂f0(v)\n∂εPs[z,ε(v)]4πv\nm/bracketleftbigg\ngs(z,v,t)−1\n2/integraldisplayπ\n0dθ′sinθ′gs(z,v′,t)/vextendsingle/vextendsingle\nv′=v/bracketrightbigg\n−∂f0(v)\n∂εPsf[z,ε(v)]4πv\nm/bracketleftbigg\ngs(z,v,t)−1\n2/integraldisplayπ\n0dθ′sinθ′g−s(z,v′,t)/vextendsingle/vextendsingle\nv′=v/bracketrightbigg\n+∂f0(v)\n∂εPsf[z,ε(v)]4πv\nm[µs(z,t)−µ−s(z,t)].(A7)\nUsing Eq. (A4), we can write Eq. (A7) in the form\n∂fs(z,v,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ncollision=−∂f0(v)\n∂εPs[z,ε(v)]4πv\nmgs(z,v,t)−∂f0(v)\n∂εPsf[z,ε(v)]4πv\nmgs(z,v,t)\n+∂f0(v)\n∂εPsf[z,ε(v)]4πv\nm[µs(z,t)−µ−s(z,t)].(A8)\nBy introducing the relaxation times\n1\nτs(v)=Ps[z,ε(v)]4πv\nm, (A9)\n1\nτsf(v)=Psf[z,ε(v)]4πv\nm, (A10)\nwhere the z-dependence of the relaxation times is ne-\nglected within the same layer, we can further write\nEq. (A8) as\n∂fs(z,v,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ncollision\n=−∂f0(v)\n∂ε/parenleftbigg1\nτs+1\nτsf/parenrightbigg\ngs(z,v,t)\n+∂f0(v)\n∂εµs(z,t)−µ−s(z,t)\nτsf.(A11)\nTaking into account the left-hand-side of Eq. (1) and in-\ntegrating over v, we can ﬁnally derive Eq. (4). Note that\nτs(v) andτsf(v) are restricted to the Fermi velocity vF\nafter the integration over vand then they are simply\nwritten as τsandτsf.3. Derivation of Eqs. (17) and (20)\nMultiplying both sides of Eq. (2) by −e/V, summing\noverv, and using Eq. (A5), we obtain\nns(z,t)−n0\ns=−eNs[µs(z,t)−µ0],(A12)\nwhere\nns(z,t) =−e\nV/summationdisplay\nvfs(z,v,t), (A13)\nn0\ns=−e\nV/summationdisplay\nvf0(v) =−ens,(A14)\nare the nonequilibrium and equilibrium charge density\nfor spins, respectively. In turn, Eq. (A12) yields\nnm(z,t) =−eNsµm(z,t), (A15)\nn(z,t)−2n0\ns=−eNs/bracketleftbig\n2µ(z,t)−2µ0/bracketrightbig\n,(A16)\nwherenm(z,t) =n+(z,t)−n−(z,t) is the spin density\nandn(z,t) =n+(z,t)+n−(z,t) the total nonequilibrium\ncharge density.10\n4. Numerical solution of Eqs. (10) and (11)\nFor the numerical solution of Eqs. (10) and (11) we\nuse the method of characteristics and Hartree’s compu-\ntational form. Following Ref. 21, the space zand time\ntare discretized into grids with equal intervals ∆ zand\n∆t, respectively. The discretized forms of Jm(z,t) and\nµm(z,t)areJn\nm,iandµn\nm,iatith spacepointand nth time\npoint, respectively. Then, Jn+1\nm,iandµn+1\nm,iat (n+ 1)th\ntime point can be calculated by the iteration relations\n/parenleftbigg\n2+∆t\nT1/parenrightbigg\nµn+1\nm,i=/parenleftbigg\n1−∆t\n2T1/parenrightbigg\n(µn\nm,i−1+µn\nm,i+1)\n−1\neNsc/parenleftbigg\n1−∆t\n2τ/parenrightbigg\n(Jn\nm,i−1−Jn\nm,i+1),(A17)\n/parenleftbigg\n2+∆t\nτ/parenrightbigg\nJn+1\nm,i=−eNsc/parenleftbigg\n1−∆t\n2T1/parenrightbigg\n(µn\nm,i−1−µn\nm,i+1)\n+/parenleftbigg\n1−∆t\n2τ/parenrightbigg\n(Jn\nm,i−1+Jn\nm,i+1)−∆t\nτ˜β(Jn+Jn+1),\n(A18)\nfor all space points except the two boundary points,\nwhich should be determined by boundary conditions.\nHere,Jnis the total current density at nth time point.\nMoreover, ∆ zand ∆tare chosen to satisfy the relation\n∆z=c∆t. Eqs. (A17) and (A18) can be iterated nu-\nmerically to yield the results presented in Sec. IV. In\nthe numerical solution, we used the following initial and\nboundary conditions for the AC spin injection and mag-\nnetization switching.\nAC spin injection : The initial conditions are µm(z,t=\n0) = 0 and Jm(z,t= 0) = 0. The boundary condition\nforµm(z,t) isµm(z=±∞,t) = 0. From Eq. (11), the\nboundary conditionJm(z=±∞,t) =˜βJ0\n1+ω2τ2[ωτcos(ωt)−sin(ωt)\n−ωτexp(−t/τ)].(A19)\nforJm(z,t) can be derived.\nMagnetization switching : The initial conditions for\nJm(z,t) andµm(z,t) are the steady-state solutions to\nEqs. (10) and (11)\nµF\nm(z,t= 0) = C0exp(−z/lF\nsf), (A20)\nJF\nm(z,t= 0) = −C0\n2eρ∗\nFlF\nsfexp(−z/lF\nsf)−˜βJ0,(A21)\nµN\nm(z,t= 0) = C0exp(z/lN\nsf), (A22)\nJN\nm(z,t= 0) =C0\n2eρ∗\nNlN\nsfexp(z/lN\nsf), (A23)\nwhereC0=−2e˜βJ0(ρ∗\nFlF\nsfρ∗\nNlN\nsf)/(ρ∗\nFlF\nsf+ρ∗\nNlN\nsf). Here,\nµF\nmandJF\nmapply to the ferromagnetic layer occupying\nz >0, whereas µN\nmandJN\nmrefer to the nonmagnetic layer\n(z <0). In deriving the initial conditions above, we have\nused the identity 1 /(2ρ∗\nN(F)) =σN(F)/2 =e2Ns¯DN(F),\nwhereσN(F)is the total conductivity of the nonmag-\nnetic (ferromagnetic) layer. The boundary condition for\nµm(z,t) isµm(z=±∞,t) = 0. Then, the boundary con-\ndition for Jm(z,t) can again be derived from Eq. (11).\nThis yields\nJm(z=±∞,t) =˜βJ0[1−2exp(−t/τ)],(A24)\nwhere˜βis the asymmetry parameter before the magne-\ntization switching. Note that ˜βbecomes −˜βwhen the\nmagnetization is switched ( t >0).\n∗Electronic address: hcsch@physik.uni-kl.de\n1I.ˇZuti´ c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n2J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nI.ˇZuti´ c, Acta Phys. Slovaca 57, 565 (2007).\n3S. Zhang and P. M. Levy, Phys. Rev. B 65, 052409 (2002).\n4E. I. Rashba, Appl. Phys. Lett. 80, 2329 (2002).\n5J. Zhang and P. M. Levy, Phys. Rev. B 71, 184417 (2005).\n6/suppress L. Cywi´ nski, H. Dery, and L. J. Sham, Appl. Phys. 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Lett.\n72, 3274 (1994).\n18L. Piraux, S. Dubois, A. Fert, and L. Belliard, Eur. Phys.\nJ. B.4, 413 (1998).\n19S. Hershﬁeld and H. L. Zhao, Phys. Rev. B 56, 3296 (1997).\n20J. Masoliver and G. H. Weiss, Eur. J. Phys. 17, 190 (1996).\n21W. F. Ames, Numerical Methods for Partial Diﬀerential\nEquations (Academic Press, San Diego, California, 1992),11\n3rd ed., see Sec. 4-15."    },    {        "title": "1909.10301v2.Coherent_spin_dynamics_of_solitons_in_the_organic_spin_chain_compounds___o__DMTTF___2X____X____Cl__Br_.pdf",        "content": "Coherent spin dynamics of solitons in the organic spin chain compounds\n(o-DMTTF) 2X(X= Cl, Br)\nJ. Zeisner,1, 2,\u0003O. Pilone,3L. Soriano,3G. Gerbaud,4H. Vezin,5\nO. Jeannin,6M. Fourmigu\u0013 e,6B. B uchner,1, 2V. Kataev,1,yand S. Bertaina3,z\n1Leibniz Institute for Solid State and Materials Research IFW Dresden, D-01069 Dresden, Germany\n2Institute for Solid State and Materials Physics, TU Dresden, D-01062 Dresden, Germany\n3Aix-Marseille Universit\u0013 e, CNRS, IM2NP UMR 7334, F-13397 Marseille, France\n4Aix-Marseille Universit\u0013 e, CNRS, BIP UMR 7281, F-13402 Marseille, France\n5Universit\u0013 e de Lille, CNRS, LASIR UMR 8516, F-59655 Villeneuve d'Ascq, France\n6Universit\u0013 e de Rennes, CNRS, ISCR UMR 6226, F-35042 Rennes, France\n(Dated: June 21, 2021)\nWe studied the magnetic properties, in particular dynamics, of the correlated spins associated\nwith natural defects in the organic spin chain compounds ( o-DMTTF) 2X(X= Br, Cl) by means\nof electron spin resonance (ESR) spectroscopy. Both materials exhibit spin-Peierls transitions at\ntemperatures around 50 K [P. Foury-Leylekian et al. , Phys. Rev. B 84, 195134 (2011)], which allow\na separation of the properties of defects inside the chains from the magnetic response of the spin\nchains. Indeed, continuous wave ESR measurements performed over a wide temperature range\nevidence the evolution of the spin dynamics from being governed by the spins in the chains at\nelevated temperatures to a low-temperature regime which is dominated by defects within the spin-\ndimerized chains. Such defects polarize the antiferromagnetically coupled spins in their vicinity,\nthereby leading to a \fnite local alternating magnetization around the defect site which can be\ndescribed in terms of a soliton, i.e. a spin 1/2 quasiparticle built of many correlated spins, pinned to\nthe defect. In addition, contributions of triplon excitations of the spin-dimerized state to the ESR\nresponse below the transition temperature were observed which provides a spectroscopic estimate\nfor the spin-gap of the studied systems. Moreover, details of spin dynamics deep in the spin-Peierls\nphase were investigated by pulse ESR experiments which revealed Rabi-oscillations as signatures\nof coherent spin dynamics. The latter is a prerequisite for a selective manipulation of the defect-\ninduced soliton spin states which is, for instance, relevant in the context of quantum computation.\nFrom a comparison of the characteristic damping times of the Rabi oscillations with measurements\nof the spin relaxation times by means of primary-echo decay and CPMG methods it becomes evident\nthat inhomogeneities in local magnetic \felds strongly contribute to the soliton decoherence.\nI. INTRODUCTION\nDefects in one-dimensional spin systems continue to\nbe an active \feld of modern solid state research as they\nare able to alter magnetic properties of the hosting ma-\nterials drastically [1]. For example, nonmagnetic defects\nintroduced into isotropic antiferromagnetic (AFM) spin\nchains lead to a breaking of the chains into \fnite seg-\nments and thereby induce additional paramagnetic-like\nmoments in the spin systems (see, e.g., Refs. [1, 2]). In\nspin-Peierls systems which feature a nonmagnetic ground\nstate due to a dimerization of the (spin) lattice [3] similar\ne\u000bects may appear, as illustrated in Fig. 1. In addition,\nsuch a nonmagnetic singlet ground state enables a de-\ntailed study of the defect-induced magnetic properties as\nthe in\ruence of the spins residing on the regular sites of\nthe chains is strongly reduced. Defects in the spin-Peierls\nchains polarize the antiferromagnetically coupled spins in\ntheir vicinity, thereby leading to a \fnite local alternating\nmagnetization around the defect site [4, 5] [see Figs. 1(c)\n\u0003j.zeisner@ifw-dresden.de\nyv.kataev@ifw-dresden.de\nzsylvain.bertaina@im2np.frand (d)]. The latter can be described in terms of a soli-\nton, i.e. a spin 1/2 quasiparticle built of many correlated\nspins, pinned to the defect or chain break [4, 5]. As\na consequence of the defect-induced soliton formation,\na spin-Peierls system can show a magnetic response at\nlow temperatures despite the nonmagnetic character of\nthe ground state and of the introduced impurity. This\nparamagnetic-like behavior of the soliton can be evi-\ndenced, for instance, by temperature-dependent suscepti-\nbility measurements at low temperatures. In contrast to\nisolated, non-interacting magnetic impurities in a three-\ndimensional lattice which are frequently observed in real\nmaterials, a soliton is of a purely quantum mechanical na-\nture. Moreover, since the solitons are formed from entan-\nglement of many spins within the dimerized chain, their\nproperties, in particular their extension along the chain\nsites as well as the coupling strength between adjacent\nsolitons, are intimately connected to the characteristics\nof the chain, i.e., the coupling Jand the dimerization\nparameter \u000e[4, 6, 7]. Theoretically, an isotropic one-\ndimensional spin system that shows a spin-Peierls tran-\nsition at low temperatures could be modeled in terms an\nS= 1=2 alternating-exchange Heisenberg chain Hamilto-arXiv:1909.10301v2  [cond-mat.str-el]  17 Dec 20192\nFIG. 1. Schematic representation of the spin-Peierls transition and defect-induced soliton formation. (a) At temperatures above\nthe spin-Peierls temperature TSPthe uniform Heisenberg chain consists of equidistant spins coupled by an isotropic nearest-\nneighbor exchange J. (b) ForT < T SPthe bond lengths within the chain lattice are modulated, eventually leading to the\nformation of spin dimers (indicated by blue circles). The antiferromagnetic intradimer interaction J1=J(1 +\u000e) is larger than\nthe interdimer interaction J2=J(1\u0000\u000e) giving rise to the spin-singlet ground state of the dimerized chain. (c) A nonmagnetic\ndefect (gray sphere) introduced into a dimerized spin system breaks the chains into \fnite segments and can be modeled, for\ninstance, by two successive weak bonds J0\n2andJ2(withJ0\n2< J 2) within the chain. The coupling constant J0\n2describes the\ne\u000bective exchange coupling between two spins across the defect site. (d) As a result of such a defect, a \fnite local alternating\nmagnetization is induced around the defect site. This can be described in terms of a quasiparticle, the pinned soliton, which\ninvolves many entangled spins in the vicinity of the defect but carries an overall spin of 1/2. The pro\fle of the local alternating\nmagnetization shown here was calculated using a density matrix renormalization group method (see Ref. [10] for further details)\nconsidering a defect at the center of a dimerized chain with a length of 201 sites and assuming J0\n2=J2as well as a dimerization\nparameter\u000e= 0:03.\nnian [8, 9]\nH=X\niJ(1 +\u000e)S2i\u00001\u0001S2i+J(1\u0000\u000e)S2i\u0001S2i+1:(1)\nThroughout this work we consider an isotropic Heisen-\nberg type exchange coupling JwithJ >0 corresponding\nto AFM coupling between nearest neighbor spins Siand\nSj. The dimerization parameter of the chain is de\fned\nsuch that\u000e= 0 recovers the Heisenberg chain above\nthe spin-Peierls transition temperature TSP[Fig. 1(a)]\nwhile the limit \u000e= 1 corresponds to isolated dimers with\na singlet ground state [Fig. 1(b)]. For real spin-Peierls\nsystems,\u000eis expected to be in between these two limits.\nNote thatJand\u000emay show a pronounced temperature\ndependence below TSPsince the magnetic transition is\ncaused by a deformation of the crystal lattice, i.e., due\nto spin-phonon interactions (see, e.g., Ref. [9]). These\ntwo parameters are, in principle, accessible via a quan-\ntitative analysis of the temperature-dependent spin sus-\nceptibility if the system can be appropriately described\nby the Hamiltonian in Eq. (1) [9]. Thus, temperature-\ndependent measurements of the static susceptibility or\nthe electron spin resonance (ESR) intensity provide a\nmeans to characterize the static magnetic properties of\nsolitons in real spin-Peierls materials. The latter are usu-\nally organic compounds (see, for instance, Refs. [11, 12]),\nthe only well-established inorganic spin-Peierls material\nup to now being CuGeO 3(see, e.g., Ref. [13]). Previous\nstudies of the magnetic properties of the organic salts(TMTTF) 2X(X= AsF 6, PF 6) [14] indeed could assign\nthe magnetic low-temperature response to the forma-\ntion of solitons pinned to defect sites which are naturally\npresent in these crystals. Moreover, a remarkable soliton\nspin dynamics was revealed featuring a substantial spin\ncoherence as evidenced by the observation of Rabi oscil-\nlations in pulse ESR measurements [14]. Such a coherent\ndynamics is a prerequisite for a selective manipulation\nof the soliton spin states. The robustness of the Rabi\noscillations with respect to the presence of anisotropic\ndipolar interactions, which usually result in decoherence\nof the spins (see, e.g., Ref. [15]), was attributed to the\nexistence of isotropic exchange interactions between the\nsolitons which, in turn, result from the fact that the soli-\ntons are built of an ensemble of strongly correlated spins\n[14]. This e\u000bect renders the spin dynamics of the solitons\nobserved in (TMTTF) 2AsF 6and (TMTTF) 2PF6compa-\nrable to the one encountered in inorganic systems of mag-\nnetic ions diluted in nonmagnetic matrices [15{20] which\nwere proposed as realizations of qubits with potential ap-\nplications in the \feld of quantum computation [16]. In\nthis respect, as well as for the sake of deepening the un-\nderstanding of spin dynamics of pinned solitons in dimer-\nized spin chains, it is important to verify whether the ex-\nistence of a coherent soliton spin dynamics represents a\ngeneric property of organic spin chain systems with natu-\nral defects. In the present work, we address this question\nby ESR studies on ( o-DMTTF) 2Cl and (o-DMTTF) 2Br\n[21], two members of a related family of one-dimensional3\norganic spin-Peierls compounds ( o-DMTTF stands for\ntheortho -dimethyltetrathiafulvalene molecule).\nThis paper is organized as follows: Details on the crys-\ntal structure of the title compounds as well as on experi-\nmental methods are provided in Sec. II. The results of our\nexperimental investigations are presented and discussed\nin Sec. III starting with the magnetic characterization of\nthe spin systems by means of temperature- and angular-\ndependent continuous wave (CW) ESR measurements in\nSec. III A. This is followed by the results of pulse ESR\nexperiments in Sec. III B which reveal the coherent char-\nacter of the spin dynamics in ( o-DMTTF) 2X. Main con-\nclusions of the present work are summarized in Sec. IV\nwhile a discussion of the temperature-dependent spin sus-\nceptibility and their analysis is provided in the Appendix.\nII. SAMPLES AND METHODS\nCrystals of ( o-DMTTF) 2Cl and (o-DMTTF) 2Br used\nin this work were synthesized by electrocrystallization\nmethods as described in Ref. [21]. Their structural, elec-\ntronic and basic magnetic characterization was reported\nin Refs. [21, 22]. The crystallographic structure of the\ntitle compounds is shown in Fig. 2 and the main features\nare brie\ry recapitulated in the following in order to fa-\ncilitate the discussion of our results. The tetragonal unit\ncell (space group I42d(no. 122) [21]) comprises eight\nplanaro-DMTTF molecules which form stacks along the\ncrystallographic caxis (Fig. 2 bottom) and which are ro-\ntated by 90\u000ewith respect to the neighboring stacks in the\nabplane (Fig. 2 top). As a consequence of this rotation,\ninteractions between neighboring stacks are weak, lead-\ning to e\u000bectively one-dimensional electronic properties\n[21, 22]. In ( o-DMTTF) 2Xtheo-DMTTF molecules are\nonly partially oxidized and each double molecule within\na stack hosts one hole (and three electrons). As a con-\nsequence, the electronic bands are quarter \flled (three-\nquarter \flled if electrons are considered instead of holes)\nwhich gives rise to a metallic behavior at room tem-\nperature and at ambient pressure [21, 22]. Resistiv-\nity measurements on ( o-DMTTF) 2Xcrystals revealed\nthe onset of charge localization below temperatures of\n\u0018150 K (X= Cl) and\u0018100 K (X= Br), respectively\n[22]. Thus, below these temperatures and above the spin-\nPeierls transition temperatures TSP\u001850 K of the respec-\ntive compounds [22] the spin systems can be described\nby anS= 1=2 Heisenberg AFM chain model in which\neach double molecule of a stack (see Fig. 2 bottom) cor-\nresponds to one site in the spin-chain lattice which is oc-\ncupied by a single spin associated with the localized hole.\n(o-DMTTF) 2Xcrystals grow in a needle-like shape with\nthe longest side of the crystal corresponding to the caxis,\nand, thus, to the stacking direction of the o-DMTTF\nmolecules, which allowed an easy orientation of the crys-\ntals in the external magnetic \feld with respect to this\naxis. Data presented in this study were collected on sev-\neral crystals of ( o-DMTTF) 2Cl and (o-DMTTF) 2Br, re-\nFIG. 2. Crystallographic structure of the ( o-DMTTF) 2Xsin-\ngle crystals. Top: View on the abplane of the unit cell with\nfouro-DMTTF molecules which are rotated by 90\u000ewith re-\nspect to each other. Bottom: Stacks of o-DMTTF molecules\nalong thecaxis. Each double molecule of such a stack hosts\none charge, thus e\u000bectively forming a spin chain along the c\naxis. Crystallographic data are taken from Ref. [21].\nspectively, in order to avoid artifacts arising from cracks\nin the crystal caused by thermal cycling. The consis-\ntency between measurement series obtained from di\u000ber-\nent crystals was always checked carefully to allow proper\ncomparisons between the di\u000berent data sets.\nContinuous wave (CW) ESR measurements were per-\nformed using Bruker EMX X-band spectrometers operat-\ning at microwave (mw) frequencies of about 9.6 GHz and\nin \felds up to 1.3 T. These spectrometers are equipped\nwith standard He-\row cryostats which allow measure-\nments in the temperature range from 4 K up to room\ntemperature. Moreover, goniometers installed at the\nspectrometers enable measurements of ESR parameters\nas a function of angle between the crystal axes and the\nexternal magnetic \feld. For pulse ESR measurements\nat X-band frequencies ( \u00189.7 GHz) Bruker Elexsys E580\nspectrometers were used. These are equipped with a He-\n\row cryostat and a cryogen-free cryostat, respectively, to\nprovide access to temperatures between 2.7 K and room\ntemperature.\nIII. RESULTS AND DISCUSSION\nIn the following, the results of CW and pulse ESR\nstudies on ( o-DMTTF) 2Cl and (o-DMTTF) 2Br are pre-\nsented and discussed. For all measurements shown here\nthe external magnetic \feld was oriented either parallel or4\nperpendicular to the caxis or was rotated in the plane\ncomprising these two directions, respectively. Measure-\nments on ( o-DMTTF) 2Br in theabplane (not shown)\ndid not reveal a substantial anisotropy of the resonance\n\felds or the linewidth, in line with expectations based on\ncrystal structure (Fig. 2 top) and with previous reports\n[21, 22]. Consequently, for measurements with H?cthe\ncrystallographic axes which are parallel to the magnetic\n\feld are not speci\fed here since they are magnetically\nequivalent.\nFirst, we will turn to the results obtained by means of\nCW ESR measurements over a wide temperature range,\nSec. III A. This will be followed by an investigation of\nthe angular dependence of the CW ESR parameters at\nvarious temperatures in the vicinity of and below the\nspin-Peierls transition, respectively. In Sec. III B, a more\ndetailed study of spin dynamics in the low-temperature\nphase by means of pulse ESR experiments will be pre-\nsented.\nA. CW ESR\n1. Temperature dependence\nThe temperature dependence of the width as well as\nthe intensity of the ESR line are summarized in Fig. 3.\nData shown were measured with the external magnetic\n\feld oriented parallel to the caxis, i.e., the spin chain\naxis. Three representative spectra obtained from mea-\nsurements on ( o-DMTTF) 2Cl at 7, 20, and 200 K are\npresented in the inset of Fig. 3(b). As will be explained\nbelow, these spectra illustrate the ESR response of the\nstudied system in di\u000berent temperature regimes which\ndi\u000ber with respect to the contributions to the resonance\nline. Note that, due to the use of the lock-in technique\nin the CW ESR measurements, the \frst derivative of the\nmicrowave power re\rected from the cavity with respect to\nthe external \feld d P/dHis measured. As a consequence,\nthe intensity of the ESR line (i.e., the area under the ab-\nsorption curve) which is proportional to the static suscep-\ntibility\u001f0, can be obtained by double integration of the\nmeasured spectra. The shown spectra consist of a single\nresonance line which can be described by a Lorentzian\npro\fle. At temperatures between room temperature and\nabove 200 K, an asymmetric Dysonian lineshape [23, 24]\nwas observed which evidences the metallic behavior of\nthe samples in this temperature region [21, 22]. From \fts\nof Lorentzian or Dysonian lines to the measured spectra\n[solid red lines in the inset of Fig. 3(b)] the ESR parame-\nters such as linewidth \u0001 H(full width at half maximum)\nand resonance position Hrescould be obtained. The lat-\nter was used to calculate the (e\u000bective) gfactor according\nto the standard frequency-\feld relation of a paramagnet\n[25]\ng=h\u0017\n\u0016B\u00160Hres; (2)where\u0017denotes the mw frequency and h,\u0016B, and\u00160are\nPlanck's constant, the Bohr magneton, and the vacuum\npermeability, respectively. As will be discussed in the fol-\nlowing section, angular dependent measurements of the g\nfactor evidence the absence of a substantial temperature\ndependence of this quantity [Fig. 4(b)].\nThe temperature dependence of the linewidth is shown\nfor (o-DMTTF) 2Cl and (o-DMTTF) 2Br in the main\npanel and in the inset of Fig. 3(a), respectively. For both\ncompounds, the linewidth reveals a maximum at tem-\nperatures slightly above TSPwhich is followed towards\nlow temperatures by a sudden drop of the linewidth be-\nlowTSPindicating a strong change in spin dynamics be-\nlow the phase transition. Furthermore, the latter is ev-\nidenced also by the steep decrease of the ESR intensity\nwhich sets in at TSP, see Fig 3(b). Down to \u001825 K the\nintensity, as a measure for the spin susceptibility, is expo-\nnentially reduced since the majority of spins approaches\nthe spin-dimerized nonmagnetic singlet ground state ex-\npected for a spin-Peierls system. Below \u001825 K the inten-\nsity shows a Curie-like increase towards lower tempera-\ntures which was previously ascribed to the presence of\nextrinsic magnetic defects with a spin quantum number\nS= 1=2 [22]. However, as will be argued in the following\nsections, the ESR response at low temperatures is, most\nlikely, related to the existence of solitons pinned to natu-\nrally present defects in the crystals which show a coherent\nspin dynamics, similar to the situation found in the re-\nlated compounds (TMTTF) 2AsF 6and (TMTTF) 2SbF 6\n[14]. The di\u000berent temperature regimes found in the\ntemperature dependences of the ESR linewidth and in-\ntensity are, naturally, as well visible in the spectra pre-\nsented in the inset of Fig. 3(b). At high temperatures\n(cf. spectrum recorded at 200 K), the ESR spectrum\nconsists of a resonance line with high intensity and a\nlinewidth of about 1.3 mT at 200 K. This line originates\nfrom the magnetic response of the spins within the reg-\nularo-DMTTF-stacks, i.e. the spin chains. Below the\nspin-Peierls transition, the simultaneous reduction of the\nlinewidth and intensity (caused by the dimerization of\nthe magnetic lattice) yields a rather narrow and weak\nESR signal at 20 K. As will be discussed in the following\nsection, angular dependent measurements reveal that at\nthese temperatures the ESR spectrum, despite featuring\nonly a single line, is composed of two contributions. The\ndominant contribution to the ESR signal is due to the\nabove-mentioned solitons while the second contribution,\narising from singlet-triplet excitations of the dimerized\nchains, is much weaker and cannot be resolved explicitly\nin the individual spectra, see below. Upon further low-\nering the temperature, the latter contribution is strongly\nreduced and the ESR signal at 7 K is fully dominated by\nthe solitons whose intensity features a Curie-like behav-\nior, thereby leading to an increased signal intensity as\ncompared to the spectrum at 20 K.\nAs mentioned in the Introduction, it is in principle\npossible to extract the characteristic parameters of the\nHamiltonian (1), such as Jand\u000e, from an analysis of the5\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s48/s32/s88 /s32/s61/s32/s67/s108\n/s32/s88 /s32/s61/s32/s66/s114\n/s84\n/s83/s80/s72 /s32/s124 /s124 /s32/s99/s44/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s69/s83/s82/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s84 /s32/s40/s75/s41/s51/s51/s56 /s51/s52/s48 /s51/s52/s50 /s51/s52/s52 /s51/s52/s54 /s51/s52/s56\n/s100 /s73/s32/s47/s32/s100 /s72 /s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s109\n/s48/s32/s72 /s32/s40/s109/s84/s41/s88 /s32/s61/s32/s67/s108\n/s72 /s32/s124/s124/s32/s99\n/s55/s32/s75\n/s50/s48/s48/s32/s75/s50/s48/s32/s75\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s48/s88 /s32/s61/s32/s66/s114/s109\n/s48/s32/s68 /s72 /s32/s40/s109/s84/s41\n/s84 /s32/s40/s75/s41/s88 /s32/s61/s32/s67/s108\n/s72 /s32/s124/s124/s32/s99/s44/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122/s72 /s32/s124/s124/s32/s99/s84\n/s83/s80\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48/s109\n/s48/s32/s68 /s72 /s32/s40/s109/s84/s41\n/s84 /s32/s40/s75/s41/s40/s98/s41 /s40/s97/s41\nFIG. 3. Temperature dependence of the ESR parameters obtained from CW ESR measurements on ( o-DMTTF) 2Xsamples\nwith the static \feld applied along the caxis. (a) ESR linewidth \u0001 Hof (o-DMTTF) 2Cl. Respective data for ( o-DMTTF) 2Br\nare shown in the inset. (b) Temperature dependence of the intensity of the ESR line for both compounds obtained from double\nintegration of the spectra. Representative spectra measured on a ( o-DMTTF) 2Cl sample at 7, 20, and 200 K are presented in\nthe inset together with \fts to the experimental data (solid red lines). At these di\u000berent temperature regimes the ESR lines are\ngoverned by the magnetic response of the solitons (7 K), by a combination of soliton and triplon contributions (20 K), and by\nthe response of the spins within the chains (200 K), respectively.\ntemperature dependent spin susceptibility. These param-\neters, in turn, determine the spin gap \u0001 of the system\nas well as the soliton properties, for instance their spa-\ntial extension [6, 7]. Moreover, temperature dependent\nmeasurements of the spin susceptibility provide a means\nfor obtaining estimates of the relative amount of soli-\ntons with respect to the majority of the spins by com-\nparing the Curie-like tail at low temperatures with the\nsusceptibility at elevated temperatures. Thus, the tem-\nperature dependence of the ESR intensity [Fig 3(b)] can\nyield, in the ideal case, valuable insights into the ba-\nsic static properties of solitons in a dimerized spin chain\nsystem. In the present case, however, such an accurate\nanalysis is hampered by several di\u000eculties as discussed\nin more detail in the Appendix where order-of-magnitude\nestimates for the above-mentioned parameters are pre-\nsented. Therefore, the temperature dependence of the\nESR intensity is discussed on a qualitative level in the\nfollowing. At temperatures above TSP, the behavior of\nthe susceptibility resembles the characteristic tempera-\nture dependence of uniform S= 1=2 Heisenberg AFM\nspin chains systems [9, 26, 27]. On the other hand, a\nmetallic behavior was reported for ( o-DMTTF) 2Cl and\n(o-DMTTF) 2Br above the charge-localization tempera-\ntures of\u0018150 and\u0018100 K, respectively [22]. This in-\ndicates a delocalized character of the charge carriers in\nthis temperature range and, thus, could imply a limited\napplicability of the model of localized spins. However,\nbelow the charge localization temperature, the modeling\nof (o-DMTTF) 2Xin terms of a spin chain with increas-ing dimerization below TSPcan be expected to be a valid\ndescription. Consequently, the respective model will be\nemployed in the remainder of this paper since the main\nfocus of interest in this work lies on the particular spin\ndynamics in the spin-gap phase. The existence of such a\nphase is unambiguously evidenced by the strong decrease\nof the spin susceptibility below the spin-Peierls transition\nas mentioned above and allows a study of the dynamic\nproperties of pinned solitons without perturbation by the\nmagnetic response of the spins within the chain.\nFinally, we note that the overall temperature depen-\ndence of the linewidth [Fig. 3(a)], in particular, the strong\nnarrowing of the ESR line as well as the temperature-\ndependent susceptibility [Fig. 3(b)] was observed as well\nin previous studies [21, 22] on samples originating from\nthe same source and demonstrates the good agreement\nbetween those studies and the measurements of the\npresent work. Thereby it is con\frmed that the basic\nmagnetic properties, in particular, the spin-Peierls tem-\nperature are the same for samples used in this work and\nin previous reports. In the following sections, we will ex-\ntend the discussion of magnetic properties contained in\nprevious studies by examining in more detail the angular\ndependence of the linewidth at temperatures below TSP\nand the spin dynamics at low temperatures.6\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s48/s32/s88 /s32/s61/s32/s67/s108\n/s32/s88 /s32/s61/s32/s66/s114/s65/s47/s66\n/s84 /s32/s40/s75/s41/s45/s53/s48 /s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s46/s48/s48/s54/s50/s46/s48/s48/s56/s50/s46/s48/s49/s48/s50/s46/s48/s49/s50/s50/s46/s48/s49/s52/s50/s46/s48/s49/s54\n/s32/s53/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s49/s53/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s53/s48/s32/s75\n/s32/s54/s48/s32/s75\n/s32/s50/s48/s48/s32/s75/s32/s103/s32 /s102/s97/s99/s116/s111/s114\n/s82/s111/s116/s97/s116/s105/s111/s110/s32/s97/s110/s103/s108/s101/s32 /s97 /s32/s40/s68/s101/s103/s41/s72 /s32/s124/s124/s32/s99/s72 /s32/s94 /s32/s99/s88 /s32/s61/s32/s67/s108\n/s45/s50/s53 /s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53 /s50/s48/s48/s48/s46/s50/s50/s48/s46/s50/s52/s48/s46/s50/s54/s48/s46/s50/s56/s48/s46/s51/s48/s48/s46/s51/s50/s48/s46/s51/s52\n/s88 /s32/s61/s32/s67/s108\n/s52/s32/s75/s109\n/s48/s32/s68 /s72 /s32/s40/s109/s84/s41\n/s82/s111/s116/s97/s116/s105/s111/s110/s32/s97/s110/s103/s108/s101/s32 /s97 /s32/s40/s68/s101/s103/s41/s72 /s32/s124/s124/s32/s99\n/s72 /s32/s94 /s32/s99/s48/s46/s50/s50/s48/s46/s50/s52/s48/s46/s50/s54/s48/s46/s50/s56/s48/s46/s51/s48\n/s88 /s32/s61/s32/s67/s108\n/s50/s48/s32/s75/s109\n/s48/s32/s68 /s72 /s32/s40/s109/s84/s41/s72 /s32/s124/s124/s32/s99\n/s72 /s32/s94 /s32/s99/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s109\n/s48/s32/s68 /s72 /s32/s40/s109/s84/s41\n/s53/s48/s32/s75/s88 /s32/s61/s32/s67/s108/s72 /s32/s124/s124/s32/s99\n/s72 /s32/s94 /s32/s99\n/s40/s99/s41/s40/s98/s41 /s40/s97/s41\nFIG. 4. Angular dependence of ESR parameters obtained from CW ESR studies at \u00189.6 GHz. In these measurements the\n\feld was rotated from HkctoH?c. (a) Angular-dependent linewidth of ( o-DMTTF) 2Cl at three di\u000berent temperatures\nbelowTSPshowing the e\u000bects of di\u000berent contributions to \u0001 H(\u000b) below the transition. Solid lines show \fts according to\nEq. (4), see text. (b) Angular dependence of the gfactor measured on a ( o-DMTTF) 2Cl crystal at temperatures between\n5 and 200 K. Within this temperature range no changes in the ganisotropy are observed. The solid black line denotes a\nrepresentative \ft to the angular dependence of the gfactor at 5 K according to Eq. (3). Corresponding measurements carried\nout on (o-DMTTF) 2Br yielded very similar results as in (a) and (b) and are not shown here. (c) Temperature dependence of\nthe ratioA=B as a measure of the relative strength of the triplon contribution to the linewidth obtained from \fts of Eq. (4) to\nthe \u0001H(\u000b) curves of ( o-DMTTF) 2Cl and (o-DMTTF) 2Br at various temperatures. Solid lines are guides to the eye.\n2. Angular dependence\nLet us now turn to the discussion of the angular-\ndependent CW ESR measurements in which the external\nmagnetic \feld was rotated from Hkc(rotation angle\n\u000b= 0\u000e) toH?c(\u000b= 90\u000e) at various temperatures.\nFigure 4(b) shows the angular dependence of the gfactor\nobtained for ( o-DMTTF) 2Cl at temperatures between\n5 and 200 K. Over the whole temperature range under\nstudy there is no signi\fcant change in the observed g(\u000b)\ndependence visible, which is an important insight for two\nreasons. First, since the symmetry of the gtensor is gov-\nerned by the local surrounding of the resonating spin, theabsence of a change in g(\u000b) shows that the local envi-\nronment remains unchanged even below the spin-Peierls\ntransition which is driven by a lattice dimerization along\nthecaxis of the crystals [22]. The angular dependence\nof thegfactor can be well described by the following\nrelation for a gtensor with uniaxial symmetry [25]\ng(\u000b) =q\ng2\nkcos2(\u000b) +g2\n?sin2(\u000b); (3)\nwheregkandg?are the principal values of the gtensor\ncorresponding to the external \feld applied parallel and\nperpendicular to the caxis, respectively. This type of\nsymmetry is consistent with the shape and the orienta-\ntion of the o-DMTTF molecules within the stacks (see7\nFig. 2) whose molecular planes are oriented perpendicu-\nlar to the stacking axis thereby leading to g?>gk. This\nis in agreement with the measured g(\u000b) dependence. A\nrepresentative \ft of Eq. (3) to the data obtained at 5 K\nis shown by a black solid line in Fig. 4(b). It is worth-\nwhile mentioning that very similar results were obtained\nfor (o-DMTTF) 2Br and are thus not shown here.\nSecond, the almost identical anisotropies of the gten-\nsor at low and high temperatures clearly evidence that\nthe magnetic response observed over the whole investi-\ngated temperature range originates from spins residing\non the sites of the spin chain lattice. Therefore, the ESR\nlines at low-temperatures are not due to magnetic im-\npurities which are unrelated to the spin chains. Instead,\nthese ESR signals can be arguably ascribed to the soli-\ntons extending over several lattice sites in the chain and\ngiving rise to a paramagnetic response on the background\nof the nonmagnetic spin-dimerized state below TSP(cf.\nthe schematic illustration in Fig. 1).\nWhile the angular dependence of the gfactor does\nnot change over the whole temperature range considered\nhere, the angular dependence of the linewidth \u0001 H(\u000b)\nreveals a very pronounced temperature dependence as\nshown in Fig. 4(a) for selected temperatures. Above\n40 K the angular dependence of the linewidth shows a\n(cos2(\u000b) + 1) behavior which is regularly encountered\nin weakly correlated or three-dimensional exchange-\nnarrowed spin systems (see, e.g., Refs. [28, 29] and refer-\nences therein). Below \u001835 K the angular dependence of\nthe linewidth changes gradually and an additional contri-\nbution of the type (3 cos2(\u000b)\u00001)2becomes visible which\nresults in a minimum at \u000b\u001960\u000eand in a local max-\nimum of \u0001 Hat\u000b= 90\u000e. Such a contribution could\nbe associated with additional broadening due to unre-\nsolved lines originating from S= 1 entities (for instance\nsinglet-triplet excitations of the dimerized ground state,\ndenoted as triplons) which are split due to anisotropic\ncouplings (e.g., a dipolar coupling between the triplons)\nand which are centered symmetrically around the reso-\nnance \feld of the main line [30{32]. As illustrated by the\nexemplary spectrum recorded at 20 K [inset in Fig. 3(b)],\nthis additional contribution to the angular dependence\nof the linewidth is, indeed, not visible as a separate fea-\nture in the spectrum and, thus, can be attributed to an\nadditional unresolved contribution to the resonance line.\nLowering the temperature further, \fnally leads to a com-\nplete \rattening of the local maximum around \u000b= 90\u000eat\n4 K. In order to analyze the described changes in the an-\ngular dependence, all \u0001 H(\u000b) dependences measured be-\ntween 4 and 50 K were \ftted with the following empirical\nexpression:\n\u0001H(\u000b) =A(3 cos2(\u000b)\u00001)2+B(cos2(\u000b)+1)+\u0001H0:(4)\nThe parameters AandBare prefactors which scale the\nrespective contribution to the angular dependence of the\nlinewidth and are thus a measure of the relative strength\nof these contributions. The o\u000bset parameter \u0001 H0de-\nscribes the isotropic part of the linewidth arising, for in-stance, from inhomogeneous broadening. From these \fts,\nthe temperature evolution of the relative strength of the\ntriplon and chain or soliton (at low temperatures) contri-\nbutions as expressed in the ratio A=B could be obtained.\nThe resulting temperature dependence of A=B is pre-\nsented in Fig. 4(c). Down to 40 K the ratio is close to zero\nwhich con\frms the pure (cos2(\u000b)+1) character of \u0001 H(\u000b)\nat these intermediate temperatures. Below 40 K, A=B in-\ncreases and reaches a maximum at around 25 and 20 K for\n(o-DMTTF) 2Cl and (o-DMTTF) 2Br, respectively. The\ndecrease of A=B below these maximum temperatures is\nin agreement with a thermally activated behavior anti-\ncipated for singlet-triplet excitations and thus supports\nthe scenario of an unresolved S= 1 contribution as an\norigin of the change in \u0001 H(\u000b). Within the framework of\nthis approach, one could relate the maximum in A=B to\nthe characteristic energy of the triplons which, in turn,\nprovides a spectroscopic estimate for the energy scale of\nthe singlet-triplet gap of the dimerized chain. The maxi-\nmum of the triplon intensity at temperatures comparable\nwith the singlet-triplet gap arises from a combination of\nthe thermally activated excitation of triplet states and\nthe Curie-Weiss-like temperature dependence of the ex-\ncited triplons (see, e.g., [32]). This characteristic maxi-\nmum energy is slightly higher for ( o-DMTTF) 2Cl com-\npared to (o-DMTTF) 2Br. Qualitatively, however, a very\nsimilar temperature dependence of A=B was found for\nboth systems indicating that the observed coexistence of\nseveral distinct excitations (i.e., triplons and solitons) is\na common feature of the investigated compounds. Note\nthat this spin-gap estimate re\rects a mean gap value av-\neraged over all temperatures since the spin gap in a spin-\nPeierls system generally features a pronounced tempera-\nture dependence [9]. Despite the fact that the spin gap\nat zero temperature \u0001(0) could be much larger than our\nspectroscopic estimates (see also the Appendix), the lat-\nter nonetheless represent an experimentally accessible,\nempirical measure of the triplon excitation energies in\n(o-DMTTF) 2X. Regarding the temperature evolution of\ntheA=B ratio at the high-temperature side of the maxi-\nmum, i.e. at temperatures above \u001830 K, we note that the\ndecrease of A=B coincides with the onset of a broaden-\ning of the main line [cf. Fig. 3(a)]. Thus, the additional\ntriplon contribution to the ESR signal, as evidenced by a\nmodulation of the angular dependence of the linewidth,\ncould be masked by the overall broadening of the ob-\nserved resonance line which hampers a detailed analysis\nof the triplon's temperature dependence at these elevated\ntemperatures.\nIt is worthwhile mentioning that in ideal one-\ndimensional spin systems an angular dependence of the\nlinewidth of the form j3 cos2(\u000b)\u00001j4=3is expected [29]\nwhich resembles the observed (3 cos2(\u000b)\u00001)2depen-\ndence. However, such a characteristic angular depen-\ndence of the linewidth due to the reduced dimension-\nality should be visible, if present at all, for all tem-\nperatures below the charge localization temperatures in\n(o-DMTTF) 2Xwhere a spin chain model could be ap-8\nplicable. This is in contrast to the observation of the\n(cos2(\u000b) + 1) dependence above \u001835 K, which could be\ncaused, for instance, by residual interchain couplings and\nconsequently rules out the low dimensionality as an origin\nof the changes in \u0001 H(\u000b).\nIn summary, temperature and angular dependent CW\nESR measurements on ( o-DMTTF) 2Xcrystals showed\nthat the magnetic response in the spin-gap phase below\nTSPis governed by the formation of solitons around defect\nsites or chain breaks and by thermally activated triplons\non the nonmagnetic background of the singlet ground\nstate. In the following section the spin dynamics of the\npinned solitons, investigated by means of pulse ESR mea-\nsurements at low temperatures, will be discussed.\nB. Pulse ESR\nIn order to shed light on the character of the spin dy-\nnamics of pinned solitons in ( o-DMTTF) 2X, nutation\nmeasurements were carried out by means of pulse ESR\nat mw frequencies of about 9.7 GHz and at low temper-\natures. In the case of ( o-DMTTF) 2Cl a free induction\ndecay (FID) detection was used because of an only weak\ninhomogeneous broadening of the ESR line. In such a\ndetection scheme a microwave pulse P R(t) of length tis\napplied to the sample. The FID signal emitted by the\nsample and setting in at the end of P Rcan be recorded\nafter the deadtime \u001cd(\u001880 ns) of the spectrometer. For\nrecording a Rabi oscillation, the pulse length tis varied\nand the intensity of the respective FID is measured (see\nalso [15]). Therefore, the time evolution of the xcompo-\nnent of the magnetization hSx(t)iunder the in\ruence of\na mw magnetic \feld was measured in these experiments.\nOn the other hand, for ( o-DMTTF) 2Br the ESR line was\nfound to be su\u000eciently inhomogeneously broadened to al-\nlow a detection of the nutation by means of primary spin\nechoes using the sequence P R(t)\u0000Tw\u0000\u0019=2\u0000\u001c\u0000\u0019\u0000echo.\nThe waiting time Twafter the end of P Rwas chosen to\nbe larger than the phase memory time Tm. The time \u001c\ndenotes the separation of the two pulses of the standard\nprimary echo sequence. For measuring a Rabi oscillation,\nthe intensity of the spin echo was recorded as a function\noft. As a consequence, the time evolution of the zcom-\nponent of the magnetization hSz(t)iwas measured. Two\nrepresentative nutation measurements are shown for ( o-\nDMTTF) 2Cl and (o-DMTTF) 2Br in Figs. 5(a) and (b),\nrespectively. For both compounds, well-de\fned Rabi os-\ncillations could be observed which is a clear-cut evidence\nfor a coherent spin dynamics in these compounds at low\ntemperatures. Since the ESR response in this temper-\nature region is dominated by solitons, such a coherent\nspin dynamics can be attributed to these quantum ob-\njects which is similar to the results obtained for other\nrelated organic spin chain compounds [14]. As can be\nseen from Figs. 5(a) and (b), the Rabi oscillations are\ndamped which is caused by growing decoherence of the\nsoliton spin nutation. The measured Rabi oscillationscan be described by the following expressions\nhSx(t)i=hSx(0)isin(\n Rt) exp(\u0000t=T\u0003\n2);\nhSz(t)i=hSz(0)icos(\n Rt) exp(\u0000t=T\u0003\n2);(5)\nwhere (\n R=2\u0019) denotes the Rabi frequency of the os-\ncillation and T\u0003\n2is the empirical damping time to ac-\ncount for decoherence [simulations of the measured os-\ncillations by the latter equations are given as red solid\nlines in Figs. 5(a) and (b)]. The two characteristic pa-\nrameters (\n R=2\u0019) andT\u0003\n2can be obtained from these\nexpressions as well as from the peak in the fast Fourier\ntransform (FFT) of oscillation measurements and a \ft\nof the oscillation envelope by an exponential decay func-\ntion, respectively. Rabi frequencies of the oscillations as\na function of mw magnetic \feld hmwat low temperatures\nare shown for ( o-DMTTF) 2Cl in Fig. 5(c) [correspond-\ning data for ( o-DMTTF) 2Br are presented in the inset].\nNote thathmwis given in arbitrary units because the mw\nmagnetic \feld at the sample position in the cavity was\nnot calibrated independently to avoid a potential in\ru-\nence of the reference signal (e.g., from a DPPH standard\nsample) on the Rabi oscillations of the investigated sam-\nples. Absolute values given for hmwin this work are es-\ntimates derived from the measured Rabi frequencies and\nthe corresponding gfactors determined separately from\nCW ESR measurements (see Sec. III A). In these cal-\nculations it is assumed that S= 1=2 for the resonating\nspins which is justi\fed by the spin-1/2 character of the\nsolitons responsible for the ESR response at low temper-\natures. As can be seen in Fig. 5(c), the Rabi frequency\nshows a linear hmwdependence which is emphasized by\nthe solid lines that represent linear \fts to the data. Such\na linear behavior proofs that the Rabi oscillations are\nindeed measured at the resonance position of the ESR\nspectrum (i.e., the static external magnetic \feld and the\nmw frequency ful\fll the resonance condition). Moreover,\nmeasurements with the external magnetic \feld oriented\nparallel and perpendicular to the caxis revealed only a\nsmall anisotropy of the Rabi frequencies, which is caused\nby the small g-factor anisotropy [see Fig. 4(b)] and the\nrelatively small microwave magnetic \feld applied [see also\nEq. (7) below].\nThe temperature dependence of T\u0003\n2measured on a\n(o-DMTTF) 2Cl and a ( o-DMTTF) 2Br sample with\nH?cis presented in Fig. 5(d). Below \u001815 K the relax-\nation times do not show a signi\fcant change upon lower-\ning the temperature. Above 15 K T\u0003\n2decreases strongly\nand above 20 K the broadening of the ESR line, as ob-\nserved in the CW ESR measurements [Fig. 3(a)], pre-\nvents the detection of an FID or spin echo signal since\nthe broadening is related to faster spin relaxation. Thus,\nthe damping of the Rabi oscillations re\rects the behav-\nior of the CW ESR linewidth because both quantities\nare governed by relaxation processes in the spin system.\nCompared to (TMTTF) 2PF6[14], the Rabi oscillations\nin both (o-DMTTF) 2Xsalts are damped more strongly\nand thus show shorter coherence times. As the deco-\nherence responsible for the damping could be caused by9\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s60 /s83\n/s120/s62/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s110/s115/s41/s88 /s32/s61/s32/s67/s108\n/s50/s46/s55/s32/s75/s44/s32 /s72 /s32/s124/s124/s32/s99\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s32/s72 /s32/s94 /s32/s99\n/s32/s72 /s32/s124/s124/s32/s99/s87\n/s82/s32/s47/s32/s50 /s112 /s32/s40/s77/s72/s122/s41\n/s104\n/s109/s119/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s88 /s32/s61/s32/s67/s108\n/s50/s46/s55/s32/s75\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53\n/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s32/s72 /s32/s94 /s32/s99\n/s32/s72 /s32/s124/s124/s32/s99/s88 /s32/s61/s32/s66/s114\n/s55/s32/s75/s104\n/s109/s119/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s87\n/s82/s32/s47/s32/s50 /s112/s32 /s40/s77/s72/s122/s41/s40/s99/s41\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s48/s32/s88 /s32/s61/s32/s67/s108\n/s32/s88 /s32/s61/s32/s66/s114/s84/s42 /s50\n/s32/s40 /s109 /s115/s41\n/s84 /s32/s40/s75/s41/s72 /s32/s94 /s32/s99\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48\n/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s87\n/s82/s32/s47/s32/s50 /s112 /s32/s40/s77/s72/s122/s41/s84/s42 /s50\n/s32/s40 /s109 /s115/s41/s72 /s32/s94 /s32/s99/s32/s88 /s32/s61/s32/s67/s108/s44/s32/s50/s46/s55/s32/s75\n/s32/s88 /s32/s61/s32/s66/s114/s44/s32/s55/s32/s75\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s60 /s83\n/s122/s62/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s110/s115/s41/s55/s32/s75/s44/s32 /s72 /s32/s124/s124/s32/s99/s88 /s32/s61/s32/s66/s114/s40/s100/s41/s40/s97/s41\n/s40/s98/s41\nFIG. 5. Rabi oscillations in ( o-DMTTF) 2Xmeasured at a mw frequency of \u00189.7 GHz. (a) Time evolution of hSx(t)iin the mw\n\feld for (o-DMTTF) 2Cl at 2.7 K with the external magnetic \feld applied along the caxis and a mw magnetic \feld hmwof about\n0.5 mT. (b) Time evolution of hSz(t)iin the mw \feld for ( o-DMTTF) 2Br at 7 K with Hkcandhmw\u00180.25 mT. Solid red lines\nare simulations of the damped oscillations according to Eq. (5). (c) Rabi frequencies (\n R=2\u0019) obtained for ( o-DMTTF) 2Cl\nat 2.7 K and various mw powers, i.e., di\u000berent hmw. Corresponding data for ( o-DMTTF) 2Br at 7 K are shown in the inset.\nNote that the mw magnetic \felds are given in arbitrary units since hmwwas not calibrated independently. Solid lines are\nlinear \fts to the data. (d) Temperature dependence of T\u0003\n2determined from the damping of the Rabi oscillations measured on\n(o-DMTTF) 2Cl and (o-DMTTF) 2Br crystals with hmw\u00180.4 mT (\n R=2\u0019\u001810.5 MHz) and hmw\u00180.6 mT (\n R=2\u0019\u001814.9 MHz),\nrespectively. The external magnetic \feld was applied perpendicular to c. The inset shows the dependence of T\u0003\n2as a function\nof Rabi frequency (and thus hmw) for both compounds measured with H?c.\ndipolar interactions between the solitons as well as by\nsuper-hyper\fne interactions between the solitons and the\nnuclei in their vicinity, a stronger damping would indi-\ncate a stronger (anisotropic) interaction of the solitons\nwith their local environment. On the other hand, it was\nproposed in Ref. [14] that isotropic exchange interactions\nbetween the solitons could stabilize the coherent Rabi\noscillations due to exchange averaging of local inhomo-\ngeneities caused by dipolar or hyper\fne interactions. In\nthis scenario, the exchange between neighboring solitons\ncould be weaker in ( o-DMTTF) 2Xfor instance due tolarger distances between adjacent solitons. Thus it can be\nconcluded that the coherent spin dynamics appears to be\na common property of pinned solitons in organic gapped\nspin chain systems while details of this particular spin\ndynamics, such as characteristic decoherence or damping\nparameters, will depend on the speci\fc structural and\nmagnetic properties of the spin chain compounds which\ndetermine the (anisotropic) interactions between the soli-\ntons as well as their lateral extensions. Moreover, one\nshould note that in addition to the above mentioned con-\ntributions, damping could be caused as well by inhomo-10\ngeneities in the mw magnetic \feld or local distributions\nofgtensors [15, 19]. In these cases, 1/ T\u0003\n2is expected\nto show a linear dependence on the mw magnetic \feld\nand thus depends linearly on the Rabi frequency [15, 19].\nAs a consequence of an inhomogeneous mw magnetic\n\feld distribution within the cavity, the relaxation time\nT\u0003\n2should decrease with increasing sample size. There-\nfore, di\u000berences between absolute values of T\u0003\n2between\n(o-DMTTF) 2Xand (TMTTF) 2X[14] could also be re-\nlated to di\u000berent sample sizes, if inhomogeneities of hmw\ncontribute to the damping. In the inset of Fig. 5(d) the\nrelaxation time T\u0003\n2as a function of Rabi frequency is\nshown for ( o-DMTTF) 2Cl and (o-DMTTF) 2Br at 2.7\nand 7 K, respectively. In these measurements, the ex-\nternal static magnetic \feld was oriented perpendicular\nto thecaxis. Measurements with Hkcyielded very\nsimilar results and are not shown. For both compounds\nunder study a decrease of T\u0003\n2with increasing Rabi fre-\nquency, i.e., with the strength of hmwwas found which\nindicates that inhomogeneities of hmwcould indeed af-\nfect the measured oscillations (although the data do not\nfollow an ideal linear dependence).\nIn order to further investigate the spin dynamics and\npotential origins of spin decoherence in the Rabi os-\ncillations, the spin relaxation times were measured on\n(o-DMTTF) 2Br crystals using pulse ESR techniques.\nIn particular, the phase memory time Tmwas mea-\nsured by means of the primary echo decay method while\nthe transverse relaxation time T2was studied using\nthe so called Carr-Purcell-Meiboom-Gill (CPMG) pro-\ntocol [33, 34]. In the former case the pulse sequence\n(\u0019=2)x\u0000\u001c\u0000(\u0019)x\u0000\u001c\u0000echo was employed and the in-\ntensity of the spin echo was measured as a function of\nthe separation between the pulses \u001c(indices of a pulse\nindicate the direction along which the mw magnetic \feld\nis applied during the pulse). For the T2measurements\nthe sequence ( \u0019=2)x\u0000f\u001c\u0000(\u0019)y\u0000\u001c\u0000echo\u0000gnwas used\nwherendenotes the number of applied \u0019pulses. In\nthe CPMG measurements a transient signal is recorded\nwhich consists, in the ideal case, of nechoes. Analyzing\nthe decrease of the echoes' heights then allows a deter-\nmination of the (e\u000bective) T2(see below). Examples of\nsuch measurements which were conducted at 9 K with\nthe external static magnetic \feld applied perpendicular\nto thecaxis are shown in Fig. 6. In these measure-\nments a\u0019=2-pulse length of 36 ns was used together with\n16-step and 2-step phase cycling in the primary echo\ndecay and CPMG measurements, respectively. For the\nCPMG sequence the number of \u0019pulsesnwas set to 50.\nFor comparison of the relaxation behavior obtained with\nthe two di\u000berent techniques, the measured spin echo in-\ntensities were normalized to the respective intensities at\ntob\u0018807 ns, i.e., the time at which the \frst echo was\ndetected in the CPMG measurement. In this represen-\ntation of the data it becomes evident that, qualitatively,\nthe relaxation in the spin system is faster when a sim-\nple primary echo decay method is employed while the\nCPMG protocol yields a slower relaxation. Such an en-\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s48/s32/s80/s114/s105/s109/s97/s114/s121/s32/s101/s99/s104/s111/s32/s100/s101/s99/s97/s121\n/s32/s67/s80/s77 /s71/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s79/s98/s115/s101/s114/s118/s97/s116/s105/s111/s110/s32/s116/s105/s109/s101/s32/s40 /s115/s41/s88 /s32/s61/s32/s66/s114\n/s57/s32/s75/s44/s32 /s72 /s32 /s32/s99\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s46/s48/s49/s48/s46/s49/s49/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s79/s98/s115/s101/s114/s118/s97/s116/s105/s111/s110/s32/s116/s105/s109/s101/s32/s40 /s115/s41FIG. 6. Comparison of primary echo decay (black\nsquares) and CPMG (open blue squares) measurements on\n(o-DMTTF) 2Br at 9 K with H?c. In both cases a \u0019=2\npulse of 36 ns was used. For the CPMG sequence the number\nof\u0019pulsesnwas set to 50. For comparison, intensities of\nthe spin echoes are normalized to the respective intensities at\ntob\u0018807 ns, the time at which the \frst echo in the CPMG\nmeasurement was detected. Dashed-dotted lines denote mo-\nnoexponential \fts to the data while the solid line represents\nthe \ft of a stretched exponential function to the CPMG data,\nsee text. The measured data are shown on a semi-logarithmic\nscale in the inset to emphasize the increase of decoherence\ntimes by the CPMG protocol as well as the deviation of the\nCPMG data from a monoexponential behavior.\nhancement has been theoretically predicted and exper-\nimentally observed, for instance, in molecular magnets\n(see, e.g., Refs. [35, 36] and references therein) and was\nattributed to contributions of certain spectral di\u000busion\nprocesses, caused by a stochastic modulation of the hy-\nper\fne interactions, to the decoherence of the spins which\nare strongly suppressed by the CPMG pulse sequence.\nIn this case, however, the shape of the relaxation curves\nof the spin echoes is expected to be converted from a\nstretched-exponential behavior in the case of the primary\necho method towards a monoexponential decay in the\ncase of CPMG measurements [36]. In the present study,\nan opposite behavior was found: While the primary echo\ndecay curves could be satisfactorily described by a mo-\nnoexponential decay, the corresponding CPMG measure-\nments clearly deviate from this behavior. In Fig. 6 \fts to\nthe data according to the following expression are shown\nI(tob) =I(0) exp(\u0000(tob=T2;m)b) +Io\u000b: (6)\nHere,I(tob) represents the echo intensity at the obser-\nvation time tob,T2;mdenotes the respective relaxation\ntime (Tmfor primary echo decay and T2for the CPMG\nmethod, respectively), bis the stretching parameter, and\nIo\u000bis an empirical constant o\u000bset to describe the mea-\nsured data. The dashed-dotted lines in Fig. 6 represent11\n\fts withbset to 1, i.e., monoexponential \fts, which yield\nTm\u00182.0\u0016s for the primary echo decay and T2\u00183.7\u0016s\nfor the CPMG measurements. As mentioned above, in\nthe latter case the monoexponential \ft does not yield a\nproper description of the data. The quality of the \ft can\nbe signi\fcantly improved by treating bas an independent\n\ft parameter which results in a strongly reduced e\u000bective\nrelaxation time of about 0.5 \u0016s andb\u00180.36 (the corre-\nsponding \ft is shown as solid line in Fig. 6). In particular\nthe smallbvalue indicates a spread of relaxation times\nwhich inhibits a meaningful quantitative comparison of\nthe relaxation times obtained by employing the two dif-\nferent pulse sequences. Note that \ftting a stretched ex-\nponential to the data of the primary echo measurements\nyields a very similar value for Tmandb\u00180.96, which\nfurther evidences the monoexponential nature of the pri-\nmary echo decay. In addition, the consequences of the\ndi\u000berent applied pulse protocols for the character of the\nspin decoherence are emphasized in the inset of Fig. 6\nwhich presents the measured data on a semi-logarithmic\nscale. In this representation, the data of the primary\necho decay measurements follow a straight line, as ex-\npected for a monoexponential decay, while the CPMG\ndata show a non-linear behavior but overall a slower re-\nlaxation of the spins as compared to the primary echo\nmethod. As pointed out, for instance, in Ref. [35] the\nCPMG method is well-suited to suppress the spin deco-\nherence due to spectral di\u000busion originating from hyper-\n\fne interactions. Despite the qualitative di\u000berences of\nthe relaxation curves observed in the present study and\nin Ref. [35], the relevance of hyper\fne interactions for the\nunderstanding of spin dynamics in ( o-DMTTF) 2Xis evi-\ndenced by the appearance of a modulation of the primary\necho decay curves (see Fig. 6), the so-called ESEEM ef-\nfect [37, 38] which is caused by dipolar (super-) hyper\fne\ninteractions between electron spins and the nuclei in\ntheir vicinity. Thus, it appears to be very instructive to\nstudy details of the hyper\fne interactions between the\nsolitons and their environment by means of more ad-\nvanced, highly sensitive techniques such as HYSCORE\nspectroscopy [38] which is, however, beyond the scope\nof the present study. It is worthwhile mentioning that,\nwithin the temperature range of our CPMG investigation\nbetween 3.6 and 16 K, the strength of deviations of the\nCPMG decay curves from a monoexponential behavior\nincrease with decreasing temperature. This could point\ntowards a temperature-dependence of the spin decoher-\nence mechanism, for instance by virtue of temperature\ndependent spin-lattice relaxation of the (electron or nu-\nclear) spins [35, 36]. In conclusion, it is possible to ef-\nfectively enhance the spin coherence of the solitons in\n(o-DMTTF) 2Br by applying a CPMG pulse sequence.\nHowever, this pulse protocol induces at the same time a\ndistribution of relaxation times which hampers a simple\ncharacterization of the spin coherence in terms of a single\nT2. Whether this peculiar behavior is an artifact of tem-\nperature dependent spin-lattice relaxation processes or\nindeed an intrinsic property of the soliton spin dynam-ics remains an open question requiring further studies.\nNonetheless, we would like to emphasize that the phase\nmemory times Tmof the order of 2 \u0016s determined in the\nmeasurements discussed here are in any case larger than\nthe damping times T\u0003\n2of the Rabi oscillations which did\nnot exceed\u00180.55\u0016s [see Fig. 5(d)]. Therefore, spectral\ndi\u000busion, although potentially active in ( o-DMTTF) 2Br\nas evidenced by the di\u000berence between primary echo and\nCPMG decay curves, appears to be not the main source\nof decoherence in the observed Rabi oscillations. Instead,\nthe relaxation of the soliton spins in the nutation experi-\nments might be dominated by inhomogeneities of the mw\nmagnetic \feld as was proposed in the discussion of the\nhmwdependence of T\u0003\n2(see above).\nIn addition to the temperature and the hmwdepen-\ndence of the nutation measurements presented in the pre-\nceding paragraphs, Rabi oscillations were measured using\ndi\u000berent excitation frequencies centered around the res-\nonance frequency in order to search for the existence of\nadditional contributions to the coherent oscillations, for\ninstance, from S= 1 triplet excitations. The FFTs of the\noscillations obtained from such detuning experiments at\n2.7 K with H?care shown for ( o-DMTTF) 2Cl and\n(o-DMTTF) 2Br in Figs. 7(a) and (b), respectively. In\nthis representation the intensity of the FFT (in arbitrary\nunits) is color coded where blue corresponds to low in-\ntensities while light colors represent regions of high FFT\nintensity. For both investigated salts very similar FFT\nspectra were obtained. In particular, the main frequency\nof the Rabi oscillation is seen as a branch of high inten-\nsity which reveals a square-root-like dependence on the\ndetuning parameter \u0001 = h(\u0017res\u0000\u0017mw). The frequencies\n\u0017resand\u0017mware the resonance frequency of the consid-\nered transition and the mw frequency used for exciting\nthe Rabi oscillations, respectively. Thus, the observed\nmain branch follows well the relation expected for the\nso-called Rabi precession (see, e.g., Ref. [39])\n\nR=2\u0019=p\n\u00012+ (g\u0016Bhmw)2=h ; (7)\nwhich describes the frequency of a transition between\nstates in a two-level system (for instance, a spin-1/2 sys-\ntem in an external static magnetic \feld) driven by an\nexternal electromagnetic \feld. The second branches of\nweaker intensity which are visible for both compounds\nabove the main branches and show the same curvature\nas the respective main branch are due to an instrumental\nartifact and are not related to the intrinsic pulse ESR re-\nsponse of the ( o-DMTTF) 2Xcrystals. Therefore, these\nmeasurements at di\u000berent excitation frequencies showed\nno evidence of further spin contributions aside from the\nS= 1=2 oscillations of the solitons.\nDespite the di\u000berences regarding details of the soli-\nton spin dynamics in organic spin systems discussed here\nand in previous studies [14], the existence of Rabi oscilla-\ntions in these compounds is remarkable, in particular, if\ncompared to other inorganic systems which show similar\ncoherent oscillations [15{20]. For observing the coherent\nspin dynamics in the latter systems, usually a dilution of12\n/s40/s98/s41\n/s88 /s32/s61/s32/s67/s108\n/s50/s46/s55/s32/s75/s44/s32 /s72 /s32 /s32/s99/s88 /s32/s61/s32/s66/s114\n/s50/s46/s55/s32/s75/s44/s32 /s72 /s32 /s32/s99/s40/s97/s41\nFIG. 7. Fourier transform of Rabi oscillations with di\u000berent excitation frequencies at 2.7 K and with H?cfor (a) (o-\nDMTTF) 2Cl and (b) ( o-DMTTF) 2Br, respectively. The intensity of the fast Fourier transform (FFT) is given as color code\nwhere blue corresponds to low and light colors represent high intensities. The external magnetic \feld was set to match\na resonance frequency of about 9.765 GHz which is indicated by gray dashed lines. These measurements were performed\nwith microwave powers which correspond to microwave magnetic \felds of about 0.7 and 0.6 mT for ( o-DMTTF) 2Cl and\n(o-DMTTF) 2Br, respectively.\nmagnetic ions is required in order to reduce anisotropic\ninteractions with the environment of the magnetic ions\nand to reduce interactions between the magnetic ions\n[15]. This emphasizes the important role played by the\nisotropic exchange between the solitons (as pointed out\nin a previous study [14]) and by the fact that solitons are\nbuilt of a highly correlated ensemble of entangled spins.\nBoth mentioned aspects appear to be crucial to protect\nthe Rabi oscillations from fast decoherence and thus ren-\nder them observable in experiments.\nIV. CONCLUSIONS\nIn conclusion, we presented a CW and pulse ESR study\nof (o-DMTTF) 2Cl and (o-DMTTF) 2Br single crystals.\nThese compounds belong to the family of organic gapped\nspin chain systems. Angular and temperature dependent\nCW ESR measurements revealed that the magnetic low-\ntemperature properties are governed by solitons pinned\nto defect sites in the (magnetic) lattice as well as by ther-\nmally activated triplons on the nonmagnetic singlet back-\nground of the dimerized spin chain. For both compounds\nvery similar characteristic temperatures of a maximal\ntriplon contribution were found which indicates that their\nspin-gaps are similar as well. The singlet ground state of\nthe majority of spins in the chains allowed a separate in-\nvestigation of the soliton spin dynamics at low tempera-\ntures by means of pulse ESR experiments. Nutation mea-\nsurements revealed the existence of well-de\fned Rabi os-\ncillations which is evidence for a coherent character of the\nspin dynamics. Studies of the spin relaxation times in the\nsystem by means of standard primary-echo decay and ad-\nvanced CPMG methods revealed an enhancement of thecoherence times as compared to the characteristic damp-\ning times of the Rabi oscillations which shows that the\ndecoherence of the latter is in\ruenced by inhomogeneities\nin local magnetic \felds, such as the mw magnetic \feld.\nAlthough increasing the spin coherence time as compared\nto the primary-echo decay measurements, the application\nof the CPMG protocol induced a spread of spin relax-\nation times in contrast to the behavior expected for spin\nsystems a\u000bected by speci\fc types of spectral di\u000busion\n[35]. Thus, our work should motivate further detailed\ninvestigations of the decoherence mechanisms relevant in\n(o-DMTTF) 2X, for instance the (super-) hyper\fne in-\nteractions between the solitons and their environment.\nTaken together with similar results obtained on other or-\nganic spin chain compounds (Ref. [14]) it appears that\nthe coherence of the spin dynamics is a speci\fc feature\nof pinned solitons in gapped spin chain systems with nat-\nurally occurring defects, potentially caused by isotropic\nexchange interactions between the solitons. As a conse-\nquence, it is possible to coherently manipulate the jmzi\nstates of these quantum objects which might be of inter-\nest for potential application in quantum computation.\nACKNOWLEDGMENTS\nJ.Z. acknowledges \fnancial support from the Ger-\nman Academic Exchange Service (DAAD). ESR mea-\nsurements at IM2NP and BIP Marseille as well as at\nLASIR Lille were supported by the CNRS research in-\nfrastructure RENARD (IR-RPE CNRS 3443).13\nAPPENDIX: ANALYSIS OF THE SPIN\nSUSCEPTIBILITY\nIn this section we analyze the temperature-dependent\nESR intensity measured on a ( o-DMTTF) 2Cl crystal\n[Fig. 3(b)] in order to obtain estimates for the parameters\nof the dimerized chain model which is used to approxi-\nmately describe the spin system in ( o-DMTTF) 2X. As\nmentioned in Sec. III A, an accurate analysis of the \u001f(T)\ndata obtained from ESR measurements is hampered by\nseveral aspects which will be discussed in the following\nbefore presenting the results of the approach employed.\nFirst, a precise evaluation would require the availabil-\nity of absolute values of the magnetic susceptibility in\norder to enable consistency checks of the obtained \ft-\nting parameters. These tests are needed to connect the\nexperimental data with the theoretical predictions, such\nas o\u000bsets to account for diamagnetic or van Vleck para-\nmagnetic contributions or proportionality factors. Al-\nthough the ESR intensity shown in Fig. 3(b) is propor-\ntional to the spin susceptibility, absolute values cannot\nbe obtained with the used setup. This situation is regu-\nlarly encountered in ESR spectroscopy and could be over-\ncome by comparison and scaling to data from static mag-\nnetic susceptibility measurements if the latter is avail-\nable (see, for instance, Refs. [40, 41]). In the case of\n(o-DMTTF) 2Xonly powder measurements of the static\nsusceptibility were reported [22] due to the weak para-\nmagnetic response of the spins in combination with a\nstrong diamagnetic background which required an in-\ncrease of sample mass used for the measurements as well\nas a background correction. Thus, the calibration of ESR\nintensities from single-crystal measurements against cor-\nrected powder measurements of the static susceptibility\nwould introduce uncertainties to the absolute values and\nwas not applied in the present case. Instead, ESR in-\ntensities were normalized to the respective 260 K value\nand consequently did not allow an evaluation of the spin\nsusceptibility's absolute value.\nSecond, when comparing experimental data with theo-\nretical curves one should take into account that measure-\nments are in most cases carried out at constant pressure\npwhile calculations are usually done for systems of con-\nstant volumes V[40]. If thermal expansion is strong,\nas it is the case in organic systems, the di\u000berence be-\ntween the magnetic susceptibility measured at constant\npressure\u001fp=const.can signi\fcantly di\u000ber from the sus-\nceptibility at constant volume \u001fV=const.which requires\ncorrections based on further measurements [40]. Thus\nthe qualitative behavior of the temperature-dependent\nsusceptibility at elevated temperature might be altered\ndue to this e\u000bect. In the present case it cannot be de-\ncided unambiguously whether the non-linear increase of\nESR intensity with increasing temperature is indeed due\nto the 1D AFM-spin-chain character of the spin system\nor due to the e\u000bect of \u001fp=const.(a transformation from\n\u001fp=const.to\u001fV=const.was shown to result in a \rattening\nof the\u001f(T) curve [40]).This aspect is related to the third di\u000eculty en-\ncountered when analyzing the ESR intensity of\n(o-DMTTF) 2Cl: the proper choice of a model to describe\nthe data. Despite the metallic behavior at temperatures\nabove\u0018150 K [22] the temperature dependence of the\nESR intensity shown in Figs. 3(b) and 8 resembles quali-\ntatively the one found for other low-dimensional spin sys-\ntems with a singlet ground state which could be described\nby an isotropic Heisenberg AFM chain above their re-\nspective transition temperatures [9, 43{45]. Although\nthe characteristic maximum in \u001f(T) is not completely\ncaptured in the temperature range up to 260 K used for\nthe measurements discussed here, a non-linear increase is\nnonetheless visible, in particular an in\rection point (at\naround 105 K) which is typical for one-dimensional AFM\nspin chains [9, 26, 27]. However, one should note that\nthe shape of the \u001f(T) curve could be in\ruenced by ther-\nmal expansion of the sample (see previous paragraph).\nThe \rattening of the temperature dependence of the ESR\nintensity after a correction for this e\u000bect was observed\nin other organic spin chain compounds [40] and could,\nin principle, lead to an almost temperature-independent\nspin susceptibility which would be expected for an ideal\nmetal (see, e.g., Ref. [46]). The situation in the systems\nunder study could also be in between these two limit-\ning cases of completely localized charges and delocalized\nmetallic behavior due to an interplay of the hopping be-\ntween the sites tand the on-site Coulomb repulsion U.\nFor particular values of the ratio t=Uthe susceptibility\nwas modeled by Seitz and Klein [47]. In particular, the\ncharacteristic maximum in \u001f(T) was found to be shifted\ntowards smaller temperatures (in units of J) with in-\ncreasingt=U. Thus, the strength of the nearest neighbor\nexchange coupling derived from the temperature of the\nmaximal ESR intensity depends on the speci\fc value of\nt=U.\nSince these parameters are currently not known for\n(o-DMTTF) 2Xwe decided to analyze the ESR intensity\nof (o-DMTTF) 2Cl as a \frst approximation in terms of\na chain of localized spins with a uniform coupling above\nand an alternating coupling below TSPas described by\nthe Hamiltonian given in the Introduction [Eq. (1)]. This\nanalysis was carried out along the lines given in Ref. [9].\nIn a \frst step we approximated the low-temperature part\n(T <20 K) of the normalized intensity curve by \ftting a\nCurie-type law,\n\u001fC(T) =C0\nT; (8)\nto the data, in order to describe the contributions of the\nsolitons to the spin susceptibility using the scaling factor\nC0[see Fig. 8(a)]. The resulting \u001fC(T) was subsequently\nsubtracted from the measured data to obtain the con-\ntribution of the spin chain for further analysis [red cir-\ncles in Fig. 8(b)]. The temperature range above TSPwas\nthen \ftted using the model of an S= 1=2 AFM uniform\nHeisenberg chain ( \u000e= 0) [9] to determine Jabove the14\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s32/s32/s67/s111/s114/s114/s101/s99/s116/s101/s100/s32/s69/s83/s82/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121 \n/s32/s32/s72/s101/s105/s115/s101/s110/s98/s101/s114/s103/s32/s65/s70/s77/s32/s99/s104/s97/s105/s110/s32/s109/s111/s100/s101/s108/s32/s40 /s100 /s32/s61/s32/s48/s41\n/s32/s32/s65/s108/s116/s101/s114/s110/s97/s116/s105/s110/s103/s32/s99/s104/s97/s105/s110/s32/s109/s111/s100/s101/s108/s32/s40 /s100 /s32/s62/s32/s48/s41\n/s32/s32/s66/s117/s108/s97/s101/s118/s115/s107/s105/s105/s32/s109/s111/s100/s101/s108/s32/s40 /s100 /s32/s62/s32/s48/s41/s67/s111/s114/s114/s101/s99/s116/s101/s100/s32/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s115/s117/s115/s99/s101/s112/s116/s105/s98/s105/s108/s105/s116/s121\n/s84 /s32/s40/s75/s41/s88 /s32/s61/s32/s67/s108\n/s72 /s32/s124/s124/s32/s99/s44/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s32/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s69/s83/s82/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s32/s67/s117/s114/s105/s101/s32/s102/s105/s116/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s115/s117/s115/s99/s101/s112/s116/s105/s98/s105/s108/s105/s116/s121\n/s84 /s32/s40/s75/s41/s88 /s32/s61/s32/s67/s108\n/s72 /s32/s124/s124/s32/s99/s44/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122/s40/s97/s41 /s40/s98/s41\nFIG. 8. Analysis of the temperature-dependent normalized susceptibility as obtained from CW ESR measurements at 9.6 GHz\non a (o-DMTTF) 2Cl crystal with external \feld applied parallel to c. (a) Fit of a Curie law (blue solid line) to the low-\ntemperature part of the ESR intensity to correct the data with respect to the soliton contribution. (b) Corrected data after\nsubtraction of the \ft shown in (a) from the data. Dashed-dotted line denotes a \ft for T >50 K using the model of a Heisenberg\nAFM chain ( \u000e= 0) from Ref. [9] in order to determine the exchange coupling constant J. The \ft in the low-temperature\nregion for\u000e>0 and \fxed Jis shown by the blue solid line. For comparison a \ft within the same temperature range using the\nBulaevskii model [42] is indicated by the orange dashed line.\nspin-Peierls transition by\n\u001fchain(T) =\u001f0+Cchain\nJ(1 +\u000e)\u001f\u0003(\u000e;T); (9)\nwhere\u001f\u0003is the theoretical spin susceptibility of the al-\nternating chain according to Eq. (56a) in Ref. [9]. The\nlatter equation is the speci\fc form of the general expres-\nsion Eq. (50a) in Ref. [9] and is the relevant equation for\nthe case of an alternating chain with \u000eas a parameter.\nSetting\u000eto zero recovers the susceptibility of the uniform\nHeisenberg chain [9]. The corresponding \ft yielded a Jof\nabout 500 K and is shown in Fig. 8(b) by a dashed-dotted\nline where the temperature range was extended to 350 K\nto visualize the maximum in \u001f(T). Considering the \ft we\nnote that there are two di\u000eculties when evaluating the\nreliability of the obtained J. First, a negative o\u000bset \u001f0\nof about -1.3 was necessary to describe the data by the\nmodel which seems to be unphysical but could be caused\nby the normalization of the ESR intensities due to the\nlack of absolute values. And second, the absence of pre-\ncise quantitative data prevents consistency checks of the\nobtained proportionality factor Cchain (not only the po-\nsition but also the height of the maximum in \u001f(T) is de-\n\fned byJ). Thus, we emphasize that the Jand\u000evalues\npresented in this section are merely \ftting parameters of\nthe simpli\fed model necessarily applied in the analysis.\nInterestingly, our value of Jis of the same order of mag-\nnitude as the exchange constants determined for other\norganic spin chain compounds such as (TMTTF) 2PF6\n[40]. Thus, J\u0018500 K appears to be a reasonable valuefor the isotropic exchange coupling in ( o-DMTTF) 2Cl\naboveTSP. By comparing the values obtained for C0and\nCchain which are proportional to the number of solitons\nand spins in the chains, respectively, the concentration of\nthe solitons csoliton =Nsoliton=Nspincould be estimated\nto be 0.0003 using a gfactor of 2.006 for this orienta-\ntion of the sample and S= 1=2. Note, however, that\nthe obtained Cchain might be in\ruenced by the o\u000bset pa-\nrameter\u001f0. Indeed, a comparison of the value of the\ncorrected normalized ESR intensity [Fig. 8(b)] at the in-\n\rection point at around 105 K with the theoretical curve\nfor the uniform chain ( \u000e= 0) [26] with J\u0018500 K yields a\nsmaller amount of spins in the chain and, thus, a value of\nthe soliton concentration of about 0.0009. Although be-\ning larger than the concentration obtained directly from\nthe \ft parameters, the concentration determined by this\nmethod is still of the same order of magnitude as the\nprevious estimate.\nWithin the framework of the approximations contained\nin the analysis of the high-temperature part of the spin\nsusceptibility, we described the ESR intensity below TSP\nby a model of an alternating spin chain to obtain an\nestimate for the dimerization parameter \u000eaccording to\nRef. [9]. For \ftting the ESR intensity below 50 K, we\nkept theJobtained in the previous step constant and\nused\u000e,\u001f0, andCchain in Eq. (9) as \ft parameters. As a\nresult we obtained \u000e\u00180.25 but also values for \u001f0(\u00180),\nandCchain which di\u000ber from the \ft above TSP[the \ft is\nshown as solid blue line in Fig. 8(b)]. Moreover, the ob-\ntained\u000erather constitutes an estimate for the dimeriza-15\ntion parameter at very low temperatures ( T <<T SP) be-\ncause\u000eas well asJcan show a temperature dependence\ndue to the gradual development of the lattice dimeriza-\ntion at the spin-Peierls transition [9]. Therefore, keeping\nJconstant during the \ftting procedure could in\ruence\nthe obtained \u000e. Furthermore, we compared \u000efrom the\n\ft according to Johnston et al. [9] with the model by\nBulaevskii [42] which has been used in previous studies\nto estimate the dimerization within the chains [40, 44].\nFrom the corresponding \ft, which is shown as dashed\norange line in Fig. 8(b), we could derive a \u000eof 0.25 in\ngood agreement with the previous value. This analysis\nrelies, however, on the same value of Jdetermined from\nthe high-temperature part of \u001f(T) and thus serves only\nas a self-consistency check of the employed \ftting pro-\ncedure. From the obtained estimates of Jand\u000eit is\npossible to calculate the spin gap of the alternating spin\nchains. Utilizing the expression \u0001 = 2 J\u000e3=4according\nto Barnes et al. [8] yields \u0001\u0018350 K, which is an esti-mate for the zero-temperature gap since the obtained \u000e\nvalue represents the zero-temperature limit of the dimer-\nization parameter. Note that the derived value of \u0001 is\nmuch larger than the spectroscopic estimate of the order\nof 25 K (see Sec. III A) which might be caused either by\nthe pronounced temperature dependence of the spin gap\n(see, for instance, Ref. [9]) or the uncertainties in the\ndetermination of Jand\u000e, respectively.\nDue to the above-mentioned approximations used and\nthe limitations caused by the normalization of the ESR\nintensity one should, faute de mieux, consider the abso-\nlute values obtained for J,\u000e, \u0001, andcsoliton as order-of-\nmagnitude estimates only. 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B 9, 2159 (1974)."    },    {        "title": "1310.7847v1.Self_Quenching_of_Nuclear_Spin_Dynamics_in_Central_Spin_Problem.pdf",        "content": "arXiv:1310.7847v1  [cond-mat.mes-hall]  29 Oct 2013Self-Quenching of Nuclear Spin Dynamics in Central Spin Pro blem\nArne Brataas1and Emmanuel I. Rashba2\n1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Physics, Harvard University, Cambridge, Mas sachusetts 02138, USA\nWe consider, in the framework of the central spin s= 1/2 model, driven dynamics of two electrons\nin a double quantum dot subject to hyperﬁne interaction with nuclear spins and spin-orbit coupling.\nThe nuclear subsystem dynamically evolves in response toLa ndau-Zener singlet-triplet transitions of\ntheelectronic subsystemcontrolled byexternalgate volta ges. Withoutnoise andspin-orbit coupling,\nsubsequent Landau-Zener transitions die out after about 104sweeps, the system self-quenches, and\nnuclear spins reach one of the numerous glassy dark states. W e present an analytical model that\ncaptures this phenomenon. We also account for the multi-nuc lear-specie content of the dots and\nnumerically determine the evolution of around 107nuclear spins in up to 2 ×105Landau-Zener\ntransitions. Without spin-orbit coupling, self-quenchin g is robust and sets in for arbitrary ratios\nof the nuclear spin precession times and the waiting time bet ween Landau-Zener sweeps as well as\nunder moderate noise. In presence of spin-orbit coupling of a moderate magnitude, and when the\nwaiting time is in resonance with the precession time of one o f the nuclear species, the dynamical\nevolution of nuclear polarization results in stroboscopic screening of spin-orbit coupling. However,\nsmall deviations from the resonance or strong spin-orbit co upling destroy this screening. We suggest\nthat the success of the feedback loop technique for building nuclear gradients is based on the eﬀect\nof spin-orbit coupling.\nPACS numbers:\nI. INTRODUCTION\nElectrical operation of electron spins in semiconduc-\ntor double quantum dots (DQD) is one of the cen-\ntral avenues of semiconductor spintronics1and quantum\ncomputing.2–5There are three basic types of electronic\nspin qubits, (i) the Loss-DiVincenzo2qubits operating\nsingle-electron spins, (ii) singlet-triplet qubits operating\na two-electron system and (iii) three electron qubits.6\nThe second type is the center of our attention. Most\nwidely explored singlet-triplet DQD qubits7are based on\nGaAs8,9andInAs.10–12BothinGaAsandInAs, thereare\nthree species of nuclei possessing non-vanishing angular\nmomenta, and the coupling between electron and nuclear\nspins (mostly through contact interaction) strongly in-\nﬂuences electron-spin dynamics. Primarily, this coupling\nhas a destructive eﬀect causing electron spin relaxation,\nand many theoretical studies have focused on the chal-\nlenging problem of determining the relaxation rate of an\nelectron spin interacting with about N≈106nuclear\nspins.13–20The problem of an electron spin interacting\nwith a bath of nuclear spins is known as the central spin\nproblem.\nHowever, a controllable nuclear spin polarization, act-\ning as an eﬀective magnetic ﬁeld, can also become a re-\nsource for manipulating electron spins.21,22In particu-\nlar, the diﬀerence (gradient) in the nuclear polarization\nof the left and right dots can be used for σxrotations\nof aS-T0singlet-triplet qubit on the Bloch sphere.8Ef-\nﬁcient control of a vast ensemble of nuclear spins is very\nchallenging, and many analytical and numerical works\nhave been carried on this subject.23–31The principal ex-\nperimental tool for polarizing nuclear spins and building\ngradients is based on driving a two-electron DQD elec-trically through the avoided crossing ( S-T+anticrossing)\nof its singlet level Sand theT+component of its elec-\ntronic triplet T= (T+,T0,T−).T+is the lowest energy\ntriplet component because the electron g-factor is nega-\ntive,g <0, both in GaAs and InAs. The width of the\nanticrossing is controlled by hyperﬁne and spin-orbit32\ninteractions. When the electron state changes from S\ntoT+orvice versa by passaging through the S-T+an-\nticrossing, and there is no spin-orbit coupling, up to one\nquantum of the angular momentum is transferred to the\nnuclear subsystem, and such transfer facilitates polariz-\ning the nuclear bath by performing multiple passages.\nUnfortunately, experimental data show that the nu-\nclear polarization saturates at a rather low level, typi-\ncally of about 1%.33The origin of this low saturation\nlevel remains unclear and constitutes the critical obsta-\ncle for achieving higher levels of nuclear spin polariza-\ntion. We recently uncovered a mechanism of dynamic\nself-quenching which, in absence of spin-orbit (SO) cou-\npling, results in fast suppression ofthe transversenuclear\npolarization under stationary pumping.30This is caused\nby screening the random ﬁeld of the initial nuclear spin\nﬂuctuation by the nuclear polarization produced through\npumping and closing the anticrossing. This conclusion is\nin a qualitative agreement with the data of Refs. 25,31\nin the strong magnetic ﬁeld limit, and the states with\nvanishing transverse magnetization are known as “dark\nstates”. Meantime, by applying feedback loops, exper-\nimenters managed to achieve considerable and control-\nlable nuclear spin polarizations.34This poses a challeng-\ning question in which way closing the S-T+intersection\ndue to the self-quenching mechanism could be avoided.\nOur data of Ref. 30 indicate that SO coupling changes\nthepatternsofself-quenchingdramatically,whichimplies\nthat it is the spin-orbit coupling that might resolve the2\nproblem. The main goal of the current paper is to shed\nmore light on the mechanisms controlling the transfer of\nangularmomentumfromtheelectronqubittothenuclear\nbath. For this purpose, we solve numerically the equa-\ntions describing coupled electron and nuclear spin dy-\nnamics for DQDs of a realistic size of more than N∼106\nnuclear spins and a shape of two overlapping Gaussian\ndistributions. These simulations follow up to 2 ×105\nsweeps and unveil intimate patterns of transferring spin\npolarizationfromtheelectronictothenuclearsubsystem.\nTo get quantitative insight onto the long time dynam-\nics of spin pumping by multiple passagesacross the S-T+\nanticrossing, we restrict ourselves to the strong magnetic\nﬁeld regime when the Zeeman split-oﬀ T0andT−compo-\nnents of the electron spin triplet are well separated from\ntheSandT+states, hence, transitionsto these statesare\ndisregarded. Therefore, with a semiclassical description\nof nuclear spins, electrons form a two-level system, and\npassages across the S-T+anticrossing are described by\nthe Landau-Zenertype theory.35,36The detailed patterns\ndepend on the shape on the pulses on the gates and the\ninstantaneousnuclearconﬁguration. Inturn, duringeach\nLZ sweep the nuclear conﬁguration changes due to the\ndirect transfer of the angular momentum and shake-up\nprocesses.29,30Between the LZ sweeps, this conﬁguration\nchanges because of the diﬀerence in the Larmor preces-\nsion rates of diﬀerent nuclear species. To follow the long\nterm evolution, we solve the problem of the coupled dy-\nnamics self-consistently. From the mathematical point of\nview, we arrive to a central spin s= 1/2 problem with a\ndriven dynamics of the electron spin. Hence, beyond the\napplication to the spin pumping problem, our results are\nof general interest for coupled dynamics of many body\nsystems.\nIn this paper we prove, both analytically and numer-\nically, that self-quenching into dark states is a generic\nproperty of the pseudospin s= 1/2 model in absence of\nSO coupling, and that self-quenching sets in after only\nabout 104sweeps. We also demonstrate that this result\nstands under a moderate noise. However, the main fo-\ncus of the paper is on the eﬀect of SO coupling. Because\nthe SO ﬁeld is static while the hyperﬁne Overhauserﬁeld\noscillates in time with the Larmor frequencies of nuclei,\nself-quenching cannot set in. Nevertheless, if the wait-\ning time between LZ sweeps coincides with the Larmor\nperiod of one of the species, self-quenching sets in stro-\nboscopically (as was demonstrated in our previous paper\nfor a single-specie model30). More speciﬁcally, the Over-\nhauser ﬁeld of the resonant specie screens the SO ﬁeld\nduring the LZ sweeps (whose duration is small compared\nwith Larmor periods). Therefore, during the sweeps the\nelectron and nuclear subsystems become decoupled. As\ndistinct from self-quenching in absence of SO coupling,\nthe stroboscopic self-quenching is fragile. Even a small\ndeviationfromtheresonance,about1%, destroysthedel-\nicatecompensationoftheSOandhyperﬁnecontributions\nduring the LZ sweeps. Moreover, we were able to observe\nthe stroboscopic self-quenching only for moderate valuesof the SO coupling that do not exceed considerably the\nrandom ﬂuctuations of the hyperﬁne ﬁeld.\nWe conclude that it is the SO coupling that endows\nthe nuclear subsystem with a long term dynamics under\nthe stationary LZ pumping. Therefore, we suggest that\nSO coupling is critical for eﬃcient operating the feedback\nloops that requireaccumulation of largepolarizationgra-\ndients at the scale of about 106sweeps.34Eﬀect of SO\ncoupling at a shorter time scale has been recently un-\nveiled by Neder et al.37by comparing with experimental\ndata of Ref. 38.\nII. OUTLINE AND BASIC RESULTS\nIn Sec. III, following two introductory sections, we\npresent the basic equations of the driven coupled\nelectron-nuclear spin dynamics of the central spin-1 /2\nproblem that is the basis for all following calculations. In\nSec. IV, we ﬁnd an analytical solution for a simple model\ndemonstrating the phenomenon of self-quenching which\nreveals basic factors controlling its rate. Our numerical\ntechnique that allows following the coupled dynamics of\nthe electron 1 /2-pseudospin and about 107nuclear spins\nduring up to 2 ×105LZ sweeps is described in Sec. V. It\nalso includes parameters of the double quantum dot and\nLZ pulses used in simulations.\nSec. VI is the central one. It opens with the nuclear\nparameters of InAs and GaAs used in simulations, and\nincludes the results of simulations and their discussion.\nIn this section, we demonstrate that self-quenching is a\ngeneric and robust property of the coupled dynamics in\nabsence of spin-orbit coupling, and analyze its speciﬁc\nfeatures in systems consisting of two and three nuclear\nspecies. Next, we introduce SO coupling and demon-\nstrate that it eliminates self-quenching and causes the\nnuclear subsystem to exhibit a persistent, but irregu-\nlar, oscillatory dynamics. We also demonstrate the phe-\nnomenonofstroboscopicself-quenchingthat sets in when\nthe waiting time between LZ sweeps is in resonance with\nthe Larmor period of one of the nuclear species and show\nthat it is very sensitive to deviations from the exact res-\nonance. Finally, we suggest that the SO induced nuclear\ndynamicsiscriticalforthe feedbacklooptechniquedevel-\noped by Bluhm et al.34for building controllable nuclear\npolarization gradients.\nWe summarize our results in Sec. VII and estimate\nthe strength of SO coupling in InAs and GaAs double\nquantum dots in Appendix A.\nIII. BASIC EQUATIONS\nHyperﬁne electron-nuclear interactions and SO cou-\nplinggovernthecouplingbetweenelectronstatesin A3B5\nquantum dots utilized for quantum computing purposes.\nNuclear spins are dynamic and can be controlled by ma-\nnipulating magnetic ﬁelds and electronic states.3\nWe consider electrons in double quantum dots inter-\nacting with nuclear spins via the hyperﬁne interaction.\nWhen there are two electrons in the dot, the orthogonal\nbasis consists of singlet and triplet spin states. Hyperﬁne\nand SO interactionscouple these states. By changing the\ngate voltages that conﬁne electrons and determine sin-\nglet and triplet energies, a transition from a singlet Sto\na triplet electron state T+(or vice versa) is accompanied\nby a change in the nuclear spin states. Our focus is on\nwhat happens to the nuclear spins as we repeat Landau-\nZener (LZ) transitions many times, up to 2 ×105, and\nhow the changes in the nuclear spin states in turn aﬀect\nelectrons in the quantum dot.\nWe deﬁne a LZ sweep in the following way. We assume\nthat the quantum dot is ﬁrst set in the singlet state, then\na change in the gate voltages drives a (partial) transition\nto the triplet state, and ﬁnally one electron is taken out\nof the system and re-inserted so that the system again\nis in its singlet state. During the sweep, the dynamics\nof the electronic qubit is controlled by the electric ﬁeld\nproduced by the gates and the nuclear polarization as\ndescribed by Eq. (5) below. In turn, semiclassical dy-\nnamics of nuclear spins is driven by the Knight ﬁelds\n∆jλ(t) arising from electron dynamics\n/planckover2pi1dIjλ\ndt=∆jλ×Ijλ, (1)\nwhere the sub index jλdenotes a nuclear specie λat a\nlattice site j. Assuming the time-scale TLZof the LZ\nsweeps is much shorter than the nuclear precession times\ntλin the external magnetic ﬁeld, the total eﬀect of the\ntime-dependent ﬁelds ∆jλ(t) on each nuclear spin can\nbe integrated over the LZ sweep. Then the change of an\nindividual nuclear spin during a sweep is\n△Ijλ=Γjλ×Ijλ, (2)\nwhereΓjλaccounts for the eﬀective magnetic ﬁeld in-\nduced by the hyperﬁne interaction during the LZ sweep\nand depends on the conﬁguration of all the nuclear spins\nbefore the sweep.\nLandau-Zener sweeps are repeated many times. Be-\ntween consecutive LZ sweeps, electrons are in the singlet\nstate and do not interact with nuclear spins. During\nthis waiting time Twbetween consecutive sweeps, nu-\nclear spins precess in an external magnetic ﬁeld Bap-\nplied along the z-direction. The changes of the nuclear\nspins between LZ sweeps are\n△Ix\njλ= cosφjλIx\njλ−sinφjIy\njλ, (3a)\n△Iy\njλ= cosφjλIy\njλ+sinφjλIx\njλ, (3b)\n△Iz\njλ= 0, (3c)\nwhere the superscripts x,y, andzdenote Cartesian com-\nponentsofthenuclearspins,thetransversephasechanges\nareφjλ=−2πTw/tλin terms of the spin precession\ntimestλ= 2π/planckover2pi1/gλµIB, wheregλ=µλ/Iλis theg-factor\nfor a nuclear specie λ,µλis its magnetic moment, and\nµI= 3.15×10−8eV/T is the nuclear magneton.We also model the inﬂuence of noise by adding phe-\nnomenologically a random magnetic ﬁeld along the z-\ndirection for each nuclear spin so that the accumu-\nlated phases in Eq. 3 change to φeﬀ\njλ=φjλ+φnoise\njλ→\n−2πTw(1/tλ+rjλ/τ), whererjλare random numbers\nin the interval from −1 to 1. This procedure simulates\na randomization of the transverse components of nuclear\nspinsafteratime ofthe order τ, andτistermedthe noise\ncorrelation time in what follows. While simulations de-\nscribed below were performed by using random sets of\nrjλ, we mention that averaging over the noise results ef-\nfectivelyinchangingthephase-dependentfactorsinEq.3\nas∝an}bracketle{tcosφeﬀ\njλ∝an}bracketri}htn= cosφjλsin(2πTw/τ)/(2πTw/τ), and sim-\nilarly for ∝an}bracketle{tsinφjλ∝an}bracketri}htn. Therefore, this model of transverse\nnoise leads to a semiclassical dephasing of the transverse\ncomponents of nuclear spins on the time scale τ.\nLet us next review how electronic Landau-Zener\nsweeps inﬂuence nuclear spins via Γjλ.29The hyperﬁne\nelectron-nuclear interaction is\nHhf=Vs/summationdisplay\nλAλ/summationdisplay\nj∈λ/summationdisplay\nm=1,2δ(Rjλ−rm)(Ijλ·s(m)),\n(4)\nwheres(m) =σ(m)/2 are the electron-spin operators\nin terms of the vector of Pauli matrices σ(m) for each\nelectronm= (1,2),Ijλare the nuclear spin operators,\nAλis the electron-nuclear coupling constant for a specie\nλ, andVsis the volume per single nuclear spin. We\nconsider GaAs or InAs quantum dots below; hyperﬁne\ncoupling parameters for them can be found in Sec. VI.\nAssuming that gate voltages keep the system close to\nthe singlet S- tripletT+transition, the eﬀective Hamil-\ntonian describing the electron qubit is\nH(ST+)=/parenleftbigg\nǫSv+\nv−ǫT+−η/parenrightbigg\n, (5)\nwhereǫSis the singlet energy and ǫT+is the triplet T+\nenergy in the external magnetic ﬁeld B=Bˆzwhen nu-\nclear spins are unpolarized. By retaining only SandT+\nstates, the problem is reduced to a 1/2 pseudospin prob-\nlem, and we apply the term the central spin problem in\nthis sense.\nThe energies ǫSandǫT+are controlled by the gate\nvoltages. The oﬀ-diagonal components v±=v±\nn+v±\nSO,\ncoupling the singlet Sand triplet T+states, contain con-\ntributions from nuclear spins\nv±\nn=Vs/summationdisplay\nλAλ/summationdisplay\nj∈λρjλI±\njλ, I±\njλ= (Ix\njλ±iIy\njλ)/√\n2,(6)\nand SO coupling v±\nSO.26,39When nuclear spins are po-\nlarized, the energy of the triplet state is aﬀected by the\nOverhauser shift\nη=−Vs/summationdisplay\nλAλ/summationdisplay\nj∈λζjλIz\njλ. (7)\nThe singlet-triplet electron-nuclear couplings are\nρjλ=/integraldisplay\ndrψ∗\nS(r,Rjλ)ψT(r,Rjλ), (8)4\nand the electron-nuclear couplings in the T+state are\nζjλ=/integraldisplay\ndr/vextendsingle/vextendsingleψT(r,Rjλ)2/vextendsingle/vextendsingle, (9)\nwhereψS(ψT) is the orbital part of the singlet (triplet)\nwavefunction. Beyondthe2 ×2S-T+model, itispossible\nto deﬁne a hyperﬁne term that determines the singlet S-\ntripletT0coupling(asinEq.4in Ref. 29), but it isnot at\nthe centerof ourattention and will not be discussed here.\nThe eﬀect of the electron spin T0andT−components is\ncritical for the development of nuclear polarization gra-\ndients and has been investigated in Refs. 25,31.\nIn terms of these parameters, the changes of the nu-\nclear spins △Ijλ=Γjλ×Ijλduring a Landau-Zener\nsweep are determined by coeﬃcients\nΓ(x)\njλ=−VsAλρjλ(Pvy+Qvx)/(2v2),(10a)\nΓ(y)\njλ=VsAλρjλ(Pvx−Qvy)/(2v2), (10b)\nΓz\njλ=VsAλζjλR/(2v), (10c)\nwithv2=|v+|2=/parenleftbig\nv2\nx+v2\ny/parenrightbig\n/2. In these expressions,\n0≤P≤1 is theS-T+transition probability, a real\nnumberQis the shake-up parameter deﬁned via29\nP+iQ=−i2v−/integraldisplayTLZ\n−TLZdt\n/planckover2pi1cS(t)c∗\nT+(t) (11)\nin terms of the singlet (triplet) amplitude cS(t) (cT(t)),\nand\nR= 2vTLZ/integraldisplay\n−TLZdt/vextendsingle/vextendsinglecT+(t)/vextendsingle/vextendsingle2//planckover2pi1 (12)\naccounts for the Overhauser shift due to the triplet\nT+component of the electron state during the interval\n(−TLZ,TLZ). In the absence of SO coupling, v±\nSO= 0,\nthe change ∆ Izof the total angular momentum of nuclei\nIz=/summationtext\njIz\njduring a single sweep equals ∆ Iz=−P, as\nfollows from the angular momentum conservation.29\nUsing the amplitudes ( cS(t),cT+(t)) found from solv-\ning the time-dependent Schr¨ odinger equation with the\nHamiltonian H(ST+)of Eq. (5) in combination with the\ndynamical equations for nuclear spins of Eq. (2) makes\nour approach completely self-consistent.\nWe assume that electrons are loaded into the singlet\n(0,2) state with energies far away from the S-T+an-\nticrossing. After loading electrons, gate voltages are\nchanged to bring the system closer to the level anticross-\ning, and this change is performed fast enough to keep\nthe system in the singlet state.40From there on, an LZ\nsweep brings the system through the anticrossing. After\nthe slow sweep, the system is moved back to the recharg-\ning point where it exchanges electrons with the reservoir.\nThis back motion is fast at the scale of the narrow anti-\ncrossingand therefore does not inﬂuence the nuclear spin\nsubsystem, but slow at the scale of the electron Zeeman\nsplitting to keep the system inside the S−T+subspace.IV. SIMPLE MODEL WITH ANALYTICAL\nSOLUTION\nLet us ﬁrst present a simpliﬁed model that can be\nsolved analytically and that manifests basic features of\nself-quenching30in the absence of SO coupling, v±\nSO= 0.\nDiﬀerent versions of this “box” or “giant spin” model\nwere applied to various problems, see Refs. 25,26,41,42.\nSubsequently, we will in Section V outline a more com-\nplex and extensive numerical procedure and discuss nu-\nmerical results for realistic models in Section VI. Re-\nstricting ourselves to a single nuclear specie, we sim-\nplify the system by modelling it as a box inside which\nthe electron wave functions are independent on posi-\ntion, all nuclei possess the same Larmor frequency, and\nall hyperﬁne coupling constants Aλand electron-nuclei\ncouplignsρjλare equal, Aλ=¯Aandρjλ= ¯ρ; typi-\ncally,¯A≈10−4eV. Then Eqs. (6) and (8) simplify to\nvα=A0Iα,α= (+,−,z) withA0=Vs¯A¯ρ∼¯A/N. Here\nNis the number of nuclei in the box, and Iα=/summationtext\njIα\nj\nare components of the “giant” collective angular momen-\ntum of nuclei. In the framework of this model, nuclear\nspin precession in the Zeeman and Overhauserﬁelds does\nnot inﬂuence the coupled electron-nuclearspin dynamics,\n∆z= 0. With these assumptions, Eq. (1) becomes\ndI+\ndt=−i\n/planckover2pi1∆+Iz,dI−\ndt=i\n/planckover2pi1∆−Iz,dIz\ndt=i\n/planckover2pi1(∆+I−−∆−I+).\n(13)\nForLZ pulses, the energylevel diﬀerence changeslinearly\nwitht,ǫS(t)−ǫT+(t) =β2t//planckover2pi1, for−TLZ≤t≤TLZ, and\nthedynamicsofthequbit iscontrolledbyadimensionless\nparameterγ=v+v−/β2. The equation of motion for γ\nfollowing from (13) is\ndγ\ndt=−A2\n0\n2β2d\ndt(Iz)2. (14)\nDuring each sweep, Izchanges by ∆ Iz=−Pandγ\nchanges by ∆ γ= (A2\n0/β2)PIz. Precession of the collec-\ntive nuclear spin Iin the external magnetic ﬁeld during\nthe waiting times between sweeps changes neither Iznor\nγand is disregarded. The discrete number of sweeps\nncan be considered as a continous variable since the\nchanges ∆γand ∆Izduring a single sweep are small as\ncompared to γandIz. We then arrive at the diﬀerential\nequations that determine the evolution of the γ(n) and\nIz(n):\ndγ\ndn=A2\n0P\nβ2Iz, (15a)\ndIz\ndn=−P. (15b)\nThis central result for the simple model clariﬁes the dif-\nferent modes of self-quenching. The evolution of the LZ\nparameterγthat controls the LZ probability Pdiﬀers\nin two scenarios that manifest themselves for opposite\nsigns ofIz. (i) When Izis initially negative, it continues5\nto decrease (becoming more negative) and magnitudes of\nbothγand the LZ probability Pdecrease, hence, the\nprocess slows down. Finally, self-quenching sets in ex-\nponentially, see Eq. 19 below. (ii) When Izis initially\npositive,γﬁrst increases so that the LZ probability P\nbecomes larger. However, since Izonly can be reduced,\nit eventually becomes negative and self-quenching of sce-\nnario (i) sets in. So, self-quenching ultimately sets in\ngenerically independent on the original sign of Iz.\nWe can get a more detailed insight into the self-\nquenchingdynamicsbyusingthe ﬁrstintegralofEq.1443\nγ=−(A2\n0/2β2)(Iz)2+γ0, (16)\nwhereγ0is an integration constant that depends on the\ninitial values of γandIz,γiandIz\ni. Obviously, Eq. 16\ndictates that γ≤γ0, andγ0≥0 becauseγ≥0 by\ndeﬁnition. Therefore,\nIz=±√\n2(β/A0)√γ0−γ, (17)\nand\ndγ/dn=±√\n2(A0/β)√γ0−γP(γ).(18)\nIn scenario (i), when Iz\ni<0, the minus sign should be\nchosen in Eqs. 17 and 18, and γ(n) decreases monoton-\nically. In scenario (ii), when Iz\ni>0, the dynamics ﬁrst\nfollows the plus branches of Eqs. 17 and 18, and γ(n)\nincreases until it reaches its maximum value γ=γ0. At\nthis point, Iz(n) vanishes, changes sign, and continues\nto decrease as follows from Eq. 16 and Eq. 15b [because\nP(γ0)>0]. At the same point, the signs in Eqs. 17\nand 18 switch from plus to minus, and afterwards γ(n)\ndecreases monotonically as follows from Eq. 15a.\nThe detailed asymptotic behavior of γ(n) forn→ ∞\ndepends on P(γ). For long LZ sweeps with 2 TLZ≫/planckover2pi1/v,\nPLZ(γ) = 1−e−2πγ≈2πγ, and\nγ(n)∝exp[−2π/radicalbig\n2γ0(A0/β)n]. (19)\nEquation(19) describesanexponentialdecaywith anon-\nuniversal exponent. The rate of decay increases with de-\ncreasingβ, when sweeps become more adiabatic. There-\nfore, in absence of SO coupling the large- nbehavior of\nγ(n) is exponential, and self-quenching sets on for arbi-\ntrary initial conditions.\nLet us make a rough estimate of the number of sweeps\nn∞before self-quenching sets in based on Eq. 19. A\ntypical original ﬂuctuation includes N1/2spins, hence,\nv∼A0√\nN. For LZ pulses with an amplitude of about v\nand duration of about /planckover2pi1/v,βis aboutβ∼v. Therefore,\nn∞∼β/A0∼√\nN, i.e., about the number of nuclear\nspins in a typical ﬂuctuation. The dependence of n∞\nonβdemonstrates the eﬀect of the sweep duration TLZ,\nn∞is smaller for longer sweeps. A similar estimate for\nthe length ∆ nof the exponential tail in Eq. (19), with\n2π√2γ0≈10, results in ∆ n∼n∞/10, i.e., it is shorter\nthann∞by a numerical factor.More detailed estimates for both regimes require spe-\nciﬁcassumptionsabouttheshapeanddurationofsweeps.\nFor suﬃciently long sweeps, PLZ(γ) can be used for P(γ)\nand Eq. (18) can be integrated. The number of sweeps\nn=n(γi,γf), inunitsof β/(√\n2A0), betweentheinitial γi\nand ﬁnalγfvalues ofγis plotted in Fig. 1 for two modes;\nthe value of γ0has been chosen equal to γ0= 2. Fig. 1(a)\nis plotted for Iz\ni<0 and Fig. 1(b) for Iz\ni>0. Front sec-\ntions ofn(γi,γf) surfaces by γf= 0 planes demonstrate\nn∞(γi), the number of sweeps before self-quenching. For\nIz\ni<0, the curve increases fast with γiand reaches its\nmaximum value at γ=γ0. It is achieved at a ridge at\nthen(γi,γf) surface that originates from the square-root\nsingularity in the dn/dγdependence and is well seen in\nFig. 1(a). For Iz\ni>0, then∞(γi) dependence is much\nslower and becomes fast only near γ=γ0. In both cases,\nn∞∼β/A0, in agreement with the previous estimate.\nTherefore,themodelnotonlyprovidesanalyticaljusti-\nﬁcation of the self-quenching phenomenon found numer-\nically in Ref. 30 for systems without SO coupling but\nrelates, for single-specie systems, two modes of behavior\n(monotonicandnonmonotonic)tothediﬀerenceininitial\nconditions. It is the ﬁrst analytical solution of the cen-\ntral spin problem (i) describing dynamical evolution of a\npumped system into a “dark state”25,44and (ii) estab-\nlishing a connection between the initial and ﬁnal states\nof the system.\nV. NUMERICAL PROCEDURE\nDuring a sweep, the diﬀerence in the singlet and triplet\nenergiesǫS−ǫT+varies linearly in time within the sweep-\ning interval −TLZ≤t≤TLZ. We impose no restrictions\nontherelativemagnitudeofthesweepduration2 TLZand\nthe inverse S-T+coupling /planckover2pi1/v, but, as stated above, TLZ\nis long as compared to the inverse electron Zeeman en-\nergy. Furthermore, it is assumed that the variationof the\nenergies of both the upper and lower spectrum branches\nis symmetric with respect to the S-T+anti-crossing for\ntheﬁrsttransition when the initial position of the T+\nlevel isη=ηi. We denote the amplitude of the change\nin the energy diﬀerence between the singlet and triplet\nenergies as ǫmax. In other words, we use\nǫS(t) =ǫmaxt/2TLZ (20a)\nǫT+(t)−η=−ǫmaxt/2TLZ−(η−ηi) (20b)\nin Eq. (5).45Note that as a result of the dynamical nu-\nclear polarization, LZ sweeps become asymmetric with\nrespect to the anticrossing point because of the changing\nOverhausershift η. There is no longer any traditional LZ\npassagewhenever |η−ηi|>ǫmax, i.e. after the anticross-\ning point passes across one of the ends of the sweeping\ninterval. This naturally implies a slowdown in accumu-\nlating dynamical nuclear polarization. Maintaining the\nLZ passages requires additional feedback mechanisms by\nchanging the energy level diﬀerence, which we introduce\nbelow by shifting the edges of the integration interval.6\nFigure 1: Number of sweeps nbetween the initial and ﬁnal\nvalues of the Landau-Zener parameter γfor two modes; nin\nunitsofβ/(√\n2A0). (a)Initial nuclear polarization is negative,\nIz\ni<0. (b) Initial nuclear polarization is positive, Iz\ni>0. In\ntheplots, thelower boundsof γiandγfwere chosen tobe 0.01\nto cut oﬀ logarithmic singularities in n(γ) developing because\nofthePLZ(γ)factor inEq.(18). Curves n∞(γi)infrontpanels\nshow the number of sweeps before the self-quenching sets in.\nSee text for details.\nAssuming |η−ηi| ≤ǫmax, theSandT+states are\ndegenerate at t∗=−TLZ(η−ηi)/ǫmax. To avoid trivial\nquenching due to the shift in ηcaused by the accumu-\nlating polarization far away from the degeneracy point,\nthe electronic energies were renormalized after every 100\nsweeps keeping η−ηi≈0 and ensuring the S-T+anti-\ncrossing be passed during all LZ sweeps, −TLZ< t∗<\nTLZ. As a result, the center of the sweep was perma-\nnently kept close to the anticrossingpoint. Such aregime\ncan be achieved experimentally by applying appropriate\nfeedback loops.\nIn order to relate the properties of the sweeps to\nthe conventional notations of the LZ transition proba-\nbilities in the limit TLZ→ ∞, it is helpful to intro-\nduce the dimensionless initial τi=−TLZ[1+t∗/TLZ]β//planckover2pi1\nand ﬁnalτf=TLZ[1−t∗/TLZ]β//planckover2pi1times, where β=\n(ǫmax/planckover2pi1/TLZ)1/2. The Landau-Zener parameter is γ=\n(v/β)2. When −τi≫√γandτf≫√γ, the transition\nprobability converges towards the Landau-Zener result\nPLZ= 1−exp(−2πγ).\nWe consider a simple model for the electron wavefunc-\ntions. The orbital part of the singlet wave function is\nψS(1,2) = cosνψR(1)ψR(2)\n+sinν[ψL(1)ψR(2)+ψL(2)ψR(1)]/√\n2 (21)and the triplet part is\nψT(1,2) = [ψL(1)ψR(2)−ψ(2)ψR(1)]/√\n2,(22)\nwhereψL(ψR) denotes the wave function in the left\n(right) dot. The angle νdepends on the electron Zeeman\nenergy. We assume the electrons are in the lowest orbital\nharmonic oscillator state so that the wave functions are\nψ(x,y,z) =exp/bracketleftbig\n−(x2+y2)/l2−z2/w2/bracketrightbig\n/radicalbig\nwl2(π/2)3/2,(23)\nwherelis the lateral size of each dot and wis its height.\nFor two dots that are separated by a distance dwe form\nan orthonormal basis set based on the functions ψ(x−\nd/2,y,z) andψ(x+d/2,y,z), that deﬁnes the above ψL\nandψR. While both dots are chosen of the same size,\nhyperﬁne couplings in them diﬀer due to the dependence\nofρjλof Eq. 8 on the mixing angle ν.\nWe solve the nuclear dynamics numerically by using\nMathematica 9. First, we include all nuclear spins that\nare in the vicinity of the double quantum dot and sat-\nisfy the condition that the electron-nuclear coupling con-\nstantsζjλ≥κMax{ζjλ}, whereκis a small parameter.\nWe checked, by changing κ, that our results converged\nand have found that reducing κbelowκ= 0.01 does not\nproduce any visible changes in the plots we present. Ini-\ntial conﬁgurations of the nuclear spin directions are cho-\nsen by a pseudo-random number generator. The initial\nnuclear spin conﬁguration determines the 2 ×2 electron\nS-T+Hamiltonian. We solve the time-dependent 2 ×2\ndiﬀerentialequationnumericallyforlinearLZsweepsand\ncompute the probability P, the shake-up parameter Q,\nand the time-integrated eﬀect of the Overhauser shift of\nthe triplet state T+described by the parameter R. We\nthen let the nuclear spins precess in the external mag-\nnetic ﬁeld and a random noise ﬁeld before the next LZ\nsweep takes place. We record all electron singlet-triplet\ncoupling parameters as a function of the sweep number,\nas well asP,Q, and the change in the total magnetiza-\ntion.\nWe choose realistic parameters for a double quantum\ndot of a height w= 3˚A, sizel= 50˚A, and distance\nd= 100˚A. We consider an external magnetic ﬁeld of\nB=10 mT. Using a cut-oﬀ κ= 0.01 implies that we\nexplicitly include in our calculations around ten millions\nspins. A single such calculation takes about one week\non our state-of-the-art workstation. We have studied the\nevolution of the nuclear spin dynamics durin up to 2 ×\n105LZ sweeps for 107spins and used various pseudo-\nrandom initial conﬁgurations of nuclear spins. While the\ndetailed pattern of the dynamics depends on the initial\nconditions, all basic regularities were exactly the same\nin all simulations. Hence, our results are representative\nfor the generic behavior of a pumped electron-nuclear\nsystem.7\nVI. DYNAMICAL NUCLEAR POLARIZATION\nWe are now ready to discuss numerical results for dy-\nnamical polarization of nuclear spins. In all our simu-\nlations we consider double dots of the size w= 3 nm,\nl= 50 nm, and d= 100 nm.\nGaAs (InAs) has 8 nuclear spins per cubic unit cell so\nthat the eﬀective volume per site is Vs=a3/8, where\nthe lattice constant is a= 5.65˚A (a= 6.06˚A). When\nall nuclear spins are fully polarized in GaAs (InAs), the\nOverhauser ﬁeld seen by the electrons is 5 .3 T (0.86\nT). We accept the following values of electron g-factors,\ngGaAs=−0.44 (gInAs=−8). The other parameters re-\nﬂecting the abundance, nuclear g-factors, and hyperﬁne\ncoupling constants are listed in Table I for GaAs and Ta-\nble II for InAs. From these values, it can be understood\nthat in our simulations GaAs behaves as a three-specie\nsystem, whereas InAs behaves as a two-specie system.\nAlthough there are three distinct species in InAs, two of\nthem behave in the same way with respect to the preces-\nsionrateinanexternalmagneticﬁeldandthecouplingto\nelectrons so that InAs is an eﬀective two-specie system.\n69Ga71Ga75As\np30%20%50%\ng1.31.70.96\nA(µeV)779994\nI3/23/23/2\nTable I: Nuclear abundances p, nuclear g-factors, hyperﬁne\ncoupling constants A, and nuclear spin in GaAs.46,47\n113In115In75As\np2%48%50%\ng1.21.20.96\nA(µeV)14014076\nI9/29/23/2\nTable II: Nuclear abundances p, nuclear g-factors, hyperﬁne\ncoupling constants A, and nuclear spin in InAs.48,49\nWe will consider systems with diﬀerent number of nu-\nclear species to deduce coupled electron-nuclear dynam-\nicsphenomenathat arerobustwith respecttoorstrongly\ninﬂuenced by the number of species. To this end, we\nchoose InAs and GaAs as model systems. These systems\nhavediﬀerentmagnitudesoftheSOsplitting; itismodest\nin GaAs but strong in InAs, see Appendix A for details.\nIn Sec. VIA, where calculations for nuclear parameters\nof InAs of Table II are carried out, we use modest val-\nues of SO coupling v±\nSOto illustrate how the dynamics\nbecomes increasinglycomplex and irregularwith increas-\ning strength of the spin-orbit interaction. Nevertheless,\nthis allows making conclusions about the expected nu-\nclear dynamics in InAs for realistic values of v±\nSO, see the\nend of Sec. VIA. The SO coupling constant v±\nSOis a com-\nplex number. Without loss of generality, we will assumein the remainder of the paper that it is real and positive,\nas well as use a simpliﬁed notation, v±\nSO=vSO.\nA. Two-specie systems: InAs\nLet us ﬁrst consider InAs which eﬀectively consists of\ntwo species because the parameters of113In and115In\npractically coincide. Therefore, species113In and115In\nbehave as a single specie and75As as a second specie.\nWe ﬁrst demonstratethat, in absenceofSO coupling, the\ndynamical evolution of nuclear spins in InAs is similar to\nthe dynamics in GaAs reported earlier.30In all our InAs\nsimulations, we startin the sameinitial (pseudo-random)\nconﬁguration of the nuclear spins. We have checked that\nsimilarresultsareobtainedwhenwestartinseveralother\nconﬁgurations. In all our simulations in this section, the\nwaiting time between LZ sweeps equals the precession\ntime of specie75As in the external magnetic ﬁeld, Tw=\nt75As.\nWe start by presenting results for a system without\nspin-orbitcoupling, vSO= 0, toprovethatself-quenching\noccurs and investigate its stability with respect to nu-\nclear noise. Fig. 2(a) shows the evolution of the magni-\ntude of the singlet-triplet coupling |v±\nn|with increasing\nnumber of sweeps n. For a sweep duration of TLZ= 40\nns, the initial LZ probability for the ﬁrst few sweeps is\nP∼0.5, see Fig. 3, and the singlet-triplet coupling is\nself-quenched already after about 20000 sweeps. The\nnumber of sweeps nrequired to reach self-quenching is\nabout the same as for GaAs.30The appearance of sev-\neral peaks of v±\nnin the rangeof 5000-20000sweeps before\nthe self-quenching sets in is typical of multi-specie sys-\ntems. In contrast to single-specie systems, and especially\nthe model of Sec. IV, in multi-specie systems ﬁnal self-\nquenching is usually preceeded by partial self-quenchings\nfollowed by revivals. We attribute this behavior to com-\npetition between subsystems with the diﬀerent Zeeman\nprecession times. As seen in Fig. 3, in each peak of |v±\nn|\ntheS-T+transitionprobability Pincreasesstrongly,near\nitIzshows a step-like behavior (not shown), and accom-\npanying peaks of Qindicate massive shakeups which ﬂop\nmany nuclear spins per LZ sweep. The model of Sec. IV\nthat only deals with the total magnetization Izdoes not\ndescribe such events and provides a smoothened picture\nof the nuclear spin evolution.\nTransverse noise transforms the dynamical evolution\ninto a dissipative one. Fig. 2 shows the eﬀect of the in-\ncrease of the level of noise from (a) through (b) to (c).\nIn Fig. 2(a), the noise correlation time is of the order of\nthe self-quenching time τ/t75As= 10000. In this case,\ntransverse noise only modestly perturbs the nuclear spin\nevolutionascomparedtothenon-dissipativeregime(sim-\nulated and analyzed, but not shown). Note the presence\nof a long slightly visible tail with irregular oscillations\nalong it. In contrast, Fig. 2(b) and (c) demonstrate that\nwhen the transverse noise correlation time τis shorter\nthan the typical self-quenching set-in time in un-noisy8\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1a/RParen1\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1b/RParen1\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1c/RParen1\nFigure 2: Hyperﬁne-induced singlet-triplet coupling |v±\nn|for\na double quantum dot with the nuclear parameters of InAs\nin absence of SO coupling with increasing level of noise as\na function of sweep number n. (a)τ/t75As= 10000, (b)\nτ/t75As= 1000, and (c) τ/t75As= 100. The other param-\neters are Tw=t75As,TLZ= 40 ns.\nsystems, self-quenchingissuppressedandeventuallydoes\nnot happen at all; in particular, Fig. 2(b) demonstrates\na possibility of revivals. We conclude that for high noise\nlevels the chaotic evolution of the nuclear spins persists,\nbut themagnitudesofthe peaksof |v±\nn|seemtogradually\ndecrease in time.\nFig. 2(a) suggests a glassy behavior of the nuclear\nsystem with an extensive manifold of dark states sepa-\nrated by low barriers. In the absence of noise, repeated\nLZ sweeps cause the system to end in one of the dark\nstates (usually after passing through several peaks of\n|v±\nn|). Weak noise produces slow diﬀusion between adja-\ncentdarkstatesacrosslowsaddle points. Duringthis dif-\nfusion, the magnetization Izchanges only slightly. With\nincreasing noise, the system experiences revivals as seen\ninFig.2(b)asasharppeakin |v±\nn|. Duringsuchevents P50000 100000n0.51.P\n50000 100000n48Q\nFigure 3: (a) Landau-Zener transition probability Pas func-\ntion of the sweep number nfor a double quantum dot with\nthe nuclear parameters of InAs in absence of SO coupling. (b)\nShake-up parameter Qas a function of sweep number n. The\nparameters are as in Fig. 2(a)\nincreasesstrongly, Izshowsstep-likebehavior, and peaks\ninQ(not shown) indicate massive shakeups, similarly to\nthe patterns discussed as applied to Fig. 2(a) above.\nWe demonstrated earlier that SO coupling is screened\nstroboscopicallyin asingle-speciesystem.30Next, wewill\ndemonstrate that SO coupling can be screened strobo-\nscopically also in multi-specie systems, and investigate\nthis phenomenon in more detail. Fig. 4 shows simula-\ntions of the singlet-triplet coupling v±\nnforvSO= 62 neV\nand three LZ sweep durations TLZ. In comparison, the\nstraight black line indicates the value of the spin-orbit\ncouplingvSO= 62 neV (which is independent of the\nsweep number n). We see that in all these simulations,\nthe spin-orbit coupling eventually becomes screened so\nthat all the colored lines approach the black line which\nimpliesthat |v±\nn|=|vSO|. ForlongerLZsweepdurations,\noscillations of v±\nnare more rapid, but screening eventu-\nally occurs faster because nuclear spins are more strongly\naﬀected during each sweep.\nScreening of the SO coupling even in multi-specie sys-\ntems sounds counter-intuitive at ﬁrst glance. Indeed, the\nspin-orbitcoupling vSOisstaticwhilethetransversecom-\nponents of the nuclearspins contributing to v±\nnprecessin\ntime. In InAs, twonuclearspecies113Inand115In precess\nat the same frequency and behave eﬀectively as a single\nspin specie whereas the third spin specie,75As, precesses\nat a diﬀerent frequency. So, while screening indicates\nthat the magnitude of the singlet-triplet coupling v±\nnre-\nmains ﬁnite, it must inevitably precess in time. There-9\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1\nFigure 4: Transverse nuclear polarization for a double quan -\ntum dot with the nuclear parameters of InAs as a function\nof sweep number nfor the spin-orbit coupling vSO= 62\nneV (black line) and the Landau-Zener sweep durations (red\ncurve)TLZ= 80 ns, (green curve) TLZ= 40 ns, and (blue\ncurve)TLZ= 20 ns. Resonant pumping with Tw=t75As, the\npolarization is plotted at multiples of Tw. See text for details.\nfore, it cannot compensate the spin-orbit coupling vSO\nat all instants of time. The screening we observe is only\npossible because the waiting time Twis exactly equal to\nthe precession time of the75As specie,Tw=t75As. The\ndata used in our plots of |v±\nn|were taken at exact mul-\ntiples of the the waiting time, which was equal to the\nprecession time of the75As specie. Therefore, the self-\nquenching that manifests itself in Fig. 4 is a stroboscopic\nself-quenching.\nStroboscopic self-quenching can be understood in the\nfollowing way. The dynamical evolution of nuclear spins\ncauses self-quenching of the sum of the contributions\nfromthetransversecomponentsofspecies113Inand115In\n(that are out of resonance with the pumping period Tw,\nhence, their contribution to v±\nnvanishes). In contrast,\nthe contribution from the specie75As tov±\nnexactly com-\npensatesthespin-orbitcoupling v±\nSOateverytimeinstant\nwhen a LZ sweep happens. In other words, the matrix\nelementsv±\nn(t) changein time harmonicallywith the am-\nplitudev±\nSOand a period t75As:\nv±\nn(t) =v±\nSOcos(2πt/t75As). (24)\nThis generalizes our previous ﬁndings of the screening\nof SO coupling in single-specie systems.30For a single-\nspecie imitation of GaAs, we found that the SO coupling\nwas screened in such a way that that the matrix element\nchanged harmonically with the amplitude vSOand a pe-\nriodtGaAs, wheretGaAsis the average precession time of\nthe three nuclear spin species in GaAs.30\nLet us now demonstrate explicitly that when self-\nquenching sets in, the sum of the contributions from the\ntransversecomponents of113In and115In to|v±\nn|vanishes\nwhile the contribution from75As equalsvSO. We show\nin Fig. 5(a) the contribution from113In and114In to|v±\nn|\nas a function of the number of sweeps n. Clearly, it van-\nishes for large n. On the other hand,75As whose nuclear\nprecession time equals the waiting time Tw, makes a con-\ntribution to |v±\nn|that exactly compensates |v±\nSO|at allintegers ofTw, see Fig. 5(b). Hence, Eq. (24) is satisﬁed.\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1/LParen1a/RParen1Sumof specie1 and2\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1/LParen1b/RParen1Specie3\nFigure 5: (a) Sum of contributions from113In and115In\nto hyperﬁne-induced singlet-triplet coupling v±\nnfor spin-orbit\ncoupling vSO= 62 neV (black line) as a function of sweep\nnumber n. (b) Contribution from75As to hyperﬁne-induced\nsinglet-triplet coupling v±\nnfor spin-orbit coupling vSO= 62\nneV (black line) as a function of sweep number n. The LZ\nsweep duration is TLZ= 80 ns.\nWe note that the contributions from113In and115In\ntov±\nnvanish not only stroboscopically but identically,\nat each instant of time (not shown). We have also\nchecked that in systems without spin-orbit coupling, self-\nquenching sets in for all species and for an arbitraryratio\nbetweenTwand the precession times of the species (not\nshown). For three-specie systems the last statement is\nproven below, see Sec. VIB.\nNow we will illustrate that stroboscopic screening of\nSO coupling can be practically achieved only for small\nand moderate magnitudes of vSO. Since stroboscopic\nscreening implies that the contribution from75As tov±\nn\ncompensates vSOwhile the combined contribution from\nspecies113In and115In vanishes, we show in Fig. 6 the\nevolution of the contribution of75As tov±\nnas a function\nof sweep number for three values of spin-orbit coupling\nvSO= 31,62 and 91 neV.50While all the results in Fig. 6\nwere found for the same value of TLZand the same initial\nconditions, screening sets in at n≈75000 forvSO= 31\nneV, is delayed to n≈125000 for vSO= 62neV, and is\nfar from complete even at n= 200000 for vSO= 93 neV.\nThese data suggest that stroboscopic self-quenching sets\nin whenvSO/lessorsimilarv0\nn, wherev0\nn≈A/√\nNis a typical ﬂuctu-\nation of the Overhauser ﬁeld, and cannot be practically\nachieved for vSO/greaterorsimilarv0\nn; see estimates of the magnitude10\nof the spin-orbit coupling in Appendix A. This criterion\nresembles the criterion of the phase transition of Ref. 26.\nOne should keep in mind that with the interval be-\ntween LZ pulses of about 1 µs, a set of n∼106pulses\ntakes about 1 s which is a typical scale of nuclear spin\ndiﬀusion51, which is not taken into account in the above\nconsiderations. Weexpect, buthavenotchecked, thatin-\nhomogeneity of magnetic ﬁeld should have a detrimental\neﬀect on stroboscopic self-quenching. Therefore, we con-\nclude that stroboscopic quenching of SO coupling is less\ngeneric and more fragile than self-quenching in systems\nwithout spin-orbit coupling.\n50000 100000 150000 200000n20406080100/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1\nFigure 6: Contribution of the specie75As to the singlet-\ntriplet coupling v±\nnas a function of sweep number nfor diﬀer-\nent values of vSO: 31 neV (red curve), 62 neV (black curve),\nand 91 neV (blue curve). Duration of LZ pulses TLZ= 40\nns. Waiting time between consecutive LZ pulses equals the\nprecession time of75As,Tw=t75As. Nuclear parameters of\nInAs.\nWe estimatein Appendix Athat SOcouplingisweaker\nor comparable to (stronger than) the typical nuclear\npolarization induced singlet-triplet coupling in GaAs\n(InAs). As a consequence, we expect the SO coupling\nmight be stroboscopically screened in GaAs systems, but\nthat stroboscopic screening is improbable in InAs sys-\ntems.\nB. Three-specie systems: GaAs\nIn this section, we present new results for GaAs\nthat complete the picture of the generic nature of self-\nquenching in multi-specie systems. Furthermore, we\nshow that screening of the SO coupling requires that the\nwaiting time Twis in resonance with the precession time\nof one of the nuclear species. When the resonance con-\ndition is not satisﬁed, screening of the SO coupling is\npartial and irregular.\nIn Sec. VIA and in Ref. 30, self-quenching in multi-\nspecie systems in absence of SO coupling was demon-\nstarted only under the conditions when the waiting time\nTwwas in resonance with the precession time t75Asof the\n75As specie,Tw=t75As. We demonstrate here that whileself-quenching is generic and independent of the waiting\ntime, the evolution towards the self-quenched states de-\npends on the waiting time.\nTo this end, we plot in Fig. 7 the evolution of the\nsinglet-triplet coupling v±\nnfor two diﬀerent values of the\nwaiting time Tw. Fig. 7(a) displays results of simulations\nfor the resonant case when Tw=t75As, in which pro-\nnounced oscillations are distinctly seen. For n/greaterorsimilar3000,\nthe plot consists of ﬁve branches that reﬂect coupled dy-\nnamics of three species. In contrast, in the absence of\nthe resonance, Fig. 7(b), the evolution is chaotic. Nev-\nertheless, self-quenching sets-in in both cases and, what\nis most remarkable, at the same time scale of n≈104.\nRemarkably, the processes of Figs. 7(a) and 7(b) ended\nin states with the same Iz(not shown). While the set\nof dark states is vast (as follows from our discussion in\nSection IV), this observation indicates that the number\nof strong attraction centers in which self-quenching ends\nis more scant.\nWe conclude that self-quenching in systems without\nspin-orbit coupling is generic and robust, at least in the\nframework of S-T+scheme.\nWe checked that not only does the total matrix ele-\nmentv±\nnvanish, but also the matrix elements for all of\nthree species contributing to it. Because between the LZ\nsweeps the electron subsystem is in its singlet state, the\nKnight shift vanishes, and according to Eq. (3) all nuclei\nbelonging to some specie precess with the same speed.\nTherefore, the self-organization of the nuclear subsystem\nthat annihilates its coupling to the electron spin persists\nduring the free precession periods.\nFinally, we demonstate that while self-quenching is a\ngenericfeatureintheabsenceofSOcouplingregardlessof\ntheratiobetweenthewaitingtimebetweentheLZsweeps\nTwand the nuclear precession times tλ, in presence of SO\ncoupling the stroboscopic self-quenching is not generic\nandhighlysensitivetothisratio. Onlymodestdeviations\nfromtheresonancedestroysthescreeningofSOcoupling.\nIn Fig. 8, we plot |v±\nn|under the conditions when the\nwaiting time is in exact resonance with the precession\ntime of75As (red curve), and when there is a 1% devi-\nation from the resonance (black curve). While the SO\ncoupling is clearly screened in resonance, only a tiny de-\nviation from resonance destroys screening.\nMore insights into the sensitivity of the screening of\nSO coupling to the deviation from the resonance can be\ngained from Fig. 9 that displays the contributions to v±\nn\nfrom eachofthe species. Using the same intial conditions\nas in Fig. 8, we plot the evolution of the matrix elements\n|v±\nn|for both the resonant and slightly oﬀ-resonance\nregimes. Initially they follow each other closely. How-\never, after a couple of thousand sweeps, the deviations\nbecome signiﬁcant. Ultimately, the contributions from\n69Ga and71Ga do not vanish in the non-resonant case,\nand the contributions from75As does not screen the SO\ncoupling.\nThe critical sensitivity of stroboscopic self-quenching\nto small deviations from resonance looks indicative of11\n2000 4000 6000 8000 10000n10203040/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1\n2000 4000 6000 8000 10000n10203040/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1\nFigure 7: Transverse nuclear polarization as a function of\nsweep number nfor a GaAs double quantum dot in absence\nof spin-orbit coupling and transverse noise. Duration of LZ\npulsesTLZ= 80 ns. (a) The waiting time is in resonance\nwith the75As precession time, Tw=t75As= 13.7µs. (b)\nThe waiting time is incommensurate with the75As precession\ntime,Tw= 1.39t75As= 19.1µs.\n10000 20000 30000n255075/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen169Ga,71Ga, and75As\nFigure 8: Transverse nuclear polarization as a function of\nsweep number nfor a GaAs double quantum dot with vSO=8\nneV. The duration of LZ pulses TLZ= 80 ns. The red curve\nshows results for the waiting time in resonance with the pre-\ncession time of75As,Tw=t75Ar, and the black curve for a\n1% deviation from the resonance.\na chaotic behavior of the system.52This is not surpris-\ningbecausethe system ofintegro-diﬀerentialequationsof\nEq. (1) is highly nonlinear because the coeﬃcients ∆jλ\ndepend through Eqs. (10) on the electronic amplitudes\ncS,cT+that, in turn, depend on all nuclear angular mo-\nmentaIjλ. In this context, we speculate that a strong\nrevival of all black curves in Fig. 9 near n≈15000 where\nall red curves saturate, and the return of black curves\nclose to their initial values near n≈28000, is reminis-cent of the strange attractor pattern.52These signatures\nofchaoticnucleardynamicinSOcoupledsystemsrequire\na more detailed study.\nWeconcludethatstroboscopicscreeningoftheSOcou-\npling is not a robust phenomenon.\nWhile the above simulations are focused on the large\nnregion, we mention that commensurability oscillations\nin the polarization accumulation per sweep were ob-\nserved experimentally38and described theoretically37in\nthe smallnregion,n/lessorsimilar104.\n10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1a/RParen169Ga\n10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1b/RParen171Ga\n10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1c/RParen175As\nFigure 9: Transverse nuclear polarization as a function of\nsweep number nfor a GaAs double quantum dot with vSO=8\nneV. The duration of LZ pulses TLZ= 80 ns. Red curves\nshow results for the waiting time Twin resonance with the\nprecession time of75As,Tw=t75As, while black curves the\ndata for 1% oﬀ-resonance regime.\nIn addition to the regular investigation of the trans-\nverse magnetization, we also followed the time de-\npendence of the longitudinal magnetization vz\nn=\nVs/summationtext\nλAλ/summationtext\nj∈λρjλIz\njλ. In presence of SO coupling, it\nshows an oscillating sign-alternating behavior, and we\nwere unable to detect any signatures of its accumulation.12\nSummarizing the results of Secs. VIA and VIB, we\nconclude that SO coupling eliminates self-quenching and\ncauses the nuclei of a pumped system to exhibit a per-\nsistent irregular dynamics. We speculate that this phe-\nnomenon is closelyrelated to the feedback looptechnique\nfor building controllable nuclear gradients which is inher-\nently based on employingsuch a dynamics.34Indeed, any\nlong-term control of the nuclear ensemble by alternating\nS→T+andT+→Ssweeps is impossible after the self-\nquenching set-in time that is of a millisecond scale in\nabsence of SO coupling. Our data, especially Fig. 8, sug-\ngest that near the resonance between the waiting time\nof LZ pulses and the Larmor frequency of one of the\nnuclear species the quasi-periods of nuclear ﬂuctuations\nbecome longer and are controlled by the deviations from\nthe exact resonance. We also expect that under these\nconditions the nuclear gradients should be dominated by\nthe resonant specie.\nVII. CONCLUSIONS\nAn analytical solution of a simpliﬁed model, and ex-\ntensive numerical simulations for a realistic geometry,\nprovethat self-quenchingis a genericpropertyofthe cen-\ntral spin-1/2 problem in absence of spin-orbit coupling.\nAs applied to a double quantum dot of a GaAs type,\nwhere electron and nuclear spins are coupled viahyper-\nﬁne interaction, pumping nuclear magnetization across a\nS-T+avoided crossing through successive Landau-Zener\nsweeps ceases after about 104sweeps. This is a result\nof the screening of the initial ﬂuctuation of the nuclear\nmagnetization by the injected magnetization and van-\nishing of the S-T+anticrossing width, and this sort of\nself-quenchingisrobust. Under the inﬂuence ofmoderate\nnoise, thesystemwandersthroughasetofdarkstatesbe-\nlongingtoawastlandscapeofthesystemincludingabout\n106nuclear spins coupled through inhomogeneous elec-\ntron spin density. With time intervals depending on the\nlevel of the noise, the system experiences revivals when\nadditional magnetization is injected, and afterwards it\nwanders through a new set of dark states.\nDue to the violation of the angular momentum conser-\nvation, spin-orbit coupling changes the situation drasti-\ncally. Self-quenching sets in only stroboscopically under\nthe condition that the waiting time between consecutive\nLandau-Zener sweeps is in resonance with the Larmor\nprecession time of one of the nuclear species. Then the\nprecessing Overhauser ﬁeld of the resonant specie com-\npensates the spin-orbit ﬁeld vSOduring the sweep, while\ncontributions of other species vanish. This sort of self-\nquenching is fragile and sensitive even to minor deviation\nfrom the resonance. Generically, injection of nuclear po-\nlarization oscillates in time and changes sign. Therefore,\nspin-orbit coupling causes the nuclear magnetization of a\npumpedS-T+doublequantumdottoexhibit apersistent\ndynamics.\nWe suggest that the feedback loop technique for build-ing controllable nuclear ﬁeld gradients34is based on the\noscillatory behavior of the nuclear spin magnetization\ncaused by spin-orbit coupling. Spin-orbit coupling is a\nnatural mechanism of overcoming self-quenching. The\ntechnique employs persistent oscillations and selects the\nsign of the pumping response to the changing magneti-\nzation gradient.\nAcknowledgments\nA.B.wouldliketothankB.I.Halperinforhishospital-\nity at Harvard University where this work was initiated.\nWe are grateful to B. I. Halperin, C. M. Marcus, L. S.\nLevitov, H. Bluhm, K. C. Nowack, M. Rudner, and L.\nM. K. Vandersypen for useful discussions. E. I. R. was\nsupported by the Oﬃce of the Director of National In-\ntelligence, Intelligence Advanced Research Projects Ac-\ntivity (IARPA), through the Army Research Oﬃce grant\nW911NF-12-1-0354and by the NSF through the Materi-\nals Work Network program DMR-0908070.\nAppendix A: Spin-orbit coupling\nThe Rashba spin-orbit Hamiltonian is\nHso=α/summationdisplay\nn=1,2[σx(n)ky(n)−σy(n)kx(n)],(A1)\nwhereαis the strength of SO interaction, and kx(n) and\nky(n) are the in-plane momenta for the electron n. The\nsinglet and triplet states are Ψ S(1,2) =ψS(1,2)χS(1,2)\nand Ψ T+(1,2) =ψT(1,2)χT+(1,2), where the orbital\ncomponents of the singlet and triplet wave functions\nhave been deﬁned in Sec. V and the spin parts of the\nwave functions are χS(1,2) = (| ↑1∝an}bracketri}ht| ↓2∝an}bracketri}ht−| ↓ 1∝an}bracketri}ht ↑2∝an}bracketri}ht)/√\n2\nandχT+(1,2) =| ↑1∝an}bracketri}ht| ↑2∝an}bracketri}ht. In terms of the orbital\nwave functions, the SO induced S-T+coupling is then\nv+\nSO=iαcosν∝an}bracketle{tψR|kx+iky|ψL∝an}bracketri}ht. This matrix element\ncan be estimated similarly to Refs. 26 and 39. It depends\nexponentially on the overlap between the wave functions\nof the dots, ψLandψR. Therefore, the SO coupling v+\nSO\ncan be tuned and strongly decreases with the interdot\ndistanced. In InAs quantum wires the SO coupling pa-\nrameter is around α∼10−11eVm53corresponding to a\nspin precession length of lSO=/planckover2pi12/(2m∗α) of around 100\nnanometers. With typical parameters of d= 100 nm\nandl= 50 nm and cos ν= 1/√\n2, we ﬁnd that vSOis\naround 4 ×10−5eV. This is about two orders of mag-\nnitude larger than the typical S-T+coupling 10−7eV\ninduced by the hyperﬁne interaction, but decreases with\ndexponentially. Theoretical estimates of vSOhold only\nwith exponential accuracy. Pre-exponential factors are\nmodel dependent, and a somewhat diﬀerent estimate was\nproposed in Ref. 54. In GaAs quantum dots,55the SO\ncoupling constant αis two orders of magnitude smaller\nthan in InAs with lSO≈30µm, so that the SO coupling13\nmay be comparable to the hyperﬁne induced coupling,\nand is usually considered as weaker than it. These esti-\nmates should be treated with caution since the SO cou-\npling is not only a function of the material but is sample\nspeciﬁc.\nFor GaAs, an estimate of a typical ﬂuctuation as v0\nn≈\nA/√\nNwithAfrom Table II and N≈106results inv0\nn≈\n100 neV. Our estimates of vSOof Ref. 30 gave vSO≈50\nneV, while the estimate of Ref. 37 is vSO≈15 neV.56Recently Shaﬁei et al.57managed to resolve the SO and\nhyperﬁne components of electric dipole spin resonance\n(EDSR)58in the same system, a GaAs double quantum\ndot. 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Acta 5, 369 (1932)."    },    {        "title": "1310.4895v1.Many_body_effects_on_electron_spin_dynamics_in_semiconductors_from_a_geometrical_viewpoint.pdf",        "content": "arXiv:1310.4895v1  [cond-mat.mes-hall]  18 Oct 2013Many-body eﬀects on electron spin dynamics in semiconducto rs from a geometrical\nviewpoint\nChunbo Zhao∗\nState Key Laboratory for Superlattices and Microstructure s, Institute of Semiconductors\nChinese Academy of Sciences, P.O.Box 912,Beijing 100083, P eople’s Republic of China\n(Dated: March 18, 2021)\nMany body eﬀects on spin dynamics in semiconductors have att racted a lot of attentions in\nrecent years. In this paper, we show why and how the many body e ﬀects have to be considered by a\nsimple Bloch sphere geometry approach. The micro dynamics h ere are viewed as a time dependent\nsequence of unitary group action on the spin density matrix. Based on this physical picture, we give\nthe explicit unitary group for conventional spin dynamics m echanism such as DP, EY, and BAP\nusing pure density matrix. And we also show the various scatt ering processes how inﬂuence the spin\nsystem via mixed density matrix and Feynman diagrams.\nI. INTRODUCTION\nSemiconductor spintronics, which aims at utilizing\nor incorporating the spin degree of freedom in elec-\ntronics, has attracted great interests in last decades\nyears1–3. Many novel spin-related phenomena and prop-\nerties, such as the spin Hall eﬀect4,5, spin Coulomb\ndrag eﬀect6,7, spin photogavanic eﬀect8,9and persistent\nspin helix eﬀect10,11, have been discovered. These novel\nspin-related physics can be partly understood well by\nsingle-particle demonstration. However, the real phys-\nical system is interacting, there may be something new\nphysics emerging only when considering many-body ef-\nfects. In fact, recently Wu, el12developed a fully micro-\nscopic many-body theory on spin dynamics in semicon-\nductors called kinetic spin Bloch equation (KSBE) using\nnon-equilibrium Green function13, considering diﬀerent\nkinds of scattering. From this theory, they predicted\nmany novel eﬀects, such as nonmontonic spin relaxation\ntime dependence of temperature or electron density in\nGaAs quantum well14, hole screeningeﬀect in hole doped\nbulkp-GaAs15and so on. And later these predictions\nhave been observed experimentally one after another by\ntime-resolved Kerr rotation technique16,17. Even though\nKSBE is useful, one cannot easily get the underlying\nphysics immediately from the numeric results of KSBE\nsince there is no analytical result generally for its compli-\ncated expression. Encouraged by the powerful of KSBE,\nit will be helpful and profound to reinterpret the many\nbody eﬀects on spin dynamics from another viewpoint if\nthis interpretation can give us a clear and insightful to\nthe questions addressed.\nIn this paper, we try to give an intuitive picture about\nthe many-body eﬀects on spin dynamics using the Bloch\nsphere geometry language. This demonstration will give\nus a general framework to interpret the explicit dynam-\nics emerging from spin-orbit coupling or/and many body\nscattering, whichwill provideus ageometricalwaytoun-\nderstand the numeric results of KSBE. The main idea is\nthat we view the spin dynamics as a time-dependent uni-\ntary group element of SU(2) action on the spin density\nmatrix. Basedonthispicture, theelectronspindynamicscan be understood as a vector rotation in Bloch sphere\nif one applies the mathematical map called Hopf map.\nTherefore, if we denote the spin ’direction’ as a point on\nthe Bloch sphere, then the resultant spin ’direction’ will\nbe the point after a SO(3) group action. Here we want\nto mention that: even though the Hopf map is not one-\nto-one, the Bloch sphere interpretation of spin dynamics\ncaptures the essential physics process that we investi-\ngated. The rest paper is organized as follows. Firstly, we\npresentthe pureandmixeddensitymatrixwith aunitary\ngroup. Secondly, we will try to transform the well stud-\nied spin dynamics mechanisms such as D’yakonov-Perel\n(DP), Elliott-Yafet (EY), and Bir-Aronov-Pikus (BAP)\nto a unitary element in SU(2). Finally, we will use the\nmixed spin density matrix and Fennyman diagram to ac-\ncount for the many body eﬀects with diﬀerent scattering\nconsidered.\nII. SPIN DYNAMICS IN SEMICONDUCTORS\nA. pure and mixed spin density matrix\nIt is known that the single electron spin space can be\nexpressed with a two components complex-valued func-\ntion|φ/an}bracketri}ht=/parenleftbigz1, z2/parenrightbigT. If we suppose the normalizing\ncondition as\n/an}bracketle{tφ|φ/an}bracketri}ht=z∗\n1z1+z∗\n2z2= 1, (1)\nthen the electron spin space is nothing but a sphere S3\nin the 4-d Euclid space with symmetry group SU(2).\nPhysically, the spin state or wave function |φ/an}bracketri}htwill be\nchangedunder the externalmagnetic ﬁeld orelectric ﬁeld\nby Rashaba spin-orbit coupling. Mathematically, the dy-\nnamicsofwavefunction |φ/an}bracketri}htcanbeinterpretedasa SU(2)\ngroup element uacting on the former wave funtion |φ′/an}bracketri}ht,\nwhich can be regarded as just a rotation of wave function\nunder the symmetry group u,|φ/an}bracketri}ht=u|φ′/an}bracketri}ht. For the con-\nvenience of the description, one usually apply the spin\ndensity operator ρ(which is isomorphism to the space of\nwavefunction) to study spin dynamics. For a pure state,2\nthe density matrix is expressed as\nρ=|φ/an}bracketri}ht/an}bracketle{tφ|=/parenleftbigg\nz1z∗\n1z1z∗\n2\nz2z∗\n1z2z∗\n2/parenrightbigg\n(2)\nAccording to this representation, one can obtain the one-\nto-one map through /an}bracketle{tφ|φ/an}bracketri}ht ↔ |φ/an}bracketri}ht/an}bracketle{tφ|between wave func-\ntion and density matrix space. Hence, we can investigate\nthe spin dynamics by density matrix or its matric repre-\nsentations of SU(2). Due to any element of SU(2) can be\ndecomposed using Pauli matrices set and identity oper-\nator, we have a compact form of the density matrix (2)\nwritten in terms of2 ×2 Pauli matricesset {I,σx,σy,σz},\nρ=x0I+x1σx+x2σy+x3σz, (3)\nwherex0,x1,x2,x3arerealnumbers, Iisthe identityma-\ntrix. Applying the Eq.(1), a constraint of density matrix\ncan be obtained:\ntrρ=z∗\n1z1+z∗\n2z2= 2x0= 1 (4)\nsox0= 1/2 is required. We rewrite the density matrix\nas\nρ=1\n2(I+/vector x·/vector σ) (5)\nwhere/vector x= (x1,x2,x3) and/vector σ= (σx,σy,σz) is the vector\nwith the three Pauli matrices as components. As for\npure state, tr(ρ2) = 118, which leads to the constraint\nof|/vector x|2=x2\n1+x2\n2+x2\n3= 1. Hence, a point in S2can\nbe a presentation of /vector x. This map of SU(2) toSO(3) is\nnothing but the Hopf map S3→S2in mathematics.\nSo far, we have introduced the pure state density ma-\ntrix, next we will discuss the more realistic case, called\nmixed state matrix, in which, the many body eﬀects will\nbe emerged naturally. Since an electron quantum state\nin semiconductor can be labeled by momentum /vectorkand\nenergy band index n, so we can denote diﬀerent state\nelectrons by quantum number /vectorkwhen only conduction\nband is considered. The whole density matrix can be\nwritten as follows18,\nρ=/summationdisplay\n/vectork|φ/vectork/an}bracketri}htp/vectork/an}bracketle{tφ/vectork| (6)\nwhere/summationtext\n/vectorkp/vectork= 1,|φ/vectork/an}bracketri}htis the spin space of state with\nmomentum k,p/vectorkis actually its corresponding probabil-\nity (statistical weights). Hence, for a physical operator\nA, the measured quantity will need to be averagedtwice,\none is the quantum mechanics average, the second is the\nstatistical average, it can be described as:\n/an}bracketle{t/an}bracketle{tA/an}bracketri}ht/an}bracketri}ht=/summationdisplay\n/vectorkp/vectork/an}bracketle{tφ/vectork|A|φ/vectork/an}bracketri}ht= tr(Aρ), (7)\nwhere tr is the trace operator. For convenience later, we\ndeﬁne single state density matrix as ρ/vectork=|φ/vectork/an}bracketri}htp/vectork/an}bracketle{tφ/vectork|for\nstate labeled with /vectork. Eq.(7) clearly indicates that theprobability p/vectorkcontaining in ρ/vectorkwill inﬂunce the phys-\nical observations. Therefore, scattering processes such\nas electron-electron, electron-phonon, electron-impurity\nscatterings, will inﬂuence the physical quantity since the\nstate at /vectorkwill be scattered to the one at/vectork′, but they\nmay not have the equal statistical probability ( p/vectork/ne}ationslash=p/vectork′).\nThere are two important properties for mixed state den-\nsity matrix, one is that the trace of ρis identity tr( ρ) =/summationtext\nkpk= 1, which is the same with pure state. The other\nspecial property is that tr( ρ2)</summationtext\n/vectorkp/vectork= 1, which is the\nonly feature of the mixed state.\nIf we know the original physical state, assuming the\nspin direction pointing to zaxis, then one can follow the\ntracks of spin dynamics via Eq.(7), Sz(t) = tr(σzρ(t)).\nSo the main task is to study the evolution of mixed den-\nsity matrix ρ. In the following part, we present how the\ndynamic process can be viewed as a simple vector rota-\ntion in Bloch sphere for single and many body eﬀects on\nelectron spin dynamics. And we also give the physical\npicture to understand the KSBE applying our method.\nB. spin relaxation mechanisms\nD’yakonov-Perel’s mechanism (DP)\nAccording to the Liouville equation of density\noperator18, we have the following motion equation for\nthe density matrix ρ:\n˙ρ=−i[H,ρ], (8)\nwhere [ ] is the commutator operator, and here we as-\nsume/planckover2pi1= 1forsimplicity. Forexample,whenthehamito-\nnianHcan be written as Dreeslhaus and Rashaba form,\nH=γ(ΩD(/vectork) +ΩR(/vectork))·/vector σ=1\n2/vectorh(/vectork)·/vector σ, the motion of ρ\ndescribes the respective scattering process:\n˙ρ=−i[γ(ΩD(/vectork)+ΩR(/vectork))·/vector σ,ρ] =−i[1\n2h(/vectork)·/vector σ,ρ] (9)\nSubstitute Eq.(5) into the Eq.(9), one can easily obtain\nthe following:\n˙/vector x=/vectorh(/vectork)×/vector x, (10)\nwhere we have used the identity ( /vector σ·/vector a)(/vector σ·/vectorb) =I(/vector a·/vectorb)+\ni/vector σ·(/vector a×/vectorb) for arbitrary vector /vector aand/vectorb. Actually, we can\nuse a unitary group element to reexpress this motion as\nρ′=u(/vectorh(t))ρ, where\nu(/vectorh(t)) = exp( −i(/vectorh(t)·/vector σ)/2). (11)\nThus the temporal evolution of density matrix can be\nunderstood as a unitary element u(/vectorh(t)) acting on the\nprevious one. If we apply the Hopf map, this process\ncan be described by Eq.(10), which is nothing but the\nfamiliarLarmorprecessionequation. Itcanbeintuitively\ninterpreted by Fig.1.3\nFIG. 1. The spin direction denoted with vector /vector xrotates\naround the eﬀective magnetic ﬁeld /vectorh, which can be used to\ndemonstrate the Dresshauls and Rashaba eﬀects on spin re-\nlaxation during the scattering.\nThe electron spin denoted as /vector xwill precess around the\nrandomly distributed eﬀective magnetic ﬁled /vectorh(/vectork) during\ntheintervalofscattering. Sincetheprocesscanbeviewed\nas a free electron motion during this time, it’s during\nthis time that the Hamiltonian has the form discussed\nabove. In the presence of momentum scattering, electron\nchanges its momentum /vectorkrandomly, hence spins precess\nrandomly between the adjacent scattering events. This\nrandom-walk-like evolution of spin phase leads to spin\nrelaxation, that is, the so called the DP mechanism19,20.\nElliott-Yafet mechanism (EY)\nAnotherimportant spinrelaxationmechanismis called\nEY mechanism21,22. It was pointed out by Elliott that\nthe spin-up and spin-down electronic eigenstates states\nmix due to the spin-orbit interaction, so the spin direc-\ntion will be ﬂipped after the scattering. Hence diﬀer-\nent kinds of spin-independent scattering can cause spin\nﬂip and thereby aﬀect the spin relaxation process. The\nspin-ﬂip matrix was usually written approximately as\nfollows23:\nΛ/vectork,/vectork′=I−iλc[(/vectork×/vectork′)·/vector σ] (12)\nwhereλcis parameters of the studied materials12. How-\never, one can describe this phenomenon geometrically as\nfollows:\nΛ/vectork,/vectork′≃u(/vector n,/vector ω) =Icos(ω/2)−i(/vector n·/vector σ)sin(ω/2)\n= exp(−i(/vector n·/vector σ)ω/2) (13)\nwhere/vector n=/vectork×/vectork′, and we assume ω= 2λc. Eq.(13) in-\ndicates that u(/vector n,/vector ω)∈SU(2), and therefore the density\nmatrix can be changed when EY process occurs, such as\nρ′=u(/vector n,/vector ω)ρ. Hence, when two electron scattering hap-\npens, we have to insert two unitary matrices Λ /vectork,/vectork′(Λ/vectork′,/vectork)\nbefore the density matrix ρ/vectork′(ρ/vectork) if considering the EY\nmechanism15. Based on the interpretation talked above,\nwecangivethespindynamicsprocesslikeinFig.2, where\nelectron spin direction /vector xof state/vectorkwill rotate an angle\nλcfrom position A to B as denoted in the sphere.\nBir-Aronov-Pikus mechanism (BAP)\nFIG. 2. The /vector nis the rotation axis of the spin direction\n/vector x,λcis the angle of rotation about the axis /vector n, during every\nscattering process, the spin-independent scattering will lead\ntoadditional spin-ﬂipbecause thisprocess proposedbyEll iott\nand Yafet, this physical picture can be described in this Blo ch\nsphere geometry like this ﬁgure. After the scattering, the\nspin will be rotated an angle λcwhich is related to speciﬁc\nmaterials. Above picture clearly indicates the ﬂip eﬀect of\nEY mechanism.\nIt was proposed by Bir, Aronov and Pikus that\nthe electron-hole exchange scattering can lead to eﬃ-\ncient electronspin relaxationin p-type semiconductors24.\nPhysically, BAP spin mechanism results from the repul-\nsion of Coulomb force and antisymmetric property of\nwavefunction. Considering both short and long range\npart of hole-electron exchange interaction, the general\nHamiltonian can be written as a compact form15,25\nHex=δ/vectorK,/vectorK′ˆJ ·ˆS, (14)\nwhere/vectorK=/vectorke+/vectorkhis the sum of electron and hole\nwavevectors which participate in the interaction. The\noperator of ˆJis a 4×4 matrix in hole spin space, ˆSis\nthe electron spin operator (also 1/2 Pauli matrix), they\ncan rotate the hole and electron spin directions, respec-\ntively. Since the exchange coupling term of Eq.(14), the\nvariation of hole spin direction will aﬀect the electron\nspin direction correspondingly. Mathematically, the hole\nspin space is equivalent to a unit sphere in 8-d Euclid\nspace,whichcanbedemonstratedusingfour-components\ncomplex-valued wave function |φ/an}bracketri}ht=/parenleftbigz1,z2,z3,z4/parenrightbigT.\nAnd if we require the wave function is normalized as\n/an}bracketle{tφ|φ/an}bracketri}ht=z∗\n1z1+z∗\n2z2+z∗\n3z3+z∗\n4z4= 1,(15)\nthen its geometry is a sphere S7in 8-d Euclid space with\nthe symmetry group SU(4). Therefore, we knowthat the\nstate|φ/an}bracketri}htcan be mapped to a normalized ﬁve-component\nreal vector in 5-d space /vector x= (x1,x2,x3,x4,x5) applying\nthe second Hopf map S7→S4. Every point in S4with\nthe components satisfying the following constraint\nx2\n1+x2\n2+x2\n3+x2\n4+x2\n5= 1 (16)\ncan be the vector of hole spin direction, which has the\nfreedom of group SO(5). So the hole spin direction’s\nvariation can be expressed as an element in SO(5) acting4\non the vector /vector xjust like the electron spin case. For a\ngiven/vectork, a ﬁxed /vector xwill single out a particular direction\nin ﬁve-dimensional vector space, the SO(5) symmetry\nwill be broken to an SO(4) symmetry. This is nothing\nbutSO(4)≃SU(2)×SU(2) symmetry of the LH (light\nhole) and the HH (heavy hole) bands26. As a general\ncase, the only subspace of HH will be important thanks\nto the much larger heavy-hole eﬀective mass12, the hole\nspin direction will be changed in this subspace. Since we\nmanily focus on the conduction electron spin dynamics\nstudy in this letter, we will only discuss the electron spin\nin the following.\nFrom another practical point, it will be convinient to\nreexpressthe spin direction or denstity matrix in another\nrepresentation when dealing with the many-body prob-\nlems (such as electron-hole exchange terms), called sec-\nond quantization representation. In fact, we can con-\nstruct the one-to-one map between the annihilation and\ncreation operators of elctron and the i-th componet of\nspin direction as follows:\nx/vectork,i=c†\n/vectorkσ(σi)σσ′c/vectorkσ′, (17)\nwherex/vectork,iis thei-th component of electron spin diren-\ntion at state /vectork,c†\n/vectorkσ,c/vectorkσare the creation and annihilation\noperators at state /vectorkwith spin σ, respectively. Similarly,\nthe density matrix may be written as\nρ=/summationdisplay\n/vectorkρ/vectork=/summationdisplay\n/vectorkσp/vectorkσc†\n/vectorkσc/vectorkσ, (18)\nwhere indices σ=↑,↓in colinear space, p/vectorkσis the statis-\ntical weight of state /vectorkwith spin σ. Therefore, generally\none can ﬁrstly analyse the physical process with anihi-\nlation/creation operators and then map them to the 3-d\nEculid space using Eq.(17). Applying this representa-\ntion, the spin ﬂip term of electron-hole exchange can be\nillustrated in Fig.3. As can be obtained in the Fenyman\nFIG. 3. Fenyman diagram of electron-heavy hole exchange in-\nteraction. The operators cke,−1/2,c†\nk+qe,1/2are electron with\nspin down annihilation and upcreation operator, respectiv ely.\nHeavy hole spin up annihilation and down creation operators\nareck′h,3/2andc†\nk′−qh,−3/2.\ndiagram, that the electron spin direction will be ﬂipped\nfrom↑at state/vectorkto↓at state/vectork+/vector qor vice versa(which is\nnotshowenhere). Thisphysicalpicturemaybedescribed\nas in Fig.4, after the exchange process, the vector of spindirection in the Bloch sphere will be reﬂected by the mir-\nror plane x3= 0. In physics, the process may be related\nFIG. 4. The schematic diagram indicate the spin ﬂip process\nof BAP mechanism. Through this mechanism the z compo-\nnent of electron spin direction is ﬂipped.\nto spin ladder operators ˆS±=ˆSx±iˆSyacting on the\ndensity matrix, but these operators are not Hermitian\noperators themselves, ˆS†\n+=ˆS−. Thus we cannot give a\nBloch sphere geometry intepretation of BAP completely\nhere.\nC. various scattering in semiconductors\nAccording to the discussions above, we know that the\ndiﬀerent scattering process will play a signiﬁcant role in\nelectron spin dynamics. They provide additional chan-\nnel for spin precession and relaxation. In this section,\nfor completeness, we review three classic scattering that\nhave been extensively used in transport theory in semi-\nconductor. We write the Hamiltonian using the second\nquantization representation, which will give us a clear\nphysical process. All of the contents discussed in this\nsection can be found in Ref.13,27.\n1.electron-phonon scattering\nA general Hamiltonian of electron-phonon interac-\ntion can be written as:\nHep=/summationdisplay\n/vectork,/vector q/planckover2pi1g/vector qc†\n/vectork+/vector qc/vectork(b/vector q+b†\n−/vector q) (19)\nwherec†\n/vectork+/vector q,c/vectorkare the fermionic electron creation\nand annihilation operators, b/vector q,b†\n/vector qare the boson op-\nerators of the phonons, g/vector qis the interaction matrix\nelement. Graphically, the electron-phonon interac-\ntion is represented by a vertex as in Fig.5.\n2.electron-electron scattering\nThe electron-electron Coumlomb scattering in\nmany-body physics can be written as:\nHee=/summationdisplay\n/vector q,/vectork,/vectork′V/vector qc†\n/vectork′+/vector qc†\n/vectork−/vector qc/vectorkc/vectork′ (20)5\nFIG. 5. Electron-phonon interaction. Left: A phonon (wavy\nline) is absorbed, while an electron (solid line) is scatter ed\nfrom state /vectorkto state/vectork+/vector q, which is represented as c†\n/vectork+/vector qc/vectorkb/vector q.\nRight: phonon emission process c†\n/vectork−/vector qc/vectorkb/vector q. The vertex is the\ninteraction g/vector q.\nwhereV/vector qis the screened Coulomb potential in the\nrandom-phase approximation27. The Fenyman di-\nagram clearly describes the scattering process as in\nFig.6.\nFIG. 6. Electron-electron interaction. Two electrons (sol id\nline) from states /vectorkand/vectork′through the Coumlomb scatter-\nring (wavy line) to the ﬁnal states /vectork−/vector q,/vectork′+/vector q, which\ncan be described by the second quantization method such as\nc†\n/vectork′+/vector qc†\n/vectork−/vector qc/vectorkc/vectork′.\n3.electron-impurity scattering\nAnothercommonscatteringprocessin semiconduc-\ntor is impurity scattering. Elastic impurity system\nwill inﬂuence the distribution ofelectron in the mo-\nmentum space. The interaction Hamiltonian reads\nHei=/summationdisplay\n/vector q,/vectorkVi(/vector q)ρi(/vector q)c†\n/vectork+/vector qc/vectork(21)\nwhereVi(/vector q) is the electron-impurity interaction\npotential with the random-phase approximation,\nρi(/vector q) is nothing but the density of the impurities\nin momentum space. This physical meaning is pre-\nsented in Fig.7.\nFIG. 7. The basic process of electron-impurity interaction .\nAn electron (solid line) from state /vectorkis scattered to state /vectork+/vector q.\nThe dotted line stands for the impuritypotential. Momentum\nis conserved at each vertex.D. The physical interpretation of KSBE\nBased on the discussion talked above, we can gen-\nerally write the dynamic equation for state of /vectorkas\nfollowing12,15:\n∂tρ/vectork=∂tρ/vectork|coh+∂tρ/vectork|scat (22)\nwhere the term ∂tρ/vectork|cohdescribe the coherent precession\ninduced by the external magnetic ﬁeld or randomly dis-\ntributed eﬀective magnetic ﬁeld induced by Dresshaules\nand Rashaba spin orbit coupling, while the scattering\nterm∂tρ/vectork|scatdepicts all kinds of scattering processes.\nThe coherenceterm of Eq.(22) describes the spin dynam-\nics during the interval of scattering. It’s during this time,\nthe electron lives in a non interacting system. So this\nprocess can be interpreted as a Larmor procession pro-\ncess in a magnetic ﬁeld or equivalent magnetic ﬁeld as in\nFig.1. The second part will mainly change the statistical\nweights of the corresponding states for spin-independent\nscattering, such as the inhomogeneous broadening mech-\nanism. And the EY spin mechanism will be contained\nwhen in the narrow gap semiconductors, we need to in-\nsert a spin ﬂip matrix when the scattering come up. The\npicture of this additional spin ﬂip can be seen in Fig.2.\nFinally, BAP spin dynamic mechanism will need to be\nconsidered when in p-type semiconductors. This term\nwill also be in the second part of Eq.(22) due to it’s a\nexchange interaction. The physical picture also can be\nunderstood in Fig.4, where the hole spin direction will\naﬀect the electron spin direction.\nThe coherent term in KSBE is given by\n∂tρ/vectork|coh=−i[Ω(/vectork)·/vector σ/2,ρ/vectork]\nthis term is nothing but the DP mechanism like term,\nthe spins precess in a random magnetic ﬁeld. Here, Ω( /vectork)\nmay contain Dresselhaus, Rashba or strain induced spin-\norbit coupling term12. The scattering term ∂tρ/vectork|scatcon-\ntains the contribution from electron-impurity ∂tρ/vectork|ei, the\nelectron-phonon scattering ∂tρ/vectork|ep, the electron-electron\nscattering ∂tρ/vectork|ee, the electron-hole Coulomb scattering\n∂tρ/vectork|eh, the electron-hole exchange scattering ∂tρ/vectork|ex,\n∂tρ/vectork|scat=∂tρ/vectork|ei+∂tρ/vectork|ep+∂tρ/vectork|ee\n+∂tρ/vectork|eh+∂tρ/vectork|ex.\nWe now give a detail description about the ﬁrst term,\n∂tρ/vectork|ei=−π/summationdisplay\n/vectork′niZiV2\n/vectork−/vectork′δ(ε/vectork′−ε/vectork)(Λ/vectork,/vectork′ρ>\n/vectork′Λ/vectork′,/vectork\n×ρ<\n/vectork−Λ/vectork,/vectork′ρ<\n/vectork′Λ/vectork′,/vectorkρ>\n/vectork)+h.c.,\nwhereniis the impurity density, Ziis the charge number\nof the impurity, ε/vectork′is the energy of conduction electron\nat state /vectork′,δfunction is the energy-conserved condition\nfor scattering processes, which is clearly indicated in the\nFenyman vertex of Fig.7, V/vector qis the screened Coulomb po-\ntential, Λ /vectork′,/vectorkis the spin ﬂip matrix which describes the6\nEY mechanism of scattering, this matrix can be removed\nif the EY mechanism isn’t included, then the scattering\ncan describe the inhomogeneous broadening mechanism\nand so on. And ρ<\n/vectork=ρ/vectorkis the electron density matrix\nwith state /vectorkand statistical probability p/vectork, soρ>\n/vectork= 1−ρ/vectork\ncan be interpreted as the statistical density matrix when\nthe state is empty. Here, the δfunction comes from the\napproximation of Markov approximation process, which\nmeans that the previous scattering process doesn’t af-\nfect the next one. The sum over diﬀerent states in the\nright hand side of equation is operable only if the scatter-\ning process satisﬁes the conservation of momentum and\nenergy. Therefore, one will get a diﬀerential equation\ngroup. By solving the equation group numerically, the\nmacroscopicphysicalquantities, such asthe electronspin\ndensity along the z-axis, can be obtained using Eq.7.\nThe other scattering terms like electron-electron,\nelectron-phonon scattering can be found in Wu’s re-lated works, they possess similar structure like electron-\nimpurity scattering above. 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Cheng, and M. Wu, Physical Review B 75,\n045305 (2007).\n15J. Jiang and M. Wu, Physical Review B 79, 125206 (2009).\n16L. Han, Y. Zhu, X. Zhang, P. Tan, H. Ni, Z. Niu, et al.,\nNanoscale research letters 6, 84 (2011).\n17C. Zhao, T. Yan, H. Ni, Z. Niu, and X. Zhang, Applied\nPhysics Letters 102, 012406 (2013).\n18K. Blum, Density matrix theory and applications , Vol. 64\n(Springerverlag Berlin Heidelberg, 2012).\n19M. D’yakonov and V. Perel, Soviet Journal of Experimen-\ntal and Theoretical Physics 33, 1053 (1971).\n20M. Dyakonov and V. Perel, Sov. Phys. Solid State 13, 3023\n(1972).\n21R. . J. Elliott, Physical Review 96, 266 (1954).\n22Y. Yafet, Solid State Physics 14, 1 (1963).\n23G. Pikus and A. Titkov, Optical Orientation 8, 73 (1984).\n24A. Aronov, G. Pikus, and A. Titkov, Sov. Phys. JETP 57,\n680 (1983).\n25M. Maialle, Physical Review B 54, 1967 (1996).\n26S. Murakami, N. Nagosa, and S.-C. Zhang, Physical Re-\nview B69, 235206 (2004).\n27H. Haug and S. W. Koch, Quantum theory of the optical\nand electronic properties of semiconductors (World Scien-\ntiﬁc Publishing Company, 2004)."    },    {        "title": "0803.4271v1.Quantized_spin_waves_and_perpendicular_standing_spin_waves_stimulated_by_current_in_a_single_layered_ferromagnetic_wire.pdf",        "content": "Quantized spin waves and perpendicula r standing spin waves stimulated by \ncurrent in a single-layered ferromagnetic wire  \n \nA. Yamaguchi1, 2, K. Motoi1, and H. Miyajima1 \n \n1 Department of Physics, Keio University, Hiyoshi, Yokohama 223-8522, Japan \n2 PRESTO, JST, Sanbancho 5, Chiyoda, Tokyo 102-0075, Japan \n          [Abstract] \n The rectifying effect of radio-frequency (RF) current is highly sensitive in terms of the \nspatial spin distribution and dynamics. It emer ged that an additional spin wave mode was \nstimulated by the direct-current (DC) current and that this spin wave was detectable via \nrectification of the RF current. A phenome nological model to describe the time-dependent \nanisotropic magnetoresistance or time-dependent planer Hall effect is proposed and found to \ncorrelate well to the experimental results . The nonlinear spin dynamics accompanying \nadditional spin waves are studied as functions of the RF and DC currents, the external magnetic \nfield, and the applied field direction.           The understanding of spin dynamics in artif icial nano-magnets is  important, not only \nin fundamental magnetism but also in the tec hnological areas. One of the distinctive spin \ndynamics in the high frequency region is the sp in wave excitation: an extended ringing of the \nmagnetization produces high frequency res onance characteristic to the devices1, 2. In nano-scale \nferromagnetic devices, both exchange and dipole ener gies contribute to the spin wave spectrum, \nwhich is strongly dependent on the system geometry1, 2. The spin wave resonance and \nmagnetization dynamics in the confined geometry  have been investigated via Brillouin light \nscattering (BLS)2, 3, ferromagnetic resonance (FMR), time -resolved magneto-optical Kerr effect4 \n- 7, the rectifying effect8 – 13 , and so on. \n When the spin-polarized current traver sing a ferromagnetic conductor transfers its \nspin-angular momentum to the magnetic system14, 15, it causes the magnetization to precess with \nstrong s-d exchange interaction between th e conduction electron and the magnetic moment14, 15. \nRecently, electrical detections of the magneti zation motion have been performed; not by \napplying an alternating-current (AC) magnetic fi eld but by using the spin-transfer torque of a \nspin-polarized AC current9 - 11. It also reveals a detailed understanding of the spin dynamics due \nto the conduction electrons and the magnetic moments. \nAs is well known, the propagation of elec tromagnetic waves though the ferromagnetic \nconductors produces some nonlinear effects, refl ecting the interaction between current and magnetization via magnetoresistance and extrao rdinary Hall effects. The non-linear effect \noccurs remarkably within the frequency region be havior of FMR and is detectable by using \nelectrical measurements. We have studied the magnetic behavior of FMR in a single layered Ni\n81Fe19 ferromagnetic wire by the simultaneous a pplication of direct-current (DC) and \nradio-frequency currents. \nIn a previous paper11, we proposed a phenomenological model for the \nmagnetoresistance response induced by the nonlin ear effect and showed a consistent view \ndescribing the DC voltage generation in a single- layered ferromagnetic wire. In this paper, we \npropose an analytical model; taking advantage of the well-resolved frequency-domain DC spectra induced by the rectifying effect. Not only the quantized spin wave excitation but also the \nperpendicular standing spin wave (PSSW)\n1 are induced by the DC current through the \nrectification of the PHE. \nThe experiments are performed on a 30nm-thick Ni 81Fe19 wire. The wire is fabricated \nonto an MgO substrate via electron beam lithography and the liftoff method. Figure 1 shows an \noptical micrograph of the wire of width 5 µm together with the electric measurement circuit. The \nsinusoidal constant wave (CW) RF current with current density of 103.0 10×  A/m2 is injected \ninto the wire by a signal generator w ith a frequency range from 10 MHz to 15 GHz. \nSimultaneously, the DC current is applied to th e wire through the bias-tee, which separates the DC- and RF-components of the current and the external magnetic field extH is applied in the \nsubstrate plane as a function of angle φ from the major axis of the wire. The precession of the \nmagnetic moment in the vicinity of FMR region generates the DC voltage  attributable to the \nmagnetoresistance oscillation. The experiment is performed at room temperature with the \nslowly sweeping frequency of the RF current that flows along th e major axis of the wire. The \nHall voltage spectra, HallV , induced across the minor axis of the wire, is also measured. \nOn the standpoint of the Mott’s two current model, the electrical field E in a \nferromagnetic metal is given in the following form: \n()() H ρρ ρ ρ⊥⊥=+⋅ ⋅ −+×Ej mjmm j& ,                 ( 1 )  \nwhere j denotes the electrical current density, mthe unit vector along the local \nmagnetization, ρ⊥ and ρ& the resistivity parallel and perpendicular to j with respect to \nm and Hρ the extraordinary Hall resistivity. Juretschke8 introduced a phenomenological \noscillating component of the magnetization to Eq. (1), and pointed out the possibility of the \ngeneration of the DC voltage when the magnetization precession was caused by the RF field. \nIn general, the magnetization process in a ferromagnetic wire is well described by the \nStoner-Wohlfarth single domain model. As shown in  Fig. 1, when the magnetization unit vector \nis located at the origin, ()sin cos ,sin sin ,cosθφθφθ =m , and the electrical current flows \nalong the longitudinal axis of the wire, (),0,0j=j , the electrical field E is given by: ()\n()\n()22\nx\n2\nyH\nzHsin cos\nsin cos sin cos\nsin cos cos sin sinE\nEj\nEρρ ρ θφ\nρρ θ φφρ θ\nρρθ θ φ ρ θ φ⊥⊥\n⊥\n⊥⎛⎞ +− ⎛⎞⎜⎟⎜⎟== − + ⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−− ⎝⎠⎝⎠E&\n&\n&.             ( 2 )  \nWhen the magnetization directs along the major axis (the magnetic easy axis), xE and yE \nalong the magnetic easy axis (x-axis) and the minor-axis (y-axis) in Eq. (2) are respectively \ngiven by: \n()()2\nx cos Ejρρρ φ⊥⊥=+ −& ,                                 ( 3 )  \n        () y cos sin Ejρρφ φ⊥ =−& .                                    ( 4 )  \nThese Eqs. (3) and (4) are known to represent the AMR effect and the planer Hall effect, \nrespectively. \nTaking into account the forced magnetization precession in the presence of a spin \ntransfer effect14, 15 and the non-uniform internal magnetic field16, the electric field components \nxE and yE are given by: \n()() ()() { }2cosxEtj tρρ ρ φ δ⊥⊥=+ − +& ,                        ( 5 )  \n()() ()()()() cos sinyEtj t tρρ φ δ φ δ⊥ =− + +& .               ( 6 )  \nBy expanding the Eqs. (5) and (6) by the add ition formula, we can approximately derive the \naverage time-dependent electric fields due to the AMR effect and planer Hall effect as: \n() () ()2\nx1cos sin 2 sin 22Et j t ρρ ρ φ φδ\n⊥ ⊥⎡⎤ ⎛⎞≈+ − −⎢⎥ ⎜⎟⎝⎠ ⎣⎦&,                 ( 7 )  \n() ()y1s i n2 c o s2 s i n22Et j t φφ δ⎡⎤≈+⎣⎦.                        ( 8 )  \nWhen the CW RF current is introduced into the wire in the FMR state, the magnetization \nmotion is a steady-state precession mode, compensating for the actual loss of the precession amplitude. Suppose that ()()0cos jt j t ω= , the frequency variation of the induced voltage, as \nderived from the AMR effect and the PHE, is given by the Fourier transformation of ()tE ; \n() ()2\nx0 sin 2 cos Ej Aω ωφ φ= ,                                             ( 9 )  \n() ()2\ny0 cos 2 cos Ej Aω ωφ φ= ,                                             ( 1 0 )  \nwhere ()Aω is the frequency spectrum given in the previous papers11, 17. \nFigure 2(a) shows the RF frequency dependence of the output signal HallV  for the DC \ncurrent DCI= +15 mA (red solid line) and DCI=0 mA (black solid line) in the external \nmagnetic field extH  = 0.5 kOe applied at the angle 45φ=D. Here, the non-resonant \nbackground, which is much stronger than the res onant signal, is subtracted for clarity. As shown \nin Fig. 2(a), the spectrum for DC0 I= contains at least two distinct modes near 6.2 GHz and \n7.1 GHz, while the spectrum for DC 15mA I=+  has a additive distinct modes near 10.8 GHz. \nFigure 2(b) shows how the relationship between DCI, and the Hall voltage difference 2V∆ \ncan be well described by 2 PHE DCVR I∆= ∆ ⋅ , where PHER∆  is the planer Hall resistance. The \nestimated PHER∆ is PHER∆ =0.032 m Ω for ()( ) ext,4 5 , 0 . 5 k O eφ= HDand PHER∆ = –0.053 \nmΩ for ()( ) ext, 120 ,0.2 kOeφ= HD. It should be noted that the sign and value of PHER∆  \nstrongly correlates with the direction of the magnetization and precessional angle. \nThe magnetic field dependence of the spin  wave frequency of each spin mode is \nshown in Fig. 3 and all observed modes are c onfirmed to be attributable to the magnetic excitations. The two modes observed at a lower fre quency (red circles and bl ue squares) and one \nmode at the higher frequency (black triangles) correspond to the quantized spin waves derived \nfrom the quantized Damon-Eshbach mode and the PSSW in a single wire, respectively1, 2, 18. We \nobtain an empirical expression describing the co mplete spin wave modes with the quantized \ninteger numbers. \nAccording to Kalinikos and Slavin1, 2, 18, the dispersion of spin waves in a confined \nmagnetic structure is given by: \n            ()22 2 2 224Sp p\nSSAAHq Hq M F q dMMωγ π⎛⎞ ⎛ ⎞=+ ++⋅⎜⎟ ⎜ ⎟\n⎝⎠ ⎝ ⎠& ,            ( 1 1 )  \nwhere  \n                  ()2\n22 2 2 2\nxyppqqq qqdπ\n⊥−∆⎛⎞=++ =+ ⎜⎟\n⎝⎠& ,                   ( 1 2 )  \nfor the normal to the film surface, q& is the in-plane wave vector. Here, since the wire length is \nmuch longer than the width, the magnetization vect or almost aligns parallel to the longitudinal \naxis of the wire (x direction). Suppose that the quantization along the x direction is neglected \nand then the y component is quantized as qn wπ=& , where w is the wire width and the \ninteger n is the quantization number for the qu antized Damon-Eshbach mode (Q-DE). The \nquantization of the z component is given by ()/ qp p d π⊥=− ∆  for the wire thickness d, \nwhile the quantized number p is the integer of a half wavelength along the zdirection, \nwhich corresponds to the PSSW. Meanwhile, the correction factor p∆ (01 p≤∆ ≤ ) is determined by the boundary condition, and () ppFq d& is the matrix element of the magnetic \ndipole interaction17. In the present analytical spin wave modes, these parameters are determined \nas (),np p−∆ =(0, 0), (1. 0) and (0, 0.47) and the calculated lines using them correlate well to \nthe experimental data. As shown in Fig. 3, the size effect is easily taken into account for the \nlong wire in a single domain state because the dipole field is homogeneous, whereipon we obtain the so-called Kittel mode\n1, 19: \n() ( )()( )22\nyx S zx S HNN M HNN Mωγ=+ − ⋅ + − ,              ( 1 3 )  \nwhere SM denotes the saturation magnetization, and ,,xyzNNN  the demagnetization \ncoefficient along the x, y, and z axes. In  the present case of the line shape geometry, 0xN≈ \nand 1zyNN≈− , and the dynamic demagnetization field is assumed to be considerably \nhomogeneous. The magnetic field dependence, as calc ulated by Eq. (13), is shown by the red \ndashed line in Fig. 3. The correlation with the experiment is convincing, indicating that the \nsimple analysis is available for the rectificati on. Furthermore, it means that the RF current \nresults in the resonant excitation of the pure uniform magnetization precession and that the \ncurrent can excite various spin wave modes characteristic to the confined structure. \n The significance of this experiment is that not only the Q-DE and PSSW modes are \ninvolved but also the DC current-induced PSSW mode. Here, we discuss the origin of the PSSW mode induced by the DC current. To simplify the analysis of the magnetization dynamics, we focus on the resonant excitation derived by th e single mode of the magnetization precession, as \ndescribed by Eqs. (8) and (10). Figure 4 show s the variation of the DC voltage difference V∆ \nwith the applied magnetic field angle φ; (a) 1V∆  for DC0m A I= , (c)2V∆  for \nDC 15mA I=+ , and (b) the non-resonant background voltage bgV for DC 15mA I=+ . \nAccording to Eqs. (8) and (10), bgV and 1V∆ vary with the angle as sin 2φ and  \ncos 2 cosφφ, respectively. The voltage bgV is derived from the time independent term sin 2φ \nin Eq. (8), while 1V∆ originates from the time dependent magnetization precession term in Eq. \n(8). Indeed, the experiment fits well to the present analytical curve, suggesting that the \nmagnetization precession at the finite frequency is the uniform mode. Coversely, the angle \ndependence of 2V∆  with DC 15mA I=+  correlates well to the sin 2φ line. This result \nindicates that the PSSW mode induced by the DC current may be related to the magnetization \ndynamics in the plane caused by the dynamic tor que perpendicular to the film surface. There are \ntwo possibilities; one is the gradient of the ma gnetic field generated by the DC current flowing \nthough the wire and electrodes11, the other is the presence of th e absorption and dissipation of \nspin angular momentum through the anisotropic energy K⊥ (see Fig. 4(d), Tatara and \nKohno20) or the spin relaxation (Zhang and Li21) derived from the spatial spin distribtuion16. In \nparticularly, there is definite potential for the latter to play an important role in the spin \ndynamics in the RF region. We showed the PHE rectification effect of the (DC+RF) current in a single layered \nferromagnetic wire, which exhibits time-depe ndent magnetoresistance and reveals the PSSW \ninduced by the DC current. Although the or igin of the DC current-induced PSSW remains \nunclear at present, the results will provide a new way for studying spin dynamics in the confined \nmagnetic structure with the correlation betw een the localized magnetic moments and the \nconduction electrons. \nThe present work was partly supported by the MEXT Grants-in-Aid for Scientific \nResearch in Priority Area, the JSPS Grants-i n-Aid for Scientific Research and the Keio \nLeading-edge Laboratory of Scie nce and Technology project 2007. \n \n         [References] \n1) B. Hillebrands and K. E. Ounadjela, Spin  Dynamics in Confined Magnetic Structures \n(Springer, Berlin, 2003), V ols. 1 - 3. 2) C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov, R. Kouba, R. Bozinoski, A. N. Slavin, \nK. Y . Guslienko, D. V . Berkov, N. L. Gorn, and M. P. Kostylev, Phys. Rev. B 72 , 064427 \n(2005).: S. O. Demokritov, B. Hillrbrands, and A. N. Slavin, Phys. Rep. 348, 441 (2001). \n3) A. Ercole, A. O. Adeyeye, J. A. C. Bland, and D. G. Hasko, Phys. Rev. B 58 , 345 (1998). \n4) A. Barman, V . V . Kruglyak, R. J. Hicken, J. M. Rowe, A. Kundrotaite, J. Scott, and M. \nRahman, Phys. Rev. B 69 , 174426 (2004). \n5) M. Belov, Z. Liu, R. D. Sydora, and M .R. Freeman, Phys. Rev. B 69 , 094414 (2004). \n6) F. Giesen, J. Podbielski, T. Korn, M. Steiner,  A. Van Staa, and D. Grundler, Appl. Phys. Lett. \n86, 112510 (2005). \n7) M. Bailleul, R. Höllinger, and C. Fermon, Phys. Rev. B 73 , 104424 (2006). \n8) H. J. Juretschke, \nJ. Appl. Phys . 31, 1401 (1960).; W. M. Moller and H. J. Juretschke, Phys. \nRev. B 2, 2651 (1970). \n9) A. A. Tulapurkar et al. , Nature  438, 339 (2005). \n10) J. C. Sankey et al., Phys. Rev. Lett . 96, 227601 (2006). \n11) A. Yamaguchi, H. Miyajima, T. Ono, Y . Suzuki, and S. Yuasa, Appl. Phys. Lett. 90, 182507 (2007); A. Yamaguchi, H. Miayajima, S. Kasai, and T. Ono, ibid. 90, 212505 (2007); A. \nYamaguchi, H. Miyajima, T. Ono, Y . Suzuki, and S. Yuasa, ibid. 91, 132509 (2007). \n12) M. V . Costache, M. Sladkov, C. H. van de r Wal, and B. J. van Wees, Appl. Phys. Lett. 89, \n192506 (2006).: M. V . Costache, S. M. Watts, M. Sladkov, C. H. van der Wal, and B. J. van \nWees, Appl. Phys. Lett. 89, 232115 (2006). \n13) Y . S. Gui, N. Mecking, X. Zhou, Gwy n Williams, and C. –M. Hu, Phys. Rev. Lett. 98, \n107602 (2007); N. Mecking, Y . S. Gui, and C. –M. Hu, Phys. Rev. B 76 , 224430 (2007). \n14) J. C. Slonczewski, J. Magn. Magn. Mater . 159, L1 (1996). \n15) L. Berger, Phys. Rev. B 54 , 9353 (1996). \n16) M. Bailluel, D. Olligs, and C. Fermon, Phys. Rev. Lett. 91, 137204 (2003). \n17) A. Yamaguchi, K. Motoi, and H. Miayjima, Y . Nakatani, arXiv:0710.2172; A. Yamaguchi, K. \nMotoi, H. Miyajima, Y . Miyashita, and Y . Sanada, arXiv:0801.2203. \n18) B. A. Kalinikos and A. N. Slavin, J. Phys. C 19 , 7013 (1986). \n19) C. Kittel, Introduction to Solid State Physics Ch. 16 (John Wiley, New York, 1986), 6th ed. \n20) G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). \n21) S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).  \n \n [Figure captions] \nFigure 1 \nSchematic diagram of RF measurement; the top vi ew of the optical micrograph of the sample, \nand the model geometry and symbol definitions. \n \nFigure 2 \n(a) Typical Hall voltage spectra with DC0m A I=  and DC 15mA I=+  measured in the \napplication of magnetic field ext0.5kOe=H  at 45φ=D. Each spectrum is vertically shifted \nfor clarity. (b) The DC current dependence of the Hall voltage difference 2V∆ . \n \nFigure 3 \n Resonant frequencies generated in the Hall volta ge spectra as a function of the applied field. \nThe lines are calculated using either a Q-DE equa tion with quantized wave vectors (black solid \nline labeled Q-DE), the Kittel formula (red dashed  line), or the first PSSW mode (black solid \nline labeled PSSW). The typical Hall voltage spectrum is shown in the inset.    Figure 4 \nMagnetic field angle dependence of (a) the Hall voltage difference 1V∆ with DC0m A I= , \n(b) the non-resonant Hall voltage bgV with DC 15mA I=+ , and (c) the Hall voltage difference \ninduced by the DC current DC 15mA I=+  in the magnetic field of 0.5kOeext=H . \n \n             Bias-tee\nSignal\ngeneratorCurrent\nSource30 nm-thick Ni81Fe19\nMgOH\nφ\nAu\nVV olt meter\n10 µm\ny\nxz\nm\nθ\nφ\n Ni81Fe19 wire \n \nFig. 1         VHall (µV)\n15 10 5 0\nω/2π (GHz)IDC=+15m A\nIDC=0mAφ=45\n0.5 kOe(a)\n∆V1∆21µV\n-1.0-0.500.51.0 ∆V2 (µV)\n-30 -20 -10 0 10 20 30IDC (mA)φ=45\n0.5 kOeφ=120\n0.2 kOe(b)V\n \nFig. 2 \n   15\n10\n5\n0ω/2π (GHz)\n1.0 0.5 0\nHext (kOe)PSSW\nQ-DEn=0n=1φ=135\nVHall (µV)\n15 10 5 0ω/2π (GHz )0.5 µV\n \nFig. 3 \n    \n \n   1000\n500\n0Vbase (µV)\nIDC = +15 mA(b)\nsin2φ\n-1.001.0∆V2 (µV)\n360 270 180 90 0\nφ (deg.)(c)\nsin2φ\nIDC = +15 mA-505∆V1 (µV)\nIDC = 0 mA\n0.5 kOe(a)\ncos2φcosφ\nM MK\nSpin transfer torque\nsscurrenteasy-axishard-axis1\nhard-axis2\nδ\u0000d\u0000\n \nFig. 4 "    },    {        "title": "0812.2509v1.Spin_Transfer_Torque_in_Helical_Spin_Density_Waves.pdf",        "content": "arXiv:0812.2509v1  [cond-mat.mtrl-sci]  13 Dec 2008Spin-Transfer Torque in Helical Spin-Density Waves\nO. Wessely,1,2,3B. Skubic,3and L. Nordstr¨ om3\n1Department of Mathematics, Imperial College, London SW7 2B Z, United Kingdom\n2Department of Mathematics, City University, London EC1V 0H B, United Kingdom\n3Department of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden\n(Dated: February 6, 2020)\nThe current driven magnetisation dynamics of a helical spin -density wave is investigated. Expres-\nsions for calculating the spin-transfer torque of real syst ems from ﬁrst principles density functional\ntheory are presented. These expressions are used for calcul ating the spin-transfer torque for the\nspin spirals of Er and fcc Fe at two diﬀerent lattice volumes. It is shown that the calculated torque\ninduces a rigid rotation of the order parameter with respect to the spin spiral axis. The torque is\nfound to depend on the wave vector of the spin spiral and the sp in-polarisation of the Fermi surface\nstates. The resulting dynamics of the spin spiral is also dis cussed.\nI. INTRODUCTION\nThe spin-transfer torque (STT) provides a means of ma-\nnipulating the magnetisation of a material using a cur-\nrent. The eﬀect was proposed from theoretical consid-\nerations by Slonczewski1and Berger2and has attracted\na lot of attention due to its potential use in applications\nwhere a magnetic state is altered by a current as opposed\nto traditional techniques involving magnetic ﬁelds.\nMost theoretical and experimental work on the STT\nconcerns layered magnetic structures where the STT ap-\npears as a consequence of angular momentum conserva-\ntion as a current traverses an interface between two re-\ngions of non-parallel magnetisation. Recently3we have\nshown that a STT also occurs in systems with a heli-\ncal spin-density wave (SDW). This was shown by a ﬁrst-\nprinciples calculation of the STT for a bulk rare-earth\nsystem (Er) in its low temperature helical spin spiral\n(SS) structure. The eﬀect of the current induced STT\nin Er is a rigid rotation of the SS order parameter. This\nphenomenon is a bulk eﬀect in contrast to STT in lay-\nered systems where the STT mainly occurs close to the\ninterfaces between the layers.\nIn this Article we present a more general discussion\non the current induced torque in a SS. First the eﬀect\nis illustrated using a one-dimensional model of a SS. We\nin the subsequent section present ﬁrst-principles calcula-\ntions of the STT for three real systems, Er and fcc Fe\norγ-Fe at two diﬀerent lattice volumes. Previous ﬁrst-\nprinciples calculations have shown that both Er and γ-Fe\nhave helical SDWs.4,5In many other respects these two\nsystems are very diﬀerent. Erbium is one of several rare-\nearth elements that exhibit a helical SDW and where the\nordering is driven by a nesting between parallel sheets of\nthe Fermi surface (FS). Iron belongs to the 3d-transition\nmetals where SDWs are less frequent. The fcc phase of\nFe, which is the phase where a SS order has been found\nboth in experiments and theory, has only been stabilised\nat very special conditions, and it is believed that the\nhelical SDW is the combined result of several parts of\nthe FS. For the one-dimensional model, as well as the\nconsidered real systems, we ﬁnd that an applied current\nFIG. 1: Local magnetisation direction of a helical spin dens ity\nwave. The spin density wave is characterised by the SS wave\nvectorq, the cone angle θ, and the rotation angle φ=q·r.\ninduces a STT generating a rigid rotation of the spiral\norder. With this theoretical prediction, new types of po-\ntential applications of the STT can be imagined, such as\ncurrent driven oscillators with tunable frequencies. The\ncurrent driven spin dynamics for such a device is investi-\ngated in section (VII)\nII. THEORY\nFor a system with a helical SDW, the direction of the\nlocal magnetisation rotates around the SS quantisation\naxis as one moves in the direction of the SS wave vector.\nWithout loss of generality, we here consider the rotation\naxis to be parallel to the SS wave vector, except when\nexplicitly stated, and will refer to it as the SS axis. Al-\nthough we here consider helical SDWs, the results are\nvalid for cycloidal SDWs as well. A SS is characterised\nby its SS wave vector qand its cone angle θ, which is\nthe angle between the SS axis and the local magnetisa-2\ntion, see Fig. 1. In this section we ﬁrst consider a one-\ndimensional model of a SS, where the spin currents and\nSTT are calculated analytically. We then discuss how\naccurate ﬁrst-principles calculations of the STT can be\nperformed.\nA. The STT in a 1D model of a spin spiral\nAs a simple one-dimensional single band model of a\nSS system, we consider independent particles in a spin-\ndependent potential. The spin-dependent potential is\nchosen such that the ground state of the system has a\nhelical magnetisation M=Mˆmwhere\nˆm= (sinθcosqz,sinθsinqz,cosθ).(1)\nHere, the SS wave vector q=qˆzandθis the cone angle.\nThe single particle Schr¨ odinger or Kohn Sham equation\nfor such a model system takes the form\n/braceleftig\n−¯h2\n2me∂2\n∂z2+U\n2/bracketleftbig\nsinθ(cosqz σx+sinqz σy)\n+cosθ σz/bracketrightbig\n−ǫ/bracerightig\nψ= 0,\nwhereUis the exchange splitting, meis the electron\nmass, ¯his the Planck constant σx, σy, σzare the Pauli\nmatrices and ǫis the one particle energy. This equation\ncanbe solvedanalytically6and theeigenfunctions arethe\nso called generalised Bloch states,\nψk=/parenleftbigg\nψnk(z,↑)\nψnk(z,↓)/parenrightbigg\n=eikz/parenleftbigg\ncos (θk/2)e−iqz/2\nsin (θk/2)eiqz/2/parenrightbigg\n.(2)\nFor the above wave function the direction of the spin ˆs\nrotates around the SS axis, with rotation angle φ=qz,\nat an angle θkwith respect to the SS axis,\nˆs(s) = (sinθkcosqz,sinθksinqz,cosθk).(3)\nNote that the single electron cone angle θkand the cone\nangleθof the SS, see Fig. 1, in general are diﬀerent. In\none dimension, the chargecurrent j, induced by applying\nan electric ﬁeld, is a scalar given by\nj(z) =−δfe¯h\nmeRe/parenleftig\nψ∗\nkFi∂\n∂zψkF−ψ∗\n−kFi∂\n∂zψ−kF/parenrightig\n,(4)\nwhereδfis the change in occupation at the Fermi level\ndue to the electric ﬁeld, kFis the Fermi wave vector and\neis the elementary charge. In a similar way, the spin\ncurrentQdue to a weak electric ﬁeld is a vector given\nby\nQ(z) =−δf¯h2\n2meRe/parenleftig\nψ∗\nkFi∂\n∂zσψkF−ψ∗\n−kFi∂\n∂zσψ−kF/parenrightig\n.\n(5)\nFrom the spin current can the rate of change of angular\nmomentum within an inﬁnitesimal region of the system\nbe calculated. The rate of change or torque on the regionbetweenz0andz0+dzis given by the spin ﬂux into the\nregion\n∂J\n∂t(z0) =Q(z0)−Q(z0+dz) =−∂Q\n∂z/vextendsingle/vextendsingle/vextendsingle\nz0dz.(6)\nBoth the spin-ﬂux ∂J/∂tand the current jdepend on\nthe change in occupation of the states at the Fermi level\ncaused by the applied electric ﬁeld. These two quantities\nare linked by the expression,\n∂J\n∂t=Cj, (7)\nwhich deﬁnes the torque current tensor C. The torque\ncurrent tensor is in three dimensions a matrix, but be-\ncomes a vector in a one-dimensional system, where also\nthe current vector is reduced to a scalar. For our model\nsystem, the torque current tensor can be calculated by\ninserting the Fermi Bloch states, ψkFandψ−kFfrom\nEq. (2) into Eq. (4)-(5), resulting in,\nC=∂Q\n∂z/vextendsingle/vextendsingle/vextendsingle\nz0dz/j(z0) =a(θ,kF)\n−qsin (qz0)\nqcos (qz0)\n0\ndz,\n(8)\nwheretheﬁrstfactor, a(θ,kF), dependsonthespiralcone\nangleθand the Fermi wave vector kF. For planar spin\nspirals, where θ=π/2, the factor reduces to a simple\nfunction of the single electron cone angle θkof the states\nat the Fermi level given by\na(π/2,kF) =¯h\n2esinθkF\n1−(q/2k)cosθkF.(9)\nAcomparisonbetween Eq.(1) andEq.(8) showsthat the\ncurrent induced torque, in our one-dimensional model, is\nperpendicular to the magnetisation direction and lies in\nthe plane of rotation of the SS, thus causing the SS to\nrotate with respect to the SS axis. It can also be seen\nfrom Eq. (8) that the magnitude of the STT is governed\nby the length of the SS wave vector qand the factor\na(θ,kF).\nB. The STT for a real system\nThe STT for a real system can be calculated by general-\nising the expressions for the charge and spin currents in\none dimension to three dimensions. In three dimensions\ncan the currents induced by an external electric ﬁeld, to\nlinear order in the ﬁeld, be calculated from a FS integral.\nFor a multiband system must the contribution from all\nbands crossing the Fermi level be taken into account. In\nthe previous section was a model system and the STT on\nan inﬁnitesimal region at an arbitrary point of the sys-\ntem considered. For a real material, where most of the\nangular momentum is localised on the atoms is the time\nevolution of the atomic moment governed by the STT on\nthe corresponding atom. The torque on an atom is given3\nby the spin-ﬂux into a sphere surrounding the atom. For\na single electron state with band index nand wavevector\nk, the ﬂux into a sphere of radius Ris given by,\n−/integraldisplayπ\n0/integraldisplay2π\n0Qnk·ˆrRsinθ dφdθ=∂Jnk\n∂t.(10)\nInthreedimensionsistheoneelectronspincurrenttensor\nQnk(r) = Re/braceleftig\nψ†\nnk(r)s⊗vψnk(r)/bracerightig\n,(11)\nwheres= (¯h/2)σ, andv= (−i¯h/me)∇. The total spin\ncurrent is the sum of the spin currents from all occupied\nstates. A external electric ﬁeld Einduces a non equi-\nlibrium spin current by changing the occupation of the\nstatesδfat the FS. The change in occupation number\nδfcan be calculated using the relaxation time approxi-\nmation and semiclassical Boltzmann theory,7\nδf(k) =−δ(ǫnk−ǫF)τe\n¯h∇kǫnk·E,(12)\nwhereτis the electron relaxation time and ǫnkis the\nband energy. The total non equilibrium torque on an\natom can be calculated from a FS integral,\n∂J\n∂t=−VCτe\n(2π)3¯h/summationdisplay\nn/integraldisplay\nFS∂Jnk\n∂t(∇kǫnk·E)dSnk\n|∇kǫnk|,(13)\nwhereVCis the volume of the system and a summa-\ntion is performed over all bands crossing the Fermi level.\nThe above equation deﬁnes a linear relation between the\ntorque and the external ﬁeld,\n∂J\n∂t=τ/summationdisplay\nnAnE. (14)\nA similar relation can be obtained for the charge current\ndensity and the electric ﬁeld,\nj=τ/summationdisplay\nnBnE (15)\n=1\n(2π)3τe2\n¯h2/summationdisplay\nn/integraldisplay\nFS∇kǫnk(∇kǫnk·E)dSnk\n|∇kǫnk|,whereBnis the conductivity tensor for band n. By\ncombining Eqs. (14)-(15) is a linear relation between the\ntorque and the current density obtained, where the un-\nknown electron relaxation time τhas been cancelled,\n∂J\n∂t= (/summationdisplay\nnAn)(/summationdisplay\nmBm)−1j=Cj.(16)\nThe above equation deﬁnes the spin current tensor Cin\n3 dimensions.\nC. The STT from the augmented plane wave\n(APW) method\nThe wave functions used in the previous section to cal-\nculate the torque current tensor Ccan be obtained from\nﬁrstprincipleselectronicstructurecalculationsusingspin\ndensity functional theory. Several electronic structure\nmethodsuse basissetswherespaceisdivided intomuﬃn-\ntin spheres surrounding the atoms and an interstitial re-\ngion. This division is introduced in order to simplify\nthe calculation of the intra and interatomic behaviour of\nthe wave functions. When calculating the STT, a nat-\nural choice is to use the atomic augmentation (muﬃn-\ntin) sphere as the surface for evaluating the STT on the\natoms. We now illustrate more in detail how the calcula-\ntionmaybecarriedoutwithin the augmentedplanewave\nmethod (APW). The APW expansion can be written as\na sum of plane waves\nψnk(r) =/summationdisplay\nG/parenleftbigg\nank,Gei(G+k−q/2)r\nbnk,Gei(G+k+q/2)r/parenrightbigg\n,(17)\nwhereGare the reciprocal lattice vectors. The plane\nwave coeﬃcients ankandbnkare obtained from the ﬁrst\nprinciples calculation. The spin-ﬂux into a sphere with\nradiusRcentred at an atom at site rmis for plane waves\ngiven by the expression\n/integraldisplayπ\n0/integraldisplay2π\n0Qnk·ˆrRsinθ dφdθ=¯hR2\nmeRe/summationdisplay\nG,G′−i4π/bracketleftig\nα†sαa∗\nnk,Gank,G′e−i(G−G′)rmj1(|G−G′|R)(G′+k−q/2)·(/hatwidestG−G′)\n+α†sβ a∗\nnk,Gbnk,G′e−i(G−G′−q)rmj1(|G−G′−q|R)(G′+k+q/2)·(/hatwidestG−G′−q)\n+β†sαb∗\nnk,Gank,G′e−i(G−G′+q)rmj1(|G−G′+q|R)(G′+k−q/2)·(/hatwidestG−G′+q)\n+β†sβ b∗\nnk,Gbnk,G′e−i(G−G′)rmj1(|G−G′|R)(G′+k+q/2)·(/hatwidestG−G′)/bracketrightig\n,(18)\nwherej1is the ﬁrst spherical Bessel function and α,β\nare the up and down spinors respectively. With this ex-pression evaluated it is straight forward to calculate the4\nSTT using Eq. (10) and Eq. (13).\nIII. MATERIALS WITH SPIN SPIRAL\nMAGNETIC ORDER\nLong ranged magnetic order in form of helical SDWs ex-\nists in several types of materials. Maybe the most fa-\nmous is the heavyrareearth elements. They havesimilar\nvalence electron structure, resulting in similar chemical\nstructure, and all the tri-valent elements order in the hcp\nlattice structure with similar lattice volumes. The main\ndiﬀerence along the series lies in the ﬁlling and magnetic\nmoment of the localised 4 f-electron shell. The FS of Er\nwhichistypicalfortheserieshasastrongnestingfeature,\ni.e. two large parallel sheets of the FS. The formation of\na SDW with a wave vector equal to the nesting vector\nallows for hybridisation removing these parts of the FS\nopening up a gap at the Fermi level lowering the total\nenergy of the system. Anisotropies in the system deter-\nmine whether the type of SDW preferred in the system,\nis a helical SDW, conical SDW or a longitudinal SDW.\nHelical SDWs also exist for various 3d-transition com-\npounds, howeveramong the elemental 3d-transition met-\nals the magnetic structure are less exotic at ambient con-\nditions. It was previously shown that if certain general\ncriteria are met, helical SDWs are formed in the transi-\ntion metal series. This means that at conditions slightly\ndiﬀerent from ambient, spin spirals occur naturally in-\ndicating that this type of magnetic order is less exotic\nthan one could believe. An example of such a system\nisγ-Fe the fcc phase of iron. Experimentally γ-Fe has\nbeen stabilised as precipitates in a Cu-matrix. The ex-\nperimental magnetic ground state structure was found to\nbe a SS with q= 2π/a(0.1,0,1) wherea=6.76 a.u. Sev-\neral theoretical studies4,5,8have mapped out the ground\nstate magnetic structure of γ-Fe which was found to be\nvery volume sensitive. In contrast to the rare-earth sys-\ntems where the SS state is driven by nesting between two\nparallel sheets of the FS, the SS state of γ-Fe is not pro-\nmoted by a single FS nesting vector. Instead there is a\nnet energy gain from many parts of the FS given by the\nhybridisation by a SS vector.\nIV. ELECTRONIC STRUCTURE\nCALCULATION\nAllthematerialsspeciﬁcparametersforErand γ-Feused\nfor the calculation of the torque current tensor Cwere\nobtained from ﬁrst principles density functional theory.\nThe calculations were made using the full-potential aug-\nmented plane waveplus localorbitals(APW+lo) method\nas described in Ref. 9. For Er, the calculations were per-\nformed in the same manner as in Ref. 3,10 and for γ-Fe\nthe calculations were performed in the same way as in\nRef. 5,8. The local spin density approximation (LSDA)\nas parametrised by von Barth and Hedin was used with-0.00 0.20 0.40 0.60 0.80 1.00\nq (π/c)0.01.02.0Energy (mRy)\n0.00 0.20 0.40 0.60 0.80 1.00\nq (2π/a)-6.0-5.0-4.0-3.0-2.0-1.00.0Energy (mRy)\n0.00 0.20 0.40 0.60 0.80 1.00\nq (π/a)-0.2-0.10.00.10.20.30.40.5Energy (mRy)Γ\nΓ XA\nW XqFe,1\nqFe,2qEr\nFIG. 2: The top panel shows the total energy of Er per hcp\nunit cell for SS wave vectors from Γ to A. The middle panel\nshows the total energy per atom of γ-Fe for SS vectors from\nΓ toXin the larger lattice volume. The bottom panel shows\nthe total energy per atom of γ-Fe for SS vectors from Xto\nWin the smaller lattice volume.\nout any shape-approximation to the non-collinear mag-\nnetisation, i.e. charge and magnetisation densities as\nwell as their conjugates potentials are allowed to vary\nfreely in space both regarding magnitude and direction.\nFor Er, a set of 12 ×12×8 k-points was used for con-\nverging the electron density and for Fe we used a set of\n20×20×20k-points. The SS was treated using the gener-\nalised Bloch theorem. The Er calculation was performed\nwith the 4f-electrons treated as spin polarised core elec-\ntronsandweusedthe experimentalErhcp latticeparam-\neters (a= 6.73 a.u. and c= 10.56 a.u.). For Er, we cal-\nculate a largenumber ofSS wavevectors, q=π/c(0,0,q)\nin the ﬁrst Brillouin zone of the hcp lattice, along the out\nof plane direction between Γ and A. The total energy for\nthese calculations are presented in Fig. 2. There is an\nenergy minimum for q= 0.40 which corresponds to the\nground state SS structure of Er. For Fe we perform cal-5\nculations at two lattice volumes. At a lattice constant\nofa= 6.82 a.u., corresponding to the lattice constant of\nCu, we do a series of calculations for q1= 2π/a(0,0,q),\nbetween Γ and Win the ﬁrst Brillouin zone, and an en-\nergy minimum is found for q= 0.59 (see middle panel in\nFig. 2). At a reduced lattice constant of a= 6.66 a.u. we\ndo a series of calculations for q2=π/a(w,0,2), between\nWandX, where there is a weak local energy minimum\natw= 0.48. This latter minimum corresponds to a spi-\nral wave vector which is close to the experimental SS of\nγ-Fe. Note that q2is non-parallel to the SS rotation axis\nwhich is in the ˆz-direction for all three calculations.\nV. RESULTS\nIn this section we present the results of the STT cal-\nculations for the three SS systems treated in the previ-\nous section. The one electron spin current tensor Qnk\nwas calculated using Eq. (18) and the Kohn-Sham eigen-\nfunctions obtained from the ﬁrst principles calculation.\nThe ﬁrst Brillouin zone was covered by a 41 ×41×41\nk-point mesh, in orderto accuratelyevaluatethe FS inte-\ngrals in Eq. (13) and Eq. (15). The conductance tensor,\nB=/summationtext\nmBm, was as expected, found to be diagonal for\nall three considered systems.\nThe torque current tensor Cwas calculated for Er hav-\ning a SS with wave vector q=π/c(0 0 0.4). We found\nthat for an Er atom situated at a site with magnetisation\ndirection (100), the spin current tensor\nC=¯h\ne\n0.0 0.0 0.0\n0.0 0.0−0.5\n0.0 0.0 0.0\n[˚A2].\nFrom the structureof the Ctensor, it canbe seen that for\na current in the (001) direction, a torque is induced in\nthe (010) direction. Since the torque is perpendicular to\nthe local magnetisation and lies in the SS rotation plane,\nwe conclude that a current along the SS axis causes a\nrigid rotation of the SS around the SS axis.\nForγ-Fe with the larger volume in the q1=\n2π/a(0,0,0.59) spiral state, the torque current ten-\nsor, evaluated for an atom with magnetisation direction\n(100),\nC=¯h\ne\n0.0 0.0 0.0\n0.0 0.0−1.4\n0.0 0.0 0.0\n[˚A2].\nAlso in this case we conclude that a current along the\nSS axis causes a rigid rotation of the SS. In this system,\nthe torque, for the same current density, is found to be\nnearly three times larger than for Er. We will return to\na discussion on the magnitude of the torque later in this\nsection.\nFinally we calculated the STT in γ-Fe with reduced\nvolume, where the SS wave vector q2=π/a(0.48,0,2) isa: Er\nn (¯h/e)(An)23[˚AeV] (¯ h2/e2)(Bn)33[eV/˚A]\n1 0.01 0.006\n2 0.01 0.01\n3 -0.04 0.03\n4 -0.008 0.007\nb: Fe q 1\nn (¯h/e)(An)23[˚AeV] (¯ h2/e2)(Bn)33[eV/˚A]\n1 -0.002 0.002\n2 0.01 0.02\n3 -0.1 0.07\n4 -0.005 0.002\nc: Fe q 2\nn (¯h/e)(An)21[˚AeV] (¯ h2/e2)(Bn)11[eV/˚A]\n1 0.005 0.006\n2 0.02 0.01\n3 0.02 0.02\n4 -0.05 0.05\n5 0.09 0.08\nTABLE I: The elements of the AnandBntensors that mainly\ncontribute to the Ctensor, see Eq. (16)\nnon parallel to the spin rotation axis ˆz. We found that\nthe torque currenttensorfor an atom with magnetisation\nin the (100) direction is given by,\nC=¯h\ne\n0.0 0.0 0.0\n0.6 0.0 0.0\n0.0 0.0 0.0\n[˚A2].\nThe structure ofthe torque currenttensorfor this system\ndiﬀers from the previous two systems. Here, a current in\nthe (100) direction is required to produce a rotation of\nthe SS. The STT from a current in the (001) direction\nvanish since γ-Fe with q2=π/a(0.48,0,2) is antiferro-\nmagnetic in the (001) direction.\nVI. DISCUSSION\nIn general, the STT depends on a balance between the\nsize of the SS wave vector and the spin polarisation of\nthe conduction electrons at the FS.\nSome insight into the origin of the STT on the single\nelectron level can be obtained by considering the semi-\nclassical expression for the spin current11\nQ(z) =¯h\n2ˆs(z)j\ne. (19)\nIf the semi-classcal spin current is combined with Eq. (8)\none obtains the following expression for the torque cur-6\nrent tensor.\nC=¯h\n2e∂ˆs(z)\n∂z/vextendsingle/vextendsingle/vextendsingle\nz0dz=¯h\n2esinθkF\n−qsin (qz0)\nqcos (qz0)\n0\ndz,\n(20)\nThe above equation gives the same result as the quan-\ntum mechanical calculation of Cin section (IIA), if the\ndenominator of the factor ain Eq. (9) is equal to one,\nwhich is approximately true if ( q/2k)cosθkFis small.\nSemi-classically, the maximum spin transferred per elec-\ntron, for a given value of q, to an atomic layer is given\nby (¯h/2)qlwherelis the inter layer distance. This maxi-\nmum can be achievedif the single electroncone angle θkF\nis equal toπ/2. For Er and γ-Fe with the larger volume\nis the semi-classical maximum of the spin-transfer, with\nthe given value of q, equal to 0.3¯hand 0.9¯h, respectively.\nA more accurate value for the average spin transferred\nper electron to an atom is obtained by multiplying the\ntorque current tensor with e/AwhereAis the area per\natom in the direction of the current. For Er, where the\narea per atom in the (001) direction A=a2√\n3/2, the\nspin-transferred per electron is 0 .05¯h. Forγ-Fe, the area\nper atom in the (001) direction is a2/2 and the spin-\ntransferred per electron is 0 .2¯h, for the larger volume.\nHence, the spin-transfer torques for Er and γ-Fe (larger\nvolume) obtained from the ﬁrst principles calculations\nare only 16% and 22%, respectively, of the their semi-\nclassical maximum. This is partially due to the fact that\nthe coneangle θkFofthe conductionelectronsislessthan\nπ/2.\nAnother reason for the reduced STT is that contribu-\ntions from diﬀerent bands partly cancel each other. The\ncontributions from the individual bands to the STT are\ngiven by the AnandBntensors shown in Table (I). For\nEr there are four bands that cross the FS and the domi-\nnant element in the Antensors is their ( An)23elements.\nAs shown in Table (Ia), the STT from band three is four\ntimes larger and in the opposite direction to the STT\nfrom band one and two. For γ-Fe (larger volume) there\nare four bands crossing the Fermi level, where the dom-\ninating contribution to the STT comes from the third\nband, see Table (Ib). For γ-Fe with the smaller volume\nthereareﬁvebandscrossingtheFermilevelandthe dom-\ninant element in the Antensors is their ( An)21elements.\nAs shown in table (Ic), the largest contribution to the\nSTT is here coming from the ﬁfth FS. Finally we would\nlike to note that semi-classically the maximal spintrans-\nfer per conduction electron is ¯ h/2, which only can be\nobtained if both ql=π/2 andθkF=π/2. An other\nroute to high spin transfer is to try to reduce the denom-\ninator in the expression for the afactor in Eq. (9). The a\nfactor becomes large for small Fermi surfaces where q/k\nis large.VII. DYNAMICS\nWe will for our SS systems represent the uniform mag-\nnetisation of an atomic plane perpendicular to the SS\naxis by a unit vector ˆmin the direction of the magneti-\nsation. The time evolution of ˆmis phenomenologically\ndescribed by the LLG equation12,13.\ndˆm\ndt=Γ−αˆm×dˆm\ndt. (21)\nThe ﬁrst term Γis the total torque exerted on the layer\nand the second term is the Gilbert damping term with\ndamping constant α. The systems under consideration\nare all having a helical SDW with an easy-plane or hard-\naxis anisotropy. The hard-axis is directed along the SS\naxis and helps maintain the planar structure of the SS.\nFor such systems is the total torque Γa sum of a current\ninduced STT and a torque resulting from the hard-axis\nanisotropy. Both of these torques are perpendicular to\nthe magnetisation ˆmand the SS axis which here is con-\nsidered to be in the ˆzdirection. Eq. (21) can therefore\nfor this system be written as\ndˆm\ndt=Aˆm×ˆz−αˆm×dˆm\ndt, (22)\nwhere\nA(j,θ) =q j a(θ,kF)\nsinθ+bm·z. (23)\nThe ﬁrst term in the above equation is due to the STT\nand the factor a(θ,kF) was introduced in Eq. (8). The\nbin the second term is the strength of the hard-axis\nanisotropy. Eq. (22) is in spherical coordinates, see\nFig. 1, given by.\ndφ\ndt(1+α2) =−A(j,θ) (24)\nand\ndθ\ndt(1+α2) =−αA(j,θ) sinθ. (25)\nEq. (24) describes a rigid rotation of the SS due to the\ncurrent induced torque where the angular velocity, as a\nfunction of the current j, is given by −A(j,θ)/(1+α2).\nIt can be seen from Eq. (25) that the eﬀect of the Gilbert\ndamping is to change the cone angle of the SS. Eventu-\nally, the spiral will reach a steady state with a cone angle\nθ0, given by\nq j a(θ0,kF)\nsinθ0=−bcosθ0, (26)\nwhen the current induced torque and the anisotropy\ntorque balance each other. At this point in time, also\nthe rotational motion of the magnetic moments with re-\nspect to the SS axis stops. The polar angle φwill thus,\nwhen a current jis passed along the SS axis, change by\n∆φ=−1\n1+α2/integraldisplayθ0\nπ\n2A(j,θ)/parenleftigdθ\ndt/parenrightig−1\ndθ(27)7\nor\n∆φ=1\nα/bracketleftbigg\nln/parenleftig\ntanπ\n4/parenrightig\n−ln/parenleftbigg\ntanθ0\n2/parenrightbigg/bracketrightbigg\n,(28)\nbefore the spiral reach the steady state with cone angle\nθ0. The SS will returns to a planar state if the current\nis switched oﬀ after the steady state is reached. During\nthe reversal, the spiral rotates in the opposite direction\nperforming the same number of rotations as required to\nreach the steady state.\nThe current induced torque has, for slowly vary-\ning magnetisation, been described by introducing an\nadiabatic and a non-adiabatic term in the LLG\nequation14,15,16,17. The adiabatic term, which for a\nSS whose axis is along ˆztakes the form ∂ˆm/∂z, has\nin Eq. (22) been replased by the STT component of\nA(j,θ). A non-adiabatic torque term proportional to\nˆm×(∂ˆm/∂z) could also be introduced in Eq. (22). The\nmain eﬀect of such a torque would be a modiﬁcation of\nEq. (25), which determines the dynamics of the cone an-\ngleθ. A non-adiabatic torque would add a term to the\nright hand side of Eq. (25) which could decrease or even\nchange the sign of dθ/dt.\nThe eﬀect of in plane anisotropy is discussed in Ref. 3,\nwhere it is suggested that the current needs to overcome\na certain critical current before the spiral start to rotate.VIII. SUMMARY AND CONCLUSIONS\nWe have shown in general, that the a current through\na SS induces a rotation of the spins, where the angular\nvelocity depends on the magnitude of the current. The\nsize of the STT has been analysed in terms of the spin-\ntransfer per conduction electron and we conclude that\nthe total STT depends on, the spin-polarisation of the\nelectrons, the SS wavevectorand howcontributions from\ndiﬀerent bands coincide. The dynamics of the SS has\nbeen calculated including the eﬀects of anisotropies and\ndamping. For a system with a hard-axisanisotropyalong\nthe SS axis, the damping leads to a steady state where\nthe rotation eventually stops.\nAcknowledgments\nThisworkhasbeensupportedbygrantsfromtheSwedish\nResearch Council (VR) and the UK Engineering and\nPhysical Sciences Research Council (EPSRC) through\nthe Spin@RT consortium. Calculations have been per-\nformedattheSwedishnationalcomputercentersHPC2N\nand NSC. We are also grateful to D. M. Edwards for very\nhelpful conversations.\n1J. C. Slonczewski, J. Magn. Magn. Mater 159, L1 (1996).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3O. Wessely, B. Skubic, and L. Nordstr¨ om, Phys. Rev. Lett.\n96, 256601 (2006).\n4O. N. Mryasov, V. A. Gubanov, and A. I. Liechtenstein,\nPhys. Rev. B 45, 12330 (1992).\n5E. Sj¨ ostedt, L. Nordstr¨ om, and D. J. Singh, Solid State\nCommun. 114, 15 (2000).\n6A. W. Overhauser, Phys. Rev. 128, 1437 (1962).\n7J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70,\n172405 (2004).\n8R. Lizarraga, Ph.D. thesis, Uppsala University (2006).\n9E. Sj¨ ostedt and L. Nordstr¨ om, Phys. Rev. B 66, 014447\n(2002).10L. Nordstr¨ om and A. Mavromaras, Europhys. Lett. 49,\n775 (2000).\n11J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73,\n054428 (2006).\n12L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Statis-\ntical Physics, part 2 (Pergamon, Oxford, 1980).\n13T. L. Gilbert, Phys. Rev. 100, 1243 (1955).\n14Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004).\n15S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n16H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Japan\n75, 113706 (2006).\n17D. M. Edwards and O. Wessely, arXiv:cond-\nmat/0811.4118v1 (2008)."    },    {        "title": "1812.11638v2.Thermalization_in_Solid_State_NMR_Controlled_by_Quantum_Chaos_in_Spin_Bath.pdf",        "content": "Thermalization in Solid-State NMR Controlled by Quantum Chaos in Spin Bath\nWalter Hahn, V. V. Dobrovitski\nQuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nWe theoretically investigate thermalization and spin diﬀusion driven by a quantum spin bath\nfor a realistic solid-state NMR experiment. We consider polycrystalline L-alanine, and investigate\nhow the spin polarization spreads among several13Cnuclear spins, which interact via dipole-dipole\ncoupling with the bath of strongly dipolar-coupled1H nuclear (proton) spins. We do this by using\ndirect numerical simulation of the many-spin time-dependent Schrödinger equation. We ﬁnd that,\nalthough the proton spins located near the carbon sites interact most strongly with the13C spins,\nthis interaction alone is not enough to drive spin diﬀusion and thermalize the13C nuclear spins.\nWe demonstrate that the thermalization within the13C subsystem is driven by the collective many-\nbody dynamics of the proton spin bath, and speciﬁcally, that the onset of thermalization among\nthe13C spins is directly related to the onset of chaotic behavior in the proton spin bath. Therefore,\nthermalization and spin diﬀusion within the13C subsystem is controlled by the proton spins located\nfar from the C sites. In spite of their weak coupling to the13C spins, these far-away protons\nhelp produce a network of strongly coupled proton spins with collective dynamics, that drives\nthermalization.\nI. INTRODUCTION\nThermalization via spin diﬀusion is ubiquitous in\nmany-spin systems, governing their most fundamental\ndynamical properties [1–3]. Understanding the dynam-\nics of thermalization and spin diﬀusion is crucial for a\nbroad range of applications, from control and protection\nof the qubit dynamics in the quantum information pro-\ncessing devices [4–7], to elucidating the structure and\nthe functions of the biomolecules via solid-state NMR\n(ssNMR) [8–14]. However, thermalization dynamics in\nquantum systems is not fully understood due to com-\nplexity of the problem. Thermalization is a result of the\nquantum non-equilibrium evolution of a large number of\ninteracting spins. Dynamics of the spin diﬀusion and\nits quenching by disorder is tightly related to important\nbut poorly studied phenomena, such as onset of quan-\ntum chaos [15–18] in many-spin systems and in few-spin\nsystems coupled to their many-spin environments.\nProgress in understanding thermalization and spin\ndiﬀusion has been achieved recently by studying well-\ncontrolled systems, such as chains of trapped ions [19, 20]\nand atomic 1D and 2D lattices [21–23]. At the same\ntime, ssNMR constitutes an important test case for ther-\nmalization and spin diﬀusion in quantum many-spin sys-\ntems[24–27]; thesephenomenaunderliemanyfundamen-\ntal ssNMR techniques such as coherence and polarization\ntransfer, measurement of the inter-nuclear distances via\nspin diﬀusion rates, and ssNMR-based structural anal-\nysis [10, 11, 28]. However, the typical systems stud-\nied in real ssNMR experiments are very complex, com-\nprising many diﬀerent nuclear species having anisotropic\nchemical shifts, subject to time-periodic modulation of\nthe Hamiltonian parameters via magic angle spinning\n(MAS), and controlled by strong external driving. Mi-\ncroscopic description of thermalization in such systems is\na challenge: it is still not clear which microscopic eﬀects\ngovern thermalization and polarization/coherence trans-\nfer, and to which extent the existing theoretical tools candescribe spin diﬀusion at the microscopic level in systems\nof such complexity [17, 29–32].\nIn this article, we theoretically study thermalization\nand spin diﬀusion in realistic ssNMR settings, consid-\nering polycrystalline powder of alanine, and taking into\naccountallrelevantexperimentaldetails(spatialarrange-\nment of the nuclei, their anisotropic chemical shifts, pe-\nriodic modulation of the Hamiltonian by MAS, etc). We\nargue that the dynamics of thermalization in ssNMR ex-\nperiments (in our case, thermalization between diﬀerent\n13C spins) is governed by collective many-body quantum\neﬀects, namely by emergence of the chaotic dynamics in\nthe surrounding spin environment (in our case, the nu-\nclear spins of the hydrogen atoms, i.e. the proton spins).\nWeshowthateventhoseprotonspinsthatarelocatedfar\naway from the13C spins, and thus are practically decou-\npled from them, still critically aﬀect thermalization, due\nto essential many-body nature of the collective chaotic\ndynamics of the proton spin environment. Tradition-\nally, spin diﬀusion in ssNMR experiments is described\nvia semi-phenomenological Bloch-Redﬁeld-type theories\n[33–35]; their validity relies on the heuristic notion of\nthe \"network of strongly coupled proton spins\". Our\nwork provides formalization for this notion in terms of\nthe spectral properties of the proton spin subsystem.\nII. QUALITATIVE CONSIDERATIONS\nThermalization in many-spin systems usually occurs\nvia ﬂip-ﬂop transitions, when two coupled spins exchange\ntheir polarization. If the local magnetic ﬁelds at the two\nspin sites diﬀer signiﬁcantly (in our case, diﬀerent13C\nspins have diﬀerent Larmor frequencies due to diﬀerent\nchemical shifts) the ﬂip-ﬂop transitions are suppressed\nbecause of the mismatch in the Zeeman energies, and\nthe spin diﬀusion may be quenched (onset of localiza-\ntion) [36, 37]. In such case, the external time-dependent\nnoise can assist spin diﬀusion: the time-varying randomarXiv:1812.11638v2  [quant-ph]  16 Jan 20192\nlocal ﬁelds occasionally bring the Larmor frequencies of\nthe spins close to each other, enabling the ﬂip-ﬂops,\nand thus promoting thermalization. Speciﬁcally, the13C\nspins occupying two chemically inequivalent sites with\ndiﬀerent chemical shifts, can equalize their polarizations\nif assisted by the noise created by the surrounding bath\nof proton spins [33, 38–40] (proton-driven spin diﬀusion,\nPDSD).\nViewed in this way, thermalization via proton-driven\nspin diﬀusion is an ssNMR representative of a typical\nsituation of a few-spin central system (13C spins) coupled\nto the large quantum spin environment (1H spins).\nThe standard theoretical analysis of the proton-driven\nspin diﬀusion [33–35]) is based on the Bloch-Redﬁeld the-\nory. It assumes that the protons form a dense network\nof strongly coupled spins, possessing fast dynamics that\nquickly erases all correlations in the proton spin bath.\nUnder this assumption, the action of the actual many-\nspin quantum system (the proton spin bath) is replaced\nby a classical time-varying random magnetic ﬁeld, which\nleads to complete thermalization of the13C spins. On\nthe other hand, it is not unusual to see the results of the\nssNMR experiments in stark disagreement with the pre-\ndictions of the standard theory [24, 32, 41–43], and the\norigins of this disagreement are not properly understood.\nOur results demonstrate an important reason for possible\nfailure of the standard spin diﬀusion theory.\nIt is natural to expect that the local noise at the C\nsites is determined by the nearby proton spins, so these\nprotons will be most essential for driving the spin dif-\nfusion, while the protons located far away from the13C\nsites, including the protons from other molecules, will\nhave negligible eﬀect. However, we show that this view\nis oversimpliﬁed because the \"far-away\" protons are crit-\nically needed for the collective ergodic dynamics of the\nnoise created by the proton spin environment [3, 24, 44].\nIn particular, we show that, if the bath is not ergodic\nthen the proton environment does not act like a noise,\nand thermalization is quenched. Thus, the \"far-away\"\nproton spins, in spite of contributing very little to the\nlocal noise, become as important as the nearby protons\nin thermalizing the13C spins.\nIn order to make conclusive statements, we perform\ntheoretical studies taking into account the conditions of\nreal ssNMR experiments. We investigate thermalization\nin L-alanine (below we often drop the \"L\" preﬁx), start-\ning with one polarized13Cspin in an otherwise unpo-\nlarized system (cf. Fig. 1), and watching the spreading\nof its polarization to the other13C spins. In compari-\nson with other systems typically studied in the context\nof thermalization, all parameters of this system are ﬁxed\n(no random potential) and well known (no ﬁtting), but,\nat the same time, the system is very complex, as is typi-\ncal for ssNMR, exhibiting non-trivial thermalization and\nspin diﬀusion controlled by the collective many-body ef-\nfects: it is 3-dimensional, includes long-range couplings,\nwith the Hamiltonian modulated in time.\nThe rest of the article is organized as follows. In the\n  \nFigure 1. Schematic structure of the L-alanine molecule [45].\nThe three carbon nuclei are chemically nonequivalent and,\ntherefore, experience diﬀerent chemical shifts. The magnetic\nﬁeld at the position of each13Cspin is governed by the1H\nspins (15Nspin has a small magnetic moment and oxygen\nnuclei have no spin).\nnext section, Sec. III, we provide a detailed description of\nthe system and present qualitative discussion. In Sec. IV,\nwe describe in detail the numerical simulation methods.\nResults of our simulations are discussed in Sec. V and\nconclusions are given in Sec. VI.\nIII. DESCRIPTION OF THE SYSTEM\nA. Nuclear spins in alanine\nAn L-alanine crystal has orthorhombic structure with\nthe space group P 212121, and has four alanine molecules\nper unit cell. The unit cell dimensions are a= 6:0Å,\nb= 12:3Å andc= 5:8Å. For the polycrystalline ala-\nnine powder, the isotropic average of individual single-\ncrystallite alignments is considered [34, 46, 47]. A\nmolecule of alanine has the chemical formula C3H7O2N;\nits zwitterionic form structure [45] is illustrated in Fig. 1.\nExcept for the oxygen nuclei which practically have no\nspin and are neglected in the following, all other nuclei\n13C,15Nand1Hhave spins ½(since nitrogen spins are\nof little interest, for simplicity below we consider iso-\ntopically puriﬁed sample without14Nspins). The three\nchemically nonequivalent carbon sites are commonly re-\nferred to as CO,C\u000b, and C\f. Among the nuclei, the pro-\ntons have the largest magnetic moment, and correspond-\ningly the largest gyromagnetic ratio \rH\u00195:6rad/(s\nT); the gyromagnetic ratios of the other spins are sig-\nniﬁcantly smaller:\n\rC\n\rH\u00190:25;\rN\n\rH\u00190:1: (1)\nB. Hamiltonian of the system and characteristic\nenergy scales\nWe consider a polycrystalline alanine sample in a\nstrong static magnetic ﬁeld (quantizing ﬁeld) B0along3\nthez-axis, which sets the proton Larmor frequency at\n400 MHz. The sample is subjected to the magic angle\nspinning (MAS), i.e. the sample is rotated with a fre-\nquency\u0017r(below we take \u0017r= 10kHz) around the rotor\naxis which makes an angle of 54:7°with thez-axis; MAS\nis often used in ssNMR to reduce the line width and in-\ncrease spectral resolution [34, 35].\nThe Hamiltonian of the system includes the single-spin\nZeeman energies and the pairwise dipolar couplings. In\nthe reference frame rotating with the Larmor frequency\naround the z-axis, the secular part of the Hamiltonian\nhas the standard form [34, 35]\nHtot(t) =HCS(t) +HDD(t); (2)\nwhere the bare Zeeman energies (the terms \riB0Siz) of\ndiﬀerent spin species (13C,15N, and1H) are eliminated\nby the rotating frame transformation. The time depen-\ndence of the Hamiltonian terms is caused by the MAS,\nand is periodic with the rotor period Tr= 1=\u0017r= 0:1ms;\ncorrespondingly, all observables in the ﬁgures below are\nshown at times commensurate with Tr.\nThetermHCS(t)describesthechemicalshift, i.e.small\ndeviation from the reference Larmor frequency of a given\nnuclear spin, caused by the ﬁnite electronic density near\nthe nucleus:\nHCS(t) =X\nj\u0001!jz(t)Sjz; (3)\nwhere \u0001!jz=\u000ezz;j\rjB0, and\rjis the gyromagnetic ra-\ntio of thej-th spin, and \u000ezzis the (z;z)component of\nthe chemical shift tensor for the j-th site. The chemical\nshift is generally anisotropic; it is described by a symmet-\nric rank-2 tensor [34]. The principal axes of this tensor\nare determined by the electronic density around the nu-\ncleus, and thus depend on the local crystalline symmetry.\nWhen the chemical shift tensor is transformed to the lab-\noratory reference frame [13, 33, 34], its (z;z)component,\nand the resulting value of \u0001!jz, depend on the orienta-\ntionofthegivencrystallitewithrespecttothequantizing\nmagnetic ﬁeld ( z-axis), and change in time because of the\nMAS [13, 33, 34]. In alanine, the chemical shift tensor for\nC\fand C\u000bsites is moderatly anisotropic, its entries are\nof the order of 6– 12kHz, while for C Osite it is strongly\nanisotropic, with the entries varying from +2 to -10 kHz\n[48]. The chemical shifts of the protons are isotropic,\nwith the magnitude of order of 1–3 kHz [49]. Here and\nbelow we follow the standard convention, setting ~= 1,\nandexpressingenergiesinrad/sorinHz(1Hzequivalent\nto 2\u0019rad/s).\nThe termHDD(t)describes the dipole-dipole interac-\ntion\nHDD(t) =likeX\ni;jAij(t)h\nSixSjx+SiySjy\u00002SizSjzi\n+unlikeX\ni;jAij(t)h\n\u00002SizSjzi\n: (4)The dipole-dipole couplings are calculated in a standard\nway using the coordinates of the atoms in the alanine\ncrystal, i.e. Aij(t) =\ri\rj~(1\u00003 cos2\u0012ij(t))=r3\nij, where\ri\nand\rjare the gyromagnetic ratios of the i-th andj-th\nspin,rijisthedistancebetweenthem, and \u0012ijisthepolar\nangleofthevectorconnectingthe i-thandj-thsites; note\nthatthisangledependsontheorientationofagivencrys-\ntallite with respect to the z-axis, and changes in time in a\nperiodicmannerduetotheMAS.Thestandardnotations\nof \"like\" or \"unlike\" spins denote the spins belonging, re-\nspectively, to the same or diﬀerent spin species (e.g.13C\nand1H), i.e. having the same or diﬀerent gyromagnetic\nratios.\nThe ﬂip-ﬂop transitions which cause the spin diﬀusion\nare mediated by the transverse dipolar coupling term\nSixSjx+SiySjy=1\n2[S\u0000\niS+\nj+S+\niS\u0000\nj], whereS+\niandS\u0000\ni\nare rising and lowering operators. These terms are ab-\nsentforthedipolarcouplingsbetweentheunlikespinsbe-\ncausethemismatchinZeemanenergiesislarge(hundreds\nof MHz), while the typical dipolar couplings rarely ex-\nceed few hundreds of kHz. In our speciﬁc system, within\nthe strongly coupled groups, such as methyl (CH 3) and\namino (NH 3) groups, the1H-1H and the13C-1H cou-\nplings are of the order of 30 kHz. The typical couplings\nbetween the protons belonging to diﬀerent groups or dif-\nferent molecules are of the order of 5–10 kHz.\nThe C-C couplings within the molecule are \u00182kHz for\nthe C\u000b-C\fand C O-C\fpairs, i.e. noticeably smaller than\nthe Zeeman energy diﬀerences caused by the chemical\nshifts. Thus, the interaction with the proton subsystem,\nwhich can absorb this energy mismatch, is critical for\nthermalization between the C spins.\nAt the characteristic temperatures of the ssNMR ex-\nperiments, the protons within each methyl and ammo-\nnium group randomly interchange their positions at the\ntimescale of the order of 1 ns [46, 50, 51]. This is much\nfaster than the characteristic times of the spin dynamics\n(microseconds to seconds). These fast jumps have to be\naccounted for by averaging Htot(t)over the correspond-\ning sites for all spins within each group.\nIV. NUMERICAL SIMULATIONS OF\nTHERMALIZATION DYNAMICS\nWe numerically simulate the dynamics of the many-\nspin system, which includes an alanine molecule and a\nnumber of protons around it; the total number of mod-\neled spins was between ns= 4(only C and N sites)\nandns= 24(one molecule and 13 protons around\nit). The surrounding protons are added individually\nfor aliphatic sites, and in groups of three for methyl\nand ammonium groups. We directly solve the time-\ndependent Schrödinger equation with the Hamiltonian\n(2) for the many-spin wavefunction, represented as a 2ns-\ndimensional vector, using the 4-th order Suzuki-Trotter\nexpansion for the evolution operator [52, 53], with the\ntimestep 2 \u0016s.4\nThe initial state of the system corresponds to the typ-\nical experiment, with one polarized13C nuclear spin and\nthe rest being unpolarized: diﬀerent C sites are address-\nableduetotheirdiﬀerentchemicalshifts[33,41], andcan\nbe individually polarized, manipulated, and measured.\nSince the typical temperatures of ssNMR correspond to\nthelarge-temperaturelimitforthenuclearspins, thenor-\nmalized relevant part of the density matrix is [34]\n\u001ainit=h\nj\"ih\"ji\nCO\nIrest; (5)\nwhere the ﬁrst term describes the COspin and Irest,\nwhich is proportional to the unit matrix (so that\nTr[Irest] = 1), describes the rest of the system. In our\nsimulationsthedensitymatrix Irestwasrepresented, with\nexponential precision, by a random wavefunction, i.e. by\nad-dimensional vector of complex numbers, with the en-\ntries drawn randomly from the uniform distribution on a\n(d\u00001)-dimensional complex unit sphere [54, 55], where\nd= 2(ns\u00001)is the dimensionality of the Hilbert subspace\ncorresponding to the unpolarized rest of the system.\nTo model the standard polycrystalline powder sam-\nple, we averaged the spin polarizations over a large num-\nber (in most cases, over 200) of randomly oriented crys-\ntallites. The quantity of interest for us is the time-\ndependent polarization Pz(t)\u0011Tr [2Sz\u001a(t)]on diﬀerent\nC sites. Since the ﬂip-ﬂops occur only between the13C\nspins, thetotalpolarizationofthe13Csubsystemremains\nconstant. In the case of complete thermalization, at long\ntimes we have\nPz[CO](t)\u0019Pz[C\f](t)\u0019Pz[C\u000b](t)\u00191\n3:(6)\nA few notes are in order. First, our test simulations\nhave shown that the results obtained with two alanine\nmolecules and several protons around them are prac-\ntically the same as the results obtained with a single\nmolecule and protons around it. The spin diﬀusion be-\ntween the C sites of diﬀerent molecules is small, and the\ndynamics of all13C spins in the sample can be repro-\nduced by modeling a single molecule. Second, we also\nsimulated the spin diﬀusion with other initially polarized\n13C spins (not shown here); the simulations show that\nour conclusions about the spin diﬀusion rates and emer-\ngence/quenching of thermalization do not depend on the\nchoice of the initially polarized C site.\nV. SPIN DIFFUSION IN ALANINE:\nNUMERICAL RESULTS\nA. Magnitude of local ﬁelds\nThe spin diﬀusion is controlled by the (quasi-)random\nmagnetic ﬁeld created by the surrounding proton spins\nat the carbon sites, e.g. at the C Osite:\nBO(t) =\u00002X\nj2PAj;O(t)Sjz; (7)\n(a)\n(b)\nBO(np) [in Hz]\n00.20.40.60.811.21.4η\ntotal n um b er np of proton spins00.20.40.60.8\n0 2 4 6 8 10 12 14 16 18 20Figure 2. (a)Eﬀective magnetic ﬁeld BO(np)created at the\nCOsite by the proton spins. The ﬁrst seven proton spins be-\nlongtothesamemoleculeas CO, thefurtherspinsfromneigh-\nboring molecules were included according to their distance to\nCO, cf. Eq. (9). The eﬀective magnetic ﬁeld BO(np)is mainly\ngovernedbytheprotonspinsfromthesamemolecule, whereas\nthe spins from neighboring molecules produce only a rather\nsmall correction to BO(np).(b)The chaoticity parameter \u0011\nintroduced in Eq. (10) signiﬁcantly decreases with increasing\nnp. In(a)and(b), dashed lines connect points to guide the\neye.\nwhere the summation is over the set Pof proton sites,\nandAj;O(t)is the parameter of the dipolar coupling be-\ntween the C Ospin and the j-th proton (including the\ntime modulation by MAS), see Eq. 4. It is instructive to\ncalculate the magnitude of this ﬁeld as a function of the\nnumbernpof protons included in the modeled system.\nThe average of this ﬁeld is zero for unpolarized protons,\nwhile the second moment\nBO2\u00111\nTrZTr\n0dtZ\n\nd\u0016RTr\u0002\n\u001a(t)B2\nO(t)\u0003\n(8)\nis ﬁnite, and includes quantum-mechanical averaging,\ntime averaging over the rotor period Tr, and averaging\nover the crystallite orientation (i.e. the integral over the\ngroup \nof all 3D rotations Rof the crystallite, weighted\nwith the ter Haar measure d\u0016R). The dispersion of the\nrandom ﬁeld BO(np)can be expressed as\nBO(np) =vuutnpX\nj=1A2\nj;O;A2\nj;O=1\nTrZTr\n0dthA2\nj;O(t)iR;\n(9)\nand can be directly calculated from the known crystal\nand molecular structure, where h:::iRdenotes average5\nover the crystallite orientations. Since the dipolar cou-\npling decays with distance ras1=r3, and its square as\n1=r6, the variance for an inﬁnite (i.e. macroscopic-sized)\nsampleBO(1)is well deﬁned, and is determined almost\ncompletely by only several (7–10) nearby protons. This\ndependence is shown in Fig. 2, where BO(np)is plotted\nfor the C Osite andnpincreases as the protons are added\none by one according to their distance from C O.\nB. Spin diﬀusion between C sites\nWhile the \"far away\" protons add very little to the lo-\ncal ﬁeld at the C sites, they play crucial role in the spin\ndiﬀusion in the13C subsystem, as seen from Fig. 3. We\nstart from simulating thermalization in a single molecule\nwith all protons removed, and observe an almost static\nPz(t)(Fig. 3, solid line), as expected: without protons\nthe diﬀerence in the chemical shifts between the C sites is\ntoo large for spin diﬀusion to happen. Next, we add pro-\ntons, and simulate a single molecule of alanine (dashed\nline), which contains seven protons. The second moment\nof the local ﬁeld, given by Eq. 9 above, for np= 7reaches\n80% of its maximum value. Initially, the polarization\nPz(t)diﬀuses from C Oto other sites, but after \u00195ms it\nsaturates at the values which are very far from thermal-\nization. With further increasing the number of proton\nspins, the saturation value of PzforCOdecreases, but\nstill does not reach its thermalized value 1/3. Only when\nwe include 7 nearest protons from othermolecules, we see\na onset of thermalization in the13C subsystem, although\nthe magnitude BOof the random ﬁeld changes very little\n00.20.40.60.81\n0 10 20 30 40 50 60 70 80 90 100\nPz(t)\nt [in ms]0H (stripp ed molecule)\n7H (molecule)\n14H (molecule + 7 H )\nFigure 3. Inﬂuence of the surrounding proton bath on spin\ndiﬀusioninalanine. Thetimeevolutionof Pz(t)forindividual\ncarbon spins is shown for a diﬀerent number of proton spins\nconsidered in the simulation. The color coding of the carbon\nspins is: CO- red, C\u000b- blue, and C\f- green. The system\nsimulated consists of three carbon spins and one nitrogen spin\n(solid line), one bare molecule (dashed line), a single molecule\nwith seven nearest proton spins from neighboring molecules\n(ﬁne dashed line). The horizontal dotted gray line indicates\nthe equilibrium value 1/3.\n00.20.40.60.81\n0 0 .5 1 1 .5 2 2 .5\nprobabilit y distribution P(s)\nenergy-lev el spacings s8H bath\n17H bath\nPP(s)\nPWD(s)Figure 4. Transition from integrability to nonintegrability in\nthe proton bath with increasing bath size np. The normal-\nized probability distribution P(s)of energy-level spacing sof\nthe Hamiltonian describing the proton bath approaches the\nWigner-Dyson distribution PWD(s)as we increase np. The\ndistributions P(s)are averaged over crystallite alignments\n(number of alignments - np= 8: 1000,np= 17: 8). The\nresults are obtained for one irreducible block of the Hamil-\ntonian taken at t= 0. We do not average the Hamiltonian\nover the proton sites in each individual group to exclude ad-\nditional eﬀective constants of motion. Lines connect point to\nguide the eye.\nin this range of np. After that, including more and more\nprotons in the simulations does not change the behav-\nior ofPz(t)much. These results clearly demonstrate our\nmain point: the \"far away\" proton spins signiﬁcantly af-\nfect spin-diﬀusion dynamics almost without aﬀecting the\nlocal ﬁeldBO(np).\nIn order to identify the key factor controlling the ther-\nmalization, we investigate an onset of chaos in the dy-\nnamics of the proton bath as the number of protons\nchanges from np= 7(single alanine molecule, no ther-\nmalization) to np= 14(a molecule with 7 additional\nprotons, thermalization). As a signature of chaos, we\nuse the statistics P(s)of the nearest level spacings s\n[17, 56–59], which is a standard spectral measure of non-\nintegrability in quantum systems. As seen in Fig. 4, for\nnp= 8protons the shape of P(s)is close to the Pois-\nson distribution PP(s) = exp(\u0000s), characteristic for in-\ntegrable quantum system. As npincreases, the shape\nofP(s)gradually changes, and for np= 17almost co-\nincides with the orthogonal Wigner-Dyson distribution\nPWD(s) =s\u0019\n2exp(\u0000s2\u0019\n4), which is a hallmark of the\nquantum chaos in the system. The closeness of the dis-\ntributionP(s)to the chaotic or integrable case can be\nquantiﬁed with the parameter\n\u0011=Rs0\n0h\nP(s)\u0000PWD(s)i\ndsRs0\n0h\nPP(s)\u0000PWD(s)i\nds; (10)\nwheres0is deﬁned as the smaller value satisfying\nPP(s0) =PWD(s0)[58, 60]; so that \u0011= 1corresponds6\nto a purely Poisson distribution, while \u0011= 0corresponds\nto the Wigner-Dyson statistics. Fig. 2(b) shows that the\nparameter\u0011monotonically decreases from 0.8 to 0.4 in\nthe same region npwhere the onset of thermalization\nwithin the C subsystem is observed.\nIt is also instructive to look closer at the initial change\nof the spin polarization Pz(t)at the C Osite, see Fig. 5.\nFornp\u00157, we observe that the initial rate of spin diﬀu-\nsion does not change much as the number of protons in\nthe simulated system increases. This means that the ini-\ntial diﬀusion rate stays approximately the same as long\nas the local-ﬁeld dispersion BO(np)does not vary much.\nThis is exactly what one would qualitatively expect from\nthe Bloch-Redﬁeld theory, which predicts that the rate \u0000\nof the polarization transfer between two C sites is [33, 41]\n\u0000/A2\n12FZQ(\u000e); (11)\nwhereA12is the dipolar coupling between the two C\nsites,\u000eis the diﬀerence in the chemical shifts, and\nF(\u000e) =R\nf1(!)f2(\u000e\u0000!)d!is the overlap of the zero-\nquantum NMR lines of the two13C spins, given by the\nconvolution of their zero-quantum lineshapes f1(!)and\nf2(!)[33]. The zero-quantum lines overlap FZQis con-\ntrolled primarily by the local random ﬁelds, so the rate\n\u0000should not change much if BOdoes not change.\nThis standard logics breaks down at later times, when\nPz(t)saturates. For small number of protons, in spite\nof the same initial depolarization rate, the curve Pz(t)\nsaturates quickly and far from the thermalized value,\nsince the overall dynamics of the proton bath is far from\nchaotic, and the local ﬁelds are far from random. In this\nregime, adding even a single extra proton changes the\nbehavior of Pz(t)att\u00154ms considerably, and notice-\nably lowers the saturation value. For larger proton baths,\nonce chaotic behavior emerges, thermalization sets in,\nandPz(t)saturates at the value 1/3. After that, adding\nmore proton spins from neighboring molecules does not\n0.70.750.80.850.90.951\n0 2 4 6 8 10\nPz[CO](t)\nt [in ms]7H (molecule)\n20H (molecule + 13 H )\nFigure 5. Initial vs. long-time behavior of the time evolution\nofz-polarization of COspin for diﬀerent sizes of the surround-\ning proton bath as indicated in the legend. In contrast to\nlonger time, the initial spin diﬀusion rate ( t\u00142ms) does\nnot depend on the size of the proton bath. For a proton\nbath consisting of less than 14 protons, Pz[CO](t)saturates\nat non-equilibrium values.\n00.20.40.60.81\n0 2 4 6 8 10 12 14 16 18 20\nPz(t)\nt [in ms]14H (molecule + 7 H )\n20H (molecule + 13 H )Figure 6. Small modiﬁcations of spin-diﬀusion behavior in\nalanine for larger spin bath. The time evolution of Pz(t)of\nindividual carbon spins is shown for a diﬀerent number of\nproton spins considered in the simulation. The color coding\nof the carbon spins is: CO- red, C\u000b- blue, and C\f- green.\nThe system simulated consists of a single molecule with 13\n(solid line) or 7 (dashed line) additional nearest proton spins\nfrom neighboring molecules. The horizontal dotted gray line\nindicates the equilibrium value 1/3.\nchangePz(t)signiﬁcantly, as seen in Fig. 6: even though\nnpincreases from 14 to 20, the changes in the Pz(t)are\nsmall even at very long times.\nVI. CONCLUSIONS\nWe conclude that the nearby protons and the far-away\nproton spins, while both essential for the spin diﬀusion\nprocess, play very diﬀerent roles. The nearby protons\ngovern the dispersion of the local ﬁelds, and thus de-\ntermine the short-time behavior 0\u0014t\u00144ms ofPz(t).\nThe long-time dynamics and thermalization between the\nC spins is governed by the far-away protons, which do\nnot directly aﬀect the local ﬁelds, but control the col-\nlective many-body dynamics of the proton bath. They\nensure emergence of quantum chaos among the proton\nspins, and, as the spectral statistics P(s)shows, it is the\nemerging chaos in the proton bath that triggers thermal-\nization among the13C spins. In this way, the far-away\nprotons ensure the very existence of the strongly coupled\nproton spin bath, which is assumed in the Bloch-Redﬁeld\ntheories of spin diﬀusion.\nThe experimental ssNMR tests of this idea are possible\nby using the deuterated alanine samples, since deutrons\nhave a noticeably smaller magnetic moment, and the\ndipolar coupling between deutron spins is about 10 times\nsmaller than the coupling between nuclear spins. How-\never, the quadrupolar coupling, and the related fast spin-\nlattice relaxation characteristic for the deutrons (which\nhave spin 1, in contrast with the proton spin 1/2), may\nalso modify the spin diﬀusion between the13C spins;\nthese eﬀects should be taken into account in the future7\nsimulations.\nACKNOWLEDGMENTS\nWe thank M. Hong and K. Schmidt-Rohr for helpful\ndiscussions. 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E 72, 026225\n(2005)."    },    {        "title": "1511.07739v1.New_pathways_towards_efficient_metallic_spin_Hall_spintronics.pdf",        "content": "New pathways towards e\u000ecient metallic spin Hall spintronics\nMatthias Benjamin Jung\reisch\nMaterials Science Division, Argonne National Laboratory,\nArgonne, Illinois 60439, USA\njung\reisch@anl.gov\nWei Zhang\nMaterials Science Division, Argonne National Laboratory,\nArgonne, Illinois 60439, USA\nzwei@anl.gov\nWanjun Jiang\nMaterials Science Division, Argonne National Laboratory,\nArgonne, Illinois 60439, USA\njiangw@anl.gov\nAxel Ho\u000bmann\nMaterials Science Division, Argonne National Laboratory,\nArgonne, Illinois 60439, USA\nho\u000bmann@anl.gov\nSpin Hall e\u000bects interconvert spin- and charge currents due to spin-orbit interaction, which en-\nables convenient electrical generation and detection of di\u000busive spin currents and even collective\nspin excitations in magnetic solids. Here, we review recent experimental e\u000borts exploring e\u000e-\ncient spin Hall detector materials as well as new approaches to drive collective magnetization\ndynamics and to manipulate spin textures by spin Hall e\u000bects. These studies are also expected\nto impact practical spintronics applications beyond their signi\fcance in fundamental research.\nKeywords : Spin Hall e\u000bect, spin dynamics, spin-transfer torques, spin-orbital e\u000bects, spin pump-\ning, magnetic skyrmions\n1. Introduction\nSpintronics relies not only on the charge degree of\nfreedom, but also utilizes the spin degree of freedom\nand enables the manipulation of material properties\nand control devices. Towards this end, creation, de-\ntection and manipulation of spin currents are at the\nheart of spintronics. Historically, the discovery of gi-\nant magnetoresistance set a milestone in spintron-\nics research. Soon after its discovery, the \feld con-\ntributed signi\fcantly to the transformation of mod-\nern information technologies, in particular with the\ndevelopment of high density magnetic recording andnew concepts for non-volatile solid state memory\ndevices [1]. Although, spin Hall e\u000bects (SHE) were\nproposed by D'yakonov and Perel' over four decades\nago, substantial experimental e\u000borts only started\nrecently [1{19] and within a short period of time,\nspin Hall e\u000bects have evolved from an academic cu-\nriosity to real applications (e.g., spin current detec-\ntion, excitation of spin dynamics and magnetization\nswitching).\nThis article will review recent experimental re-\nsearch e\u000borts towards new, metallic spin-Hall phe-\nnomena and spintronics applications. Section 1 will\n1arXiv:1511.07739v1  [cond-mat.mes-hall]  24 Nov 20152M. B. Jung\reisch et al.\nintroduce basic theoretical concepts related to spin\nHall e\u000bect physics including spin pumping and spin\nSeebeck e\u000bect. Following this, in Section 2, we will\ndiscuss novel spin Hall detector materials in light of\nhigh e\u000eciencies of spin-to-charge conversion. Sec-\ntion 3 will present new approaches to drive col-\nlective spin dynamics by means of SHEs in het-\nerostructures consisting of a normal metal in con-\ntact with a magnetic material. Lastly, in Section 4,\nwe will focus on the manipulation of topologically\nstable spin textures by SHEs.\n1.1. Spin Hall e\u000bects\nIn 1879, Edwin E. Hall discovered the e\u000bect that\nwas named after him [20]. The Hall e\u000bect describes\nthe generation of a transverse voltage in a conduc-\ntor in the presence of a perpendicular magnetic \feld\ndue to the Lorentz force on the electric charge cur-\nrent. In ferromagnetic metals, however, the voltage\nis not directly proportional to the magnetic \fled. It\nalso depends on the magnetization and this e\u000bect\nis known as the anomalous Hall e\u000bect [21, 23]. De-\npending on their spin orientation, electrons \rowing\nin a ferromagnetic conductor will acquire di\u000berent\ntransverse velocities and since the electric charge\ncurrent in a ferromagnet is polarized, the spin-\ndependent velocity leads to a transverse voltage.\nThe SHE is closely related to the anomalous Hall\ne\u000bect since the development of the spin-dependent\ntransverse velocities is also present in non-magnetic\nmaterials. If we consider an electric charge current\n\rowing through a non-ferromagnetic material, such\nas a paramagnetic metal or a doped semiconductor,\nelectrons will experience an asymmetric scattering\ndepending on the orientation of their spin: Electrons\nwith spin-up will be scattered preferably in one\ndirection perpendicular to the \row of the electric\ncharge current and electrons with spin-down in the\nopposite direction [2, 22]. This spin current evolv-\ning transversally to the charge current with a spin\norientation perpendicular to the plane, spanned by\nthe two currents, is called SHE [2, 22, 24{26]. The\ngeneration of charge current due to a gradient in the\nspin accumulation is called inverse spin Hall e\u000bect\n(ISHE) [3]. The conversion from a spin- into charge\ncurrent is described by:\n~JC/\u0012SH~JS\u0002~ \u001b: (1)\nHere,~JCis the charge-current density, \u0012SHis the\nspin Hall angle, ~JSis the spin-current density and\n~ \u001bis the spin-polarization vector. The spin Hall an-gle is a material-speci\fc parameter and the absolute\nvalue and the polarity of \u0012SHdepends on the band\nstructure of the spin Hall detector material [27].\nThere are mainly three possible contributions\nto anomalous and SHEs [2]: (1) Skew scattering de-\nscribes the asymmetric scattering of spin-up and\nspin-down electrons on a spherical potential due to\nthe spin-orbit interaction which leads to di\u000berent\n\fnal momenta. (2) Side jump is caused by a spin-\ndependent di\u000berence in acceleration and decelera-\ntion during scattering. This leads to a \fnite spin-\ndependent displacement of the unpolarized charge\ncurrent, and (3) intrinsic e\u000bects, which are given by\nthe electronic band structure and due to a combi-\nnation of spin-orbit coupling and virtual interband\ntransitions that result in spin-dependent transverse\nvelocities.\nAs shown in Fig. 1 the spin Hall angle can ei-\nther be positive ore negative. Theoretical models\nshow that the intrinsic spin Hall conductivity de-\npends on the spin-orbit polarization at the Fermi\nlevel, which is positive in metals with more than half\n\flling and negative for less than half \flling [28,29].\nA similar dependence has also been found for the\nextrinsic SHE [30].\nThe direct SHE and ISHE are not only intrigu-\ning from a fundamental physics point of view, but\nthey also have become essential for many spintron-\nics applications and devices. The ISHE is oftentimes\nused for a convenient spin-current detection in bi-\nlayers consisting of a ferromagnetic material close\nto a heavy metal. Prominent experimental exam-\nples are the combination of spin-pumping [3{19] or\nFig. 1. Spin Hall angles \u0012SHof 3dand 5dmetals as a func-\ntion of the total number of dandselectronsnd+s. The green\nopen circles represent theoretical calculations of the spin Hall\nconductivity (SHC) for 5 dmetals and a good agreement, both\nin sign and magnitude, is found. Reprinted with permission\nfrom Ref. [12]. Copyright 2015, AIP Publishing LLC.New pathways towards e\u000ecient metallic spin Hall spintronics 3\nspin-Seebeck [31{45] experiments with the\nISHE.\n1.2. Spin pumping\nSpin pumping describes the generation of a spin-\npolarized electron current in a normal metal in the\nvicinity of a ferromagnet that undergoes a resonant\nexcitation (e.g., ferromagnetic resonance, excitation\nof spin waves) [46,47]. Spin pumping can be under-\nstood as a transfer of spin-angular momentum from\na ferromagnet into a normal metal, thus reducing\nthe magnetization. As a result of this process, the\nGilbert damping of the magnetization precession is\nenhanced. This enhancement of the Gilbert damp-\ning parameter \u0001 \u000bis given by:\n\u0001\u000b=g\u0016B\n4\u0019MSdFMg\"#\ne\u000b; (2)\nwheregis theg-factor,\u0016Bis the Bohr magneton,\nMSis the saturation magnetization, dFMis the fer-\nromagnetic \flm thickness, and g\"#\ne\u000bis the real part\nof the e\u000bective spin mixing conductance [46].\n1.3. Spin Seebeck e\u000bect\nThe \feld of spin caloritronics [48], investigating\nthe interplay between heat, charge and spin, has\nbecome a substantial research area in spintron-\nics [31{37, 43{45]. Analogous to the Seebeck e\u000bect\nwhere a temperature di\u000berence between two dissim-\nilar electrical conductors or semiconductors gives\nrise to a voltage di\u000berence between the two ma-\nterials, the spin Seebeck e\u000bect (SSE) describes the\ngeneration of a \\spin voltage\" when a magnetic ma-\nterial in contact with a normal metal is exposed to\na temperature di\u000berence [31,32]. Depending on the\nexperimental con\fguration, the spin current gener-\nated by the SSE can either be perpendicular or par-\nallel to the temperature gradient. The \frst arrange-\nment is called transverse [31], the latter one longi-\ntudinal con\fguration [33]. The transverse SSE can\nbe observed in both metals and insulators, whereas\nthe longitudinal spin Seebeck e\u000bect can be detected\nunambiguously only in insulators because of the ab-\nsence of the anomalous Nernst e\u000bect [42]. The theo-\nretical understanding of the SSE is still under a con-\ntroversial debate. While some theories argue that\nthe spin current results from the temperature dif-\nference between the ferromagnetic insulator and the\nmetallic layer which results in a thermal interfacial\nspin pumping, e.g., Refs. [33, 34, 35], other authors\npropose that the longitudinal SSE originates fromthermally-excited bulk magnons \rowing across the\nthickness, e.g., Refs. [38, 39, 40, 41]. At the same\ntime, it has recently been shown that SSE is not\nlimited to ferromagnetically ordered systems, but\ncan also be observed with antiferromagnets [49] as\nwell as paramagnets [50].\nIndependent of the possible explanation for the\nspin-current generation in these heterostructures,\nthe ISHE is generally used to detect the spin cur-\nrent.\n2. Route towards energy-e\u000ecient\nspintronics devices utilizing novel\nspin Hall detector materials\nThis Section is dedicated to investigations of novel,\ne\u000ecient spin Hall detector materials. First, we will\nreview elementary materials (transition metals) as\nspin-to-charge current converters, followed by prox-\nimity induced ferromagnetically ordered Pt and Pd.\nLastly, very recent studies utilizing antiferromag-\nnets, materials showing magnetic ordering, but zero\nnet magnetization, will be discussed.\nFig. 2. Illustration of the chemical structure of CuAu-\nI-type antiferromagnets (X = Fe, Pd, Ir, Pt) and the\nspin pumping and spin Hall e\u000bect experiment for the\nPy/Cu/antiferromagnet structures. Reprinted \fgure with\npermission from Ref. [7]. Copyright 2014 by the American\nPhysical Society.\nThe inverse spin Hall e\u000bect in ferromagnet/Pt\nbilayers [3{10, 31, 51] is widely used for spin cur-\nrent detection generated by either magnetization\ndynamics (e.g., ferromagnetic resonance) or a ther-\nmal gradient by the spin Seebeck e\u000bect [31]. Af-4M. B. Jung\reisch et al.\nter the experimental discovery of the inverse spin\nHall e\u000bect in ferromagnetic metal/normal metal\n(Ni80Fe20/Pt) bilayers [3, 52], a variety of di\u000ber-\nent metals and alloys were considered as poten-\ntially e\u000ecient spin Hall detectors. Mosendz et al.\ncarried out combined spin pumping/ISHE measure-\nments on Ni 80Fe20/normal metal bilayers and re-\nported spin Hall angles for Pt, Pd, Au, and Mo [5].\nThe analysis of these measurements for the spin\nHall angle was later revised by Zhang et al. by de-\ntermining the spin di\u000busion length of Pt through\nthickness dependent measurements [53].\nFurthermore, Wang et al. investigated bilay-\ners of the ferrimagnetic insulator yttrium iron gar-\nnet (YIG) and various metals such as Cu, Ag, Ta,\nW, Pt, and Au with varying spin-orbit coupling\nstrengths [6]. The spin Hall angles scale roughly\nasZ4, whereZis the atomic number, corroborat-\ning the spin-orbit coupling as underlying physical\nreason for the spin-charge interconversion. Figure 1\nshows how the spin Hall angles \u0012SHof various 3 d\nand 5dmetals vary as a function of the total num-\nber ofdandselectronsnd+s[12]. A good agree-\nment with theoretical calculations of the spin Hall\nconductivity (SHC), both in sign and magnitude,\nis found [54]. This result also highlights the impor-\ntance of the electron count of the d-bands [28,30,54]\nand con\frms previous experimental investigations\nby Morota et al., where the spin Hall conductivities\nof the 4dand 5dtransition metals Nb, Ta, Mo, Pd,\nand Pt were systematically studied by nonlocal spin\nvalve measurements [29].\nAnother pathway to design magnetic multi-\nlayers with optimal spin current e\u000eciencies, is to\nstudy how magnetic ordering a\u000bects spin transport\nphenomena. In particular, it is interesting to ad-\ndress this question in widely used ferromagnetic\nmetal/normal metal bilayers because many of the\nnominally non-magnetic materials are highly sus-\nceptible to magnetic proximity e\u000bects. Recently, it\nwas shown that for Pt and Pd increased proximity\ninduced magnetic moments can be correlated with\nstrongly reduced spin-Hall conductivities [17]. This\nobservation is in agreement with the energy depen-\ndence of the intrinsic SHE determined by \frst prin-\nciple calculations and can be understood, in a sim-\nple picture, as the development of a spin splitting\nof the chemical potential [17].\nBesides normally non-magnetic metals, which\nshow proximity induced magnetic polarization,\nmetals exhibiting magnetic ordering by themselvesare interesting candidates as possible spin Hall de-\ntectors. In this context, antiferomagnets have re-\ncently attracted increased attention. In contrast to\nferromagnets, antiferromagnets exhibit remarkable\nproperties such as zero net magnetization, nontriv-\nial spin-orbit coupling, and nonlinear magnetism.\nIn addition, their excitations are at higher frequen-\ncies beyond ferromagnetic resonance. Due to their\nzero net magnetization additional ferromagnetic or-\ndering is absent and, thus, other confounding ef-\nfects of anisotropic magnetoresistance (AMR) can\nbe ruled out. It was proposed theoretically that \r-\nFeMn, IrMn 3and Cr show a large anomalous Hall\ne\u000bect and a SHE due to the large spin-orbit cou-\npling and the Berry phase of the noncollinear spin\ntextures in these antiferromagnets [55{57].\nFig. 3. Anisotropic magnetoresistance-inverse spin Hall ef-\nfect spectra measured at\n9 GHz of the Py(15 nm)/Cu(4 nm)/antiferromagnet(5 nm)\nstructure for FeMn, PdMn, IrMn, and PtMn at room temper-\nature. Reprinted \fgure with permission from Ref. [7]. Copy-\nright 2014 by the American Physical Society.\nMendes et al. investigated experimentally the\nantiferromagnetic metal Ir 20Mn80, a material that\nis commonly used in spin-valve devices, as spin Hall\nmaterial [58]. In agreement with theoretical predic-\ntions, they found a large SHE as big as in Pt. Here,\nthe spin-current injection in Ir 20Mn80is achieved\nby microwave spin pumping as well as by the lon-\ngitudinal spin-Seebeck e\u000bect from a single crystal\nYIG.\nMore systematic studies were reported by\nZhang et al., where di\u000berent CuAu-I-type metal-New pathways towards e\u000ecient metallic spin Hall spintronics 5\nlic antiferromagnets were tested for their potential\nas spin-current detectors using spin pumping from\na Ni 80Fe20layer and the inverse spin Hall e\u000bect [7].\nThe investigated antiferromagnets feature the same\nchemical structure, i.e., X 50Mn50where X = Fe,\nPd, Ir, and Pt (with increasing atomic number) as\nillustrated in Fig. 2. By thickness-dependent mea-\nsurements, the spin di\u000busion length of the investi-\ngated materials is found to to be all rather short,\non the order of 1 nm. Figure 3 shows typical volt-\nage spectra for the di\u000berent types of investigated\nantiferromagnets. The estimated spin Hall angles\nof the four materials corroborate the importance\nof spin-orbit coupling of the heavy metals for the\nproperties of the Mn-based alloys through orbital\nhybridization [7]. First-principles calculations of or-\ndered alloys showed that the value of the spin Hall\nconductivity depends strongly on the crystal ori-\nentation and staggered antiferromagnetic magneti-\nzation. A follow-up work by Zhang and co-workers\nindeed showed a growth-orientation dependence of\nspin torques by studying epitaxial samples [59,60],\nwhich may be correlated to the anisotropy of the\nSHE [61].\nBesides being promising materials for spin cur-\nrent generation and detection, some antiferromag-\nnetic materials also appear to be good candidates\nfor transmitting spin currents. It was shown that\nspin currents can be injected e\u000eciently from a mag-\nnetic insulator (YIG) into an antiferromagnetic in-\nsulator (NiO) [11]. The insertion of a thin NiO layer\nbetween YIG and Pt enhances the spin-current in-\njection into Pt, which suggests a high spin-transfer\ne\u000eciency at both YIG/NiO and NiO/Pt interfaces\nand spin transport in NiO mediated by antifer-\nromagnetic magnons or antiferromagnetic \ructua-\ntions [11].\nFig. 4. (a) Scanning electron microscope image of spin-\ntorque nano-oscillator. (b) Power spectral density of the emit-\nted microwave signal at H= 700 Oe,I= 20 mA,T= 6 K\nand at an angle \u0012= 60\u000e. Reprinted \fgure with permission\nfrom Ref. [72]. Copyright 2013 by the American Physical So-\nciety.3. New approaches to drive\ncollective magnetization\ndynamics by spin Hall e\u000bect\nMagnetization dynamics is conventionally driven at\nrf frequencies by the Oersted \feld generated by a\nmicrowave signal. However, over the past decades\nnew methods were developed using dc current in-\nduced spin currents to exert spin torques on the\nmagnetization. In this Section, we will give a very\nbrief overview over the more traditional ways to ex-\ncite spin dynamics by dc currents and review alter-\nnative approaches on basis of SHEs reported very\nrecently.\nPioneering work by Slonczewski and Berger\npredicted that a \row of spin angular momentum\ncan exert a spin-transfer torque [62] on the magne-\ntization of a ferromagnet and drive it out of equilib-\nrium [63,64]. This spin torque can act as a negative\nmagnetic damping and even excite auto-oscillations\nof the magnetization [65, 66]. Spin-torque nano-\noscillators have been demonstrated in nanopillars\nand spin valves [67,68], point contacts [69] and mag-\nnetic tunnel junctions [70]. The basic idea of all\nthese structures is to generate a spin polarization\nof the current that is noncollinear with the back-\nground magnetization. Usually, this is achieved by\npatterning of multilayered \flms. These multilayered\n\flms consist of a \\reference layer\" acting as a po-\nlarizer that spin-polarizes an electric charge current\nand a \\free layer\". If the reference and free layer\nmagnetizations are noncollinear, the spin-polarized\ncurrent will destabilize the magnetization in the free\nlayer bysd-exchange interaction and eventually ini-\ntiate auto-oscillations in the free layer when a cer-\ntain threshold current is reached.\nRecently, a new type of spin-torque nano-\noscillators based on spin-orbit torques from a charge\ncurrent was realized in Ni 80Fe20/Pt bilayers [71{\n73]. Here, the spin current is generated by means\nof the SHE which makes simpli\fed device struc-\ntures without complicated multilayer stacks pos-\nsible. Figure 4(a) shows a possible realization of\nspin Hall nano-oscillator, which comprises a 4 \u0016m\nPy(5 nm)/Pt(4 nm) disk and Au electrodes with\na separation of 70 nm. A typical power spectral\ndensity spectrum of the emitted microwave signal\nis shown in Fig. 4(b). Even more important than\nthe simpler device geometry is that the utiliza-\ntion of the SHE allows the realization of magnetic\nnano-oscillators based on conducting andinsulating\nmaterials that do not require charge current \row6M. B. Jung\reisch et al.\nthrough the active device area [71].\nFirst approaches to drive auto-oscillations by a\ndc current in bilayers of the ferrimagnetic insula-\ntor YIG and Pt did not succeed because the esti-\nmated threshold current that is required to trigger\nthe self-oscillations exceeded the experimental feasi-\nbility [9,10]. These experiments were performed on\nlarge, macroscopic structures, where many nearly\ndegenerate spin-wave modes are present. When the\ndc source is applied, all of these modes are driven\nsimultaneously and, thus, they compete with each\nother. This leads to a self-limitation since none of\nthese modes can overcome the threshold for the on-\nset of auto-oscillations [51, 74]. In order to over-\ncome these di\u000eculties, it is crucial to reduce the\ndensity of the mode spectrum, and, ideally, to iso-\nlate one single mode [8,51]. This discretization can\nbe achieved by reducing either the thickness [8]\nand/or the lateral dimensions [51, 74] of the YIG\npattern, because quantization leads to increased fre-\nquency gaps between the modes. For this purpose,\nnanometer-thin, ultra-low damping YIG \flms are\nindispensable [75{77].\nHamadeh et al. use magnetic resonance force\nmicroscopy (MRFM) to show damping compensa-\ntion in 5\u0016m diameter YIG(20 nm)/Pt(7 nm) disks\n[51]. They demonstrate that the magnetic losses of\nspin-wave modes existing in the magnetic insulator\ncan be reduced or enhanced by a factor of 5 depend-\ning on the polarity and magnitude of an in-plane dc\ncurrent \rowing through the adjacent normal metal\nwith strong spin-orbit interaction. Figure 5(a) illus-\ntrates a density plot of MRFM spectra as a func-\ntion of \feld and current. The signal is asymmet-\nric in the applied dc current. The signal broadens\nand the amplitude decreases as the current is in-\ncreased to positive values and almost disappears\nat +8 mA. For negative currents, it becomes nar-\nrower and the amplitude is maximal for currents\nsmaller than -10 mA. The normalized integrated\npower increases by a factor of 5 from +12 mA to\n-12 mA, see Fig. 5(b). As is apparent from Fig. 5(c),\nHamadeh and co-workers observe an increase of\nthe linewidth from 6 Oe at 0 mA up to 14 Oe at\n12 mA and they \fnd a minimum of about 2 Oe\nbetween -8 and -11 mA. In order to check if this\ncurrent can be identi\fed as the threshold current\nfor the onset of self-oscillations, they show MRFM\nspectra in an experiment where no rf excitation\nis applied [Fig. 5(d)]. Strikingly, the narrow peak\nshifts linearly in dc current with the applied \feld,\nFig. 5. (a) Density plot of the MRFM spectra at 4.33 GHz\nas function of \feld and current. The color scale represents\nthe change in the magnetization (white: 0 G, black: 1.5 G).\n(b) Integrated power versus applied current. (c) Dependence\nof linewidth on current. (d) Di\u000berential measurements mod-\nulated by 0.15 mA, no rf excitation, versus current at six\ndi\u000berent values of the in-plane magnetic \feld. Reprinted \fg-\nure with permission from Ref. [51]. Copyright 2014 by the\nAmerican Physical Society.\nwhich is in agreement with the theoretically ex-\npected behavior for auto-oscillations.\nAnother interesting approach to drive uniform\nmagnetization dynamics is spin-torque ferromag-\nnetic resonance (ST-FMR) which was originally de-\nveloped for all-metallic systems [78]. The basic idea\nis as follows: If a \row of alternating charge current\nis passed through a heavy normal metal/metallic\nferromagnet bilayer, the SHE generates an oscillat-\ning transverse spin current in the normal metal.\nThis results in a spin accumulation at the inter-\nface and, thus, it leads to a transfer of spin an-\ngular momentum to the ferromagnetic layer that\ncan induce ferromagnetic resonance. In metallic fer-\nromagnets, the resonance is detected by a homo-\ndyne voltage from anisotropic magnetoresistance\n[79,80]. The ST-FMR technique allows for a conve-\nnient determination of the spin Hall angle by a sim-\nple lineshape analysis, which is in agreement with\nmeasurements of the dc-current dependence of the\nresonance linewidth [78]. Besides investigations on\nPy/Pt bilayers, ST-FMR metrology was also used\nin ferromagnet(Py)/topological insulator(Bi 2Se3)\nsystems [81] and at a ferromagnet(Py)/Rashba-\ninterface(Ag/Bi) [82].\nRecently, it was predicted that the concept of\nST-FMR, where a ac spin Hall e\u000bect mediated spin\naccumulation drives magnetization oscillations, can\nbe extended to insulating systems [83,84]. Here, weNew pathways towards e\u000ecient metallic spin Hall spintronics 7\nhave to consider two di\u000berent contributions to the\nmeasured voltage signal: (1) spin pumping and (2)\nspin Hall magnetoresistance (SMR) [85]. In these\nkind of magnetic insulator/normal metal bilayers,\nSMR mixes with the microwave signal rather than\nthe anisotropic magnetoresistance. SMR stems from\nthe spin-current back\row at a ferromagnetic insu-\nlator/normal metal interface: If a charge current is\npassed through the normal metal, it is converted\ninto a spin current that accumulates at the interface\nto the ferromagnetic insulator. If the spin polariza-\ntion\u001band the magnetization Mare noncollinear,\nspin-\rip scattering can occur. This leads to a par-\ntial absorption of the spin current at the interface\nwhich results in the excitation of magnetization\nprecession.\nFig. 6. (a) and (b) Geometric relation between the \row of\nelectrons and accumulated spins in the normal metal and\nthe magnetization in the magnetic insulator. Reprinted \fg-\nure with permission from Ref. [85]. Copyright 2013 by the\nAmerican Physical Society.\nConsequently, less spin current is back-\nre\rected. This absorption is maximized when Mis\nperpendicular to \u001band zero when Mis parallel to\n\u001b[85], see Fig. 6. Thus, the conductivity enhance-\nment due to SHE and ISHE is maximized (mini-\nmized) when Mis perpendicular (parallel) to the\ncharge current Je, becauseJeis perpendicular to \u001b\nand the Pt resistance depends on the magnetization\ndirection in the YIG. The basic idea of SMR was al-\nready proposed by D'yakonov in 2007 [86]. Here, an\nexternally applied magnetic \feld destroys the edge\nspin polarization resulting in a positive magnetore-\nsistance. In the work by Nakayama et al. exchange\ninteraction dephasing the spin accumulation is the\nreason for the observed magnetoresistance.\nChiba et al. developed a model of the ac\nspin Hall magnetoresistance in a bilayer system\nconsisting of a magnetic insulator and a heavy\nmetal [83, 84]. As is shown in Refs. [83, 84], it is\npossible to derive expressions for dc voltages under\nST-FMR in these systems using the drift-di\u000busion\nspin model and quantum mechanical boundary con-dition at the interface. Figure 7 shows the calculated\ndc voltage spectra according to their model. The\ninset illustrates that the spin-pumping contribution\nis purely symmetric and the SMR features a super-\nimposed symmetric and antisymmetric Lorentzian\nlineshape. Very recent experimental studies show\nthat the concept of ST-FMR can indeed be applied\nto insulating systems [87{89].\nFig. 7. Calculated dependence of the ST-FMR spectra on\nthe ferromagnetic layer thickness dFat a driving frequency of\n9 GHz and an in-plane angle of 45\u000e. Inset: Contributions by\na recti\fcation due to spin Hall magnetoresistance and spin\npumping. Reprinted \fgure with permission from Ref. [83].\nCopyright 2014 by the American Physical Society.\n4. Manipulation of spin textures by\nspin Hall e\u000bects\nIn this Section, we will provide a short summary of\ndi\u000berent, widely known spin textures and introduce\nnovel ideas, which make use of SHEs to dynamically\ncreate and manipulate magnetic skyrmion bubbles.\nCompeting magnetic interactions, most impor-\ntantly exchange interactions at short length scales\nand magnetostatic dipole interactions at long length\nscales, result in the formation of a variety of magne-\ntization states and spin textures resulting in many\nfascinating physical phenomena. These spin tex-\ntures exhibit technologically relevant features in-\ncluding emergent electromagnetic \felds and e\u000e-\ncient manipulation [90{95].\nOne of the easiest examples are magnetic do-\nmain walls, the boundaries between magnetic do-\nmains with di\u000berent magnetization con\fgurations.\nMagnetic domain walls exhibit interesting physical\nphenomena including complex dynamics at di\u000berent\ntime scales [96, 97]; typical domain wall widths lie\nin the nanometer-range. Another prominent exam-\nple of spin textures are magnetic vortices that are8M. B. Jung\reisch et al.\nstabilized by magnetostatic interactions. Magnetic\nvortices are planar spin textures with an in-plane\nmagnetization curling leading to two possible chi-\nralities and a central singularity that is called vortex\ncore. A vortex core points perpendicular to the \flm\nplane and can thus have two polarities. They have\ntypically a diameter of a few nm [98].\nFig. 8. (a) Illustration of the formation of magnetic\nskyrmions from chiral stripe domains through a geometrical\nconstriction. (b) Kerr Microscopy image taken after applying\na short current pulse, showing a large number of skyrmions\nformed on the right side of the constriction. From Ref. [95].\nReprinted with permission from AAAS.\nVery recently, a similar, but more complex type\nof topologically stable spin textures has gained\ngreat attention in the scienti\fc community: mag-\nnetic skyrmions. They were \frst observed by neu-\ntron scattering at the border between paramag-\nnetism and long-range helimagnetic order perpen-\ndicular to small applied magnetic \felds [99]. The\nnovelty of magnetic skyrmions and what sets them\napart from the above mentioned spin textures is\nthat they are three-dimensional in character. There\nare two types of magnetic skyrmions: (1) \\hedge-\nhog skyrmions\", where the progression of the mag-\nnetization is cycloidal across the diameter and (2)\n\\vortex skyrmions\", where the progression of the\nmagnetization is helical. The ability to controllably\ncreate and move magnetic skyrmions is of funda-\nmental importance for technological implementa-\ntion of skyrmion-based spintronics. Here, the sta-\nbility of skyrmions at room temperature is key and\nhas been challenging in the past. Recently, it was\nshown that magnetostatically stabilized skyrmion\nstructures, magnetic bubbles, can form in magneticthin \flms with perpendicular magnetic anisotropy\nat room temperature [95]. The electric-current gen-\neration of skyrmions is achieved by adding an addi-\ntional layer with strong spin-orbit coupling to the\nferromagnet which results in an interfacial broken\ninversion symmetry and, thus, the generation of in-\nterfacial Dzyaloshinskii Moriya interaction (DMI).\nDMI stabilizes chiral magnetic domain walls around\nthe bubble that lead to a skyrmion spin struc-\nture. These skyrmion bubbles can then be electri-\ncally manipulated by SHEs. Jiang et al. demon-\nstrated this in a Ta/CoFeB/TaO xtrilayer, where\nskyrmions can be generated via laterally inhomoge-\nneous current-induced spin-orbit torques in a pro-\ncess analog to the droplet formation in surface-\ntension driven \ruid \row [95].\nIn the Ta/CoFeB/TaO xheterostructure the\nelectric current \rowing through the heavy metal\ngenerates a transverse spin current due to the SHE,\nwhich results in spin accumulation at the interface\nwith the ferromagnetic layer [95]. This spin accumu-\nlation exerts a spin-orbital torque on the chiral do-\nmain wall. If the current \row is homogeneous, a chi-\nral spin-orbital torque enables e\u000ecient domain-wall\nmotion [100,101]. Because of symmetry reasons the\ntorques on a stripe domain with chiral walls cancel\non the sides parallel to the current and, thus, only\nthe end of the stripe domain is moved. For a pinned\ndomain, this results in an elongation of the stripe.\nHowever, by introducing a geometrical constriction\ninto the current-carrying trilayer wire, a transverse\ncurrent component around the narrow neck can be\nachieved.\nAs a result an inhomogeneous e\u000bective force\ncaused by the spin Hall \feld is created and extends\nthe end on the domain. The surface tension in the\ndomain wall increases because of the continuously\nexpanding radius, which \fnally results in breaking\nthe stripes into circular domains [see Fig. 8(a)].\nDue to the presence of interfacial DMI, the circular\ndomains maintain a well-de\fned chirality and once\nformed, these created synthetic hedgehog (N\u0013 eel)\nskyrmions are stable due to topological protection.\nThey can be moved very e\u000eciently in the direc-\ntion of the charge current, Fig. 8(b). Depending on\nthe strength of the external magnetic \feld, these\ndynamically created skyrmions feature a variable\nsize between 700 nm and 2 mm and are stable for\nat least 8 hours [95]. Figure 9 shows an experi-\nmentally determined electric current vs. magnetic\n\feld phase diagram for the skyrmion formation. InNew pathways towards e\u000ecient metallic spin Hall spintronics 9\nprinciple, the size of the skyrmions could be scaled\ndown [102,103] by engineering material-speci\fc pa-\nrameters that control the various competing mag-\nnetic interactions which might lead to sophisticated\nskyrmionic device concepts such as a spin Hall con-\ntrolled skyrmion racetrack memory [95].\nFig. 9. Phase diagram for skyrmion formation. The shaded\narea illustrates \feld-current combinations that result in the\npersistent generation of skyrmions after each current pulse.\nFrom Ref. [95]. Reprinted with permission from AAAS.\nAs compared to the extensively studied mag-\nnetic domains and domain walls in the metallic\nsystems, another interesting, but yet missing as-\npect is the electrical control of magnetic domains\nand domain walls in magnetic insulators such as\nYIG [105]. This is made possible by the SHE in\nclose contact to the magnetic insulator. Historically,\nit is well known that rare-earth doped YIG \flms\ncontain periodic magnetic bubble domains [106].\nFigure 10 shows such periodic bubble domains in a\n6\u0016m thick Bi doped YIG \flm. It may be that these\nmagnetic bubbles in YIG \flms are topologically dif-\nferent from coherent skyrmion spin textures in B20\ncompounds, due to the absence of DMI. By em-\nploying the SHE of various heavy metals in heavy\nmetal/YIG hybrids, it would be intriguing to see\nthe motion of these bubbles. These investigations\ncould shed light on the question if the direction of\nbubble motion may serve as a criterion to select\nthe bubbles with di\u000berent skyrmion numbers. On\nthe other hand, as compared to lithographically\npatterned micro-disks of a magnetic insulator, it is\nalso possible that these naturally existing bubbles\ncan be driven into auto-oscillation by means of SHE\nand spin-transfer torque.\nFig. 10. Polar magneto-optical Kerr e\u000bect image of periodic\nmagnetic bubble domains in the 6 \u0016m thick Bi doped YIG\n\flm grown on GGG (111) substrate by liquid phase epitaxy.\nThe diameter of each magnetic bubbles is around 1.3 \u0016m.\n5. Conclusion\nMore than two decades after the discovery of the\ngiant magnetoresistance, spintronics is still a vivid\n\feld of contemporary magnetism research that con-\nstantly grows and develops new ideas. In particular,\nit gained new momentum in recent years after the\nexperimental demonstration of many spin Hall ef-\nfect related e\u000bects such as the spin Seebeck e\u000bect,\ndetection of spin-pumping driven spin currents or\nspin-torque ferromagnetic resonance to name only\na few.\nPredicting future research directions is cer-\ntainly an impossible task, but we want to give a brief\noutlook on possible pathways to novel spin Hall ef-\nfect related spintronics applications.\nThe search for optimal spin Hall detector mate-\nrials has de\fnitely not yet come to an end. The uti-\nlization of antiferromagnets in spintronics devices is\na promising route; in particular taking into account\ntheir high-frequency characteristics beyond ferro-\nmagnetic resonance, which makes them interesting\nfor information technologies. Besides that topolog-\nical insulators [81,107] might revolutionize the e\u000e-\nciency charge-spin current interconversion, and lead\nto new spin Hall driven spin-torque devices.\nThe implementation of magnetic insulators in\nspintronics has the potential for the development\nof novel low-energy consuming devices. Many of\nthe above mentioned e\u000bects were \frst shown in all-\nmetallic systems, and later on in insulators. Along\nthose lines, it would be very interesting to demon-\nstrate the movement of skyrmions bubbles in insu-\nlators by spin Hall e\u000bect driven spin torques or even10 M. B. Jung\reisch et al.\nthe onset of auto-oscillations.\nBeyond bulk e\u000bects such as spin Hall e\u000bects,\none might think about the utilization of interface-\nor surface e\u000bects such as the Rashba-Edelstein e\u000bect\n(REE). First experimental studies show that REEs\ncan indeed be used for the spin- to charge-current\ninterconversion [82,108,109].\nMany surprises can be expected in this rapidly\nevolving \feld and we look forward to exciting dis-\ncoveries in the future.\nAcknowledgments\nThis work was supported by the U.S. Department\nof Energy, O\u000ece of Science, Materials Science and\nEngineering Division.\nPreprint of an article published in\nSPIN 05, 1530005 (2015) [13 pages]\nDOI: 10.1142/S2010324715300054 c\r\ncopyright World Scienti\fc Publishing\nCompany, http://www.worldscientific.com/\nworldscinet/spin .\nReferences\n1. A. Ho\u000bmann and S.D. Bader, New Opportunities\nat the Frontier of Spintronics, Phys. Rev. Appl. , in\npress.\n2. A. Ho\u000bmann, Spin Hall E\u000bects in Metals, IEEE\nTrans. Magn. 49, 5172 (2013).\n3. E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara,\nConversion of spin current into charge current at\nroom temperature: inverse spin-Hall e\u000bect, Appl.\nPhys. 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Di Ventra4\n1Department of Physics, University of Nizhny Novgorod,\n603950 Gagarin Avenue 23, Nizhny Novgorod, Russian Federat ion\n2Department of Physical Chemistry, The University of the Bas que Country, 48080 Bilbao, Spain\n3IKERBASQUE Basque Foundation for Science, Bilbao, Spain\n4Department of Physics, University of California, San Diego ,\n9500 Gilman Drive, La Jolla, CA 92093-0319, USA\nThe coupling of orbital and spin degrees of freedom is the sou rce of many interesting phenomena.\nHere, we study the electron dynamics in a quantum billiard dr iven by a periodic electric ﬁeld–a\nmesoscopic rectangular quantum dot– with spin-orbit coupl ing. We ﬁnd that the spatial and tem-\nporal proﬁles of the observables demonstrate the transitio n to chaotic dynamics with qualitative\nmodiﬁcations of the power spectra and patterns of probabili ty and spin density. The time depen-\ndence of the wavefunctions and spin density indicates spin- charge separation seen in the decay of\nthe spin-charge density correlators. Experimental veriﬁc ation of this spin chaos eﬀect can lead to\na better understanding of the interplay between spin and spa tial degrees of freedom in mesoscopic\nsystems.\nPACS numbers: 72.25.Dc,72.25.Pn,73.63.Kv,75.70.Tj\nI. INTRODUCTION\nThe emergence of stochasticity is of fundamental im-\nportance for classical and quantum physics with broad\ninterdisciplinary connections and applications [1–5]. Fas-\ncinating examples of irregular dynamics can be found in\nmeso- and nanoscale systems, including quantum dots\n[4, 6, 7]. While the understanding of charge transport\nin such systems is already quite deep [8], the knowledge\nof the chaotic spin evolution is still poor. Since the re-\nlated branch of physics, “spintronics” is among the most\npromising research ﬁelds [9–11], it is of importance and\nappliedinterest tostudy thespin dynamicsin mesoscopic\nsystems with coupled charge and spin degrees of free-\ndom. A natural example for these studies is provided by\nsemiconductor structures where the spin-orbit coupling\n(SOC) plays a signiﬁcant role in dynamics, including the\nability of spin manipulation by electric ﬁeld [12–16]. It\nhas been predicted that for a suﬃciently strong driving\nﬁeld the dynamics of charge and spin can become unex-\npectedly complicated for electron in a double quantum\ndot [17–19]. However, it is not known what kind of irreg-\nular coupled dynamics one may expect in a mesoscopic\nstructure such as a semiclassical quantum billiard.\nWithout SOC such structures may demonstrate cer-\ntain classical traits of the transition to chaos due to the\nhigh density of states, including the formation of irregu-\nlar wavefunctions inside the billiard [4, 6, 20, 21].\nIt is known that the eigenstates distributions in rect-\nangular billiards with SOC demonstrate ﬁngerprints of\nchaos such as the Wigner statistics, not expected for in-\ntegrable systems [22]. Since chaos driven by an exter-\nnal periodic ﬁeld is a common phenomenon in nonlinear\n∗Electronic address: khomitsky@phys.unn.rusystems, it is natural to ask whether an irregular mo-\ntion arises in such a billiard for coupledcharge and spin\nchannels under a periodic driving. The problem itself is\ncomplicated since there are no explicit criteria for iden-\ntifying the chaotic regimes in the dynamics of quantum\nspin observables not having classical counterparts.\nIn this paper, we present a model for exploring dy-\nnamical regimes of coupled charge and spin degrees of\nfreedom for two-dimensional (2D) electrons conﬁned in a\nrectangular quantum billiard with Rashba SOC and an\nin-plane magnetic ﬁeld. We consider charge and spin dy-\nnamics driven by a periodic electric ﬁeld resonating with\ntransitions between two nearest semiclassical size quanti-\nzation levels. We observe strong indications of transition\nto irregular dynamics both for spatial coordinates and\nspins asstudied using the Floquet stroboscopictechnique\n[2, 4, 18, 23]. Although the quantum nature of our sys-\ntems hampers pure classical manifestation of chaos like\nthe Lyapunov exponents, the scenarios for both charge\nand spin evolution suggest the transition to irregular\nregimes. We found that the most sensitive observables\nare the densities of charge and spin, where the textures\nof diﬀerent shapes determined by the driving frequency\nand amplitude are formed. The results can be useful for\ntheunderstandingofthequantumchaosinvolvingspinas\nwell as for the design of semiconductor-based spintronics\ndevices.\nII. MODEL\nOur Hamiltonian, H(t) =H0+V(t), consists of the\nunperturbed part H0describing an electron conﬁned in\nthe 2D rectangular billiard with sides aandb, hard-wall\nboundary conditions, Rashba SOC and Zeeman interac-\ntion to the in-plane magnetic ﬁeld Bx:2\nH0=p2\n2m+Hso+1\n2gµBσxBx, (1)\nwhere we take Hso=αR(σxky−σykx) in the Rashba\nform. Here m,αR,andgare the electron eﬀective\nmass, Rashba SOC constant, and g-factor, respectively.\nThe in-plane magnetic ﬁeld lifts the Kramers degener-\nacy, however, does not cause the diamagnetic coupling.\nSince the σxoperator is present in both the SOC and\nZeeman terms, the Zeeman splitting is coupled to the\norbital quantization making it orbital state-dependent.\nThe spin-coordinate entangled wavefunction of the n-th\nstate ofH0can be constructed as a superposition of the\norbital wavefunctions multiplied by the two-component\nspinors,\nψn(r) =/summationdisplay\nlx,ly/bracketleftBigg\nζ↑\nlx,ly\nζ↓\nlx,ly/bracketrightBigg\nn·2√\nabsinπlxx\nasinπlyy\nb,(2)\nwhere the n-dependent coeﬃcients ζ↑\nlx,lyandζ↓\nlx,lyare\ndetermined from the matrix eigenvalue problem, and\nr= (x,y). The SOC leads to the spin-coordinate en-\ntanglement of the state ψn(r), while at αR= 0 the cor-\nresponding state is the product of the orbital state and\neigenstate of σx.\nThe typical level splitting in a µm-size billiard, being\nsigniﬁcantlylowerthaninananoscalequantumdot,leads\nto a better possibility to reproducethe quantum-classical\ncorrespondence in the regular and stochastic evolution.\nBerggren and Ouchterlony proved that the SOC leads to\nthe non-Poissonian level statistics, indicating the pres-\nence of quantum chaos [22]. However, this well-known\npicture of static eigenstate description leaves open the\nquestion on the dynamical characteristics of the associ-\nated evolution. In particular, on the diﬀerence between\nsimply irregular and chaotic behavior for coupled spin\nand charge degrees of freedom, which we consider in our\npaper. The external driving is needed since it allows\nmanipulating the perturbation amplitude in a controlled\nmanner, which is required in most of the applications to\ntransfer and keep the system in a required state in the\npresence of inevitable momentum and spin relaxation.\nWe will see that the quantum-classical correspondence\nwith the generation of the chaotic behavior of observ-\nables will be maintained mainly in the initial period of\nthe evolution, and after that more regular quantum dy-\nnamics is achieved both for charge and spin degrees of\nfreedom. This is in agreement with general properties of\nquantum chaos [1–5].\nThe driving term V(t) =eE0xcosωt, whereeis the\nfundamental charge, is chosenasthe monochromaticuni-\nform electricﬁeld ofamplitude E0which induces the local\nresonance between the level pair split by En0−En0−1=\n/planckover2pi1ω0for a given n0. Due to the spin-coordinate entan-\nglement in Eq. (2), the electric ﬁeld causes transitionsbetween states with diﬀerent expectation values of spin,\nand leads to nontrivial dynamics as discussed below.\nTocharacterizethe couplingsin the system, weapply a\nperturbativeapproachand ﬁnd the dimensionless param-\neters for spin-orbit coupling and electric ﬁeld strength.\nWe begin with the unperturbed spectrum:\nE(lx,ly,σx) =π2/planckover2pi12\n2m/parenleftBigg\nl2\nx\na2+l2\ny\nb2/parenrightBigg\n+1\n2gµBσxBx(3)\nand take the main semiclassical term (assuming lx≫\n1,ly≫1) in the energy diﬀerence of the same-spin states\n∆E≡E(lx+1,ly,σx)−E(lx,ly,σx) =π2/planckover2pi12\nma2lx.(4)\nFor spin-orbit coupling we take the ratio\nfso≡|/angbracketleftlx+1,ly,1|Hso|lx,ly,−1/angbracketright|\n∆E=2\nπ2αR\nama2\n/planckover2pi12,(5)\nthatis, essentially,theratioofthebilliardsizetothespin\nprecession length /planckover2pi12/mαRas a dimensionless strength\nof spin-orbit coupling. For the driving ﬁeld we proceed\nsimilarly and deﬁne\nfE≡eE0|/angbracketleftlx+1,ly,1|x|lx,ly,1/angbracketright|\n∆El−1x=2\nπ4eE0ama2\n/planckover2pi12.(6)\nIII. LEVEL VARIANCE EVOLUTION AND\nFOURIER POWER SPECTRA\nWe begin by solving the nonstationary Schr¨ odinger\nequation with the driving V(t) in the basis of the eigen-\nstates (2) to obtain the spinor wavefunction\nΨ(r,t) =/summationdisplay\nnCn(t)ψn(r), (7)\nwhere the time-dependent coeﬃcients Cn(t) are solu-\ntions of a system of ordinary diﬀerential equations. The\ninitial condition is taken as the single level occupancy,\nCn(0) =δnn0. The initial single level state can be\nprepared in a mesoscopic billiard by resonant tunneling\nof an electron with required energy, entering from at-\ntached leads [28]. If the initial state is a superposition of\nthe eigenstates, its dynamics can be found as the cor-\nresponding superposition of the time-dependent states\ndemonstrated below.\nFor numerical calculations we consider a GaAs billiard\nwith (if not stated otherwise) αR= 5 meVnm, a= 2.1\nµmandb= 1.5µm (sameasin Ref.[22]). Here /planckover2pi12/ma2=\n0.26µeV, and fso≈2.0. We assume a magnetic ﬁeld\nBx= 500 Gs with the Zeeman splitting of 1.3 µeV. The3\n0 100 200 300 400 500\nN510\n0 100 200 300 400 500\nN40444852\n0 10 2000.15I   ( )\n0 50 10000.01I   ( )∆L\n∆Lσz\nσz\nω/ω0ω/ω0ω ω(a) (b)\n(c) (d)\nFIG. 1: (Color online) (a),(b) Stroboscopic evolution of th e\nvariance ∆ Lof the level number eﬀectively involved into the\ndynamics for N= 500 periods, T, of driving ﬁeld, (a) weak\ndriving electric ﬁeld with amplitude E0= 0.14 V/cm and\n(b) moderate driving ﬁeld E0= 0.70 V/cm. After a short\nperiod of diﬀusive growth the evolution of ∆ Lapproaches the\nstationary regime with stable average value. (c),(d): Four ier\npower spectra (in dimensionless units) for the mean values o f\nσzspin component, (c) E0= 0.14 V/cm and (d) E0= 0.70\nV/cm. The total observation time in (c) and (d) is Ttot=\n100T.\ntypical driving frequency here is ν=ω0/2π= 0.78 GHz,\nand the initial state is on the level n0= 200.\nWe calculate the evolution of quantum observables\nusing the wavefunction (7). The parameters describ-\ning the evolution in the Hilbert space can be chosen\nas the mean level number ¯L(t) =/summationtext\nLL|CL(t)|2and\nits dispersion /angbracketleft∆2L(t)/angbracketright=/summationtext\nL/parenleftbig\nL−¯L(t)/parenrightbig2|CL(t)|2. It\nis known that during the initial stages of the develop-\nment of quantum chaos the variance of the level num-\nber ∆L(t)≡ /angbracketleft∆2L(t)/angbracketright1/2grows with time [2, 23]. This\ncorresponds to the diﬀusion in the Hilbert space with in-\ncreasing number of levels being involved, and represents\ncounterpartoftheclassicalchaoticdynamics. After some\ntime, this stage transforms into the stabilized quantum\nevolution, where ∆ L(t) saturates, thus showing the sup-\npression of the Hilbert space diﬀusion [2, 5, 23].\nThe stroboscopic dynamics of level variance ∆ L(t) is\nshown in Fig.1 for 500 periods T, of driving (a) for the\nmoderate ﬁeld E0= 0.14 V/cm ( fE≈2.3) and (b) for\nstronger ﬁeld E0= 0.70 V/cm ( fE≈11.5). It is clear\nthat the initial fast growth in the number of involved lev-\nels ceases after N= 10...30 periods of driving ﬁeld, and\nafter that ∆ L(t) demonstrates the oscillating behavior\naroundtheaverage∆ Lav≈9inFig. 1(a)and∆ Lav≈46\nin Fig. 1(b). This behavior corresponds to the expected\ngrowth in the number of participating states with in-creasing driving amplitude. We may conclude that the\ndynamics ratherquicklyreachesa stabilized regimewith-\nout further diﬀusion in the Hilbert space, indicating that\neven a large billiard with hundreds of levels involved in\nthe dynamics behaves essentially as a quantum system\nwithout the long-lasting Hilbert space diﬀusion.\nQuantum evolution can be described in terms of the\npower spectrum, deﬁned for an observable ξ(t) as\nIξ(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞\n−∞ξ(t)e−iωtdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n. (8)\nFigures 1(c),(d) show power spectra for the spin com-\nponentσzfor the same driving ﬁelds, plotted in dimen-\nsionless units for mutual comparison. While for Fig. 1(c)\nthe spectrum has the form of discrete harmonics, the\nstronger driving ﬁeld leads to qualitatively richer spec-\ntrum as shown in Fig. 1(d) where the lower band is ﬁlled\ncontinuously. According to the basic concepts of quan-\ntum evolution,[1] this may be considered as the onset of\nchaos. It should be mentioned that the long-period\npatterns in level variations visible in Fig.1(a),(b) provide\nsizable contributions into the Fourier power spectra for\nthe spin component in Fig.1(c),(d) as the strong peaks\nat the left part of the frequency axis near ω=ω0. How-\never, the main diﬀerence between the regular and chaotic\ndynamics is in the mid- and high-frequency part of the\nspectrum, as it can be seen by comparing Figs.1(c) and\n(d), where a densely ﬁlled frequency band in Fig.1(d)\nrepresents the chaotic behavior.\nIV. POINCAR ´E SECTIONS\nIn order to gain insight into mutual impact of orbital\nand spin motion, we plot the evolution of expectation\nvalues in the pair of canonical variables ( x(t),px(t)) and\nin the non-canonical pairs ( Sβ(t),Sγ(t)), where βand\nγare Cartesian coordinates. For a periodic driving it\nis of interest to consider the Poincar´ e sections at stro-\nboscopic times t=NT(withNan integer) such as\n(x(N),px(N)) or (Sβ(N),Sγ(N)). To obtain the evo-\nlution with high accuracy, we use the Floquet strobo-\nscopic technique which requires the direct integration of\nthe time-dependent Schr¨ odinger equation only at a sin-\ngle period Tof the perturbation V(t). After that the\nstate of the system at any t=NTcan be obtained by\na ﬁnite algebraic procedure [18, 23]. In Fig. 2 we show\nthe stroboscopic Poincar´ e plots for the same ﬁelds as in\nFig.1.\nDue to the Rashba SOC the coordinate degree of free-\ndom evolves in the same manner as the spin one, indi-\ncating an irregular regime of dynamics like a “stochastic\nsea”[1, 2] without any “islands of regularity” with peri-\nodicorbits. Thisexampleofdynamicsmayserveasatool\nfor observing the spin chaos in quantum systems. In the\nabsence of SOC, the states would remain the eigenstates4\n-0.2 -0.1 0 0.1 0.2-4-20246px\n-0.2 -0.1 0 0.1 0.2-6-4-2024px\n-0.06-0.04 -0.02 0\nSx-0.0400.04Sy\n-0.05 00.05\nSx-0.100.1 Sy\n-0.04 0 0.04\nSy-0.02-0.010Sz\n-0.1-0.0500.050.1\nSy-0.0200.02Sz\n(c)(a)\n(b)(d)\n(e)\n(f)x [10-4 cm] x [10-4 cm]\nFIG.2: (Color online)stroboscopic Poincar´ e plots: (a,d) coor-\ndinate mean values ( x(N),px(N)) and (b) and (e) spin mean\nvalues (Sx(N),Sy(N)) and (c) and (f) - spin mean values\n(Sy(N),Sz(N)), shown for the driving ﬁeld strength ((a)-(c))\nE0= 0.14 V/cm and ((d)-(f)) E0= 0.70 V/cm. The coor-\ndinate degree of freedom evolves in the same manner as the\nspin degree of freedom due to Rashba SOC, both indicating\nan irregular regime of dynamics. Momentum pxis measured\nin units of /planckover2pi1/µm.\nofσx, and the Poincar´ e sections would be reduced to the\npointsSx(N) =±1 andSy(N) =Sz(N) = 0. Small\nexpectation values of spin components are due to the\nspin-coordinate entanglement of the states in Eq. (2). It\ncanbeseenthatthespindynamicsisstronglysensitiveto\nthe number of levels involved each having a diﬀerent spin\npolarization, so the amplitudes for the mean values also\ngrow at higher electric ﬁelds. By looking at Fig. 2 one\nalso notices that the dynamics in coordinate space is not\nso sensitive to the driving ﬁeld strength as the evolution\nof spin variables. This can be explained by taking into\naccount the structure of eigenstates and the initial con-\ndition of our evolution model which is an eigenstate with\na rather high number of spatial harmonics, and is com-\npletely delocalized in the billiard. The volume spanned\nby the evolution of the ( x,px) pair of mean values does\nnot expand greatly with increasingdriving strength since\nthese variables have comparable expectation values for\ntwo diﬀerent numbers of levels eﬀectively involved intothe dynamics for weak (Fig. 1(a)) and moderate (Fig.\n1(b)) driving amplitudes.\nIt should be mentioned also that the average value of\nspin for the weaker driving is shifted from zero more visi-\nbly, as it can be seen in Fig. 2. We explain this behavior\nby distinct, but generally alternating spin polarizations\nof the basis states, where the greater number of states\ninvolved at stronger driving leads to more eﬀective can-\nceling of non-zero contributions to the midpoint of spin\ndensity stroboscopic ensemble.\nV. SPIN TEXTURES AND SPIN-CHARGE\nSEPARATION\nAtypical experiment with scanning ofthe billiard with\nelectron gas measures local spin density integrated over\nthe “spot” under the probe [24]. Thus, the evolution of\nlocal spin density in the billiard can be of interest for fur-\nther experimental advances in exploring and controlling\nvarious regimes of the driven spin dynamics. We then\nlook at the spatial distributions of spin density in the\nwhole billiard, or spin textures, considered at the strobo-\nscopic time t=NTwith arbitrary N, together with the\ncharge density contained in the spinor components.\nThe charge- ρ(r,N) and spin density components\nSβ(r,N) are found with the wavefunction (7) as\nρ(r,N) =Ψ†(r,N)Ψ(r,N), (9)\nSβ(r,N) =Ψ†(r,N)σβΨ(r,N),(10)\nrespectively. In Fig.3 we show the probability distri-\nbutions for the charge and the Szspin component for\nthe initial eigenstate ( n= 200) with high number of\nspatial harmonics, and after N= 500 periods of driv-\ning. The other components of spin density as well\nas the probability density have similar patterns. The\ncharge and spin densities calculated at N <500 indi-\ncate that the picture is stabilized after several hundreds\nof driving periods, similar to the mean level dynamics in\nFig.1(a),(b). We have found that the distributions pre-\nsented in Figs.3(b),(c),(e),(f) are formed in the evolution\nin the electric ﬁeld as the interplay of two distinct pat-\nterns, which modify the regular structure of the initial\nstate in Fig.3(a),(d). One pattern is the average-scale\nand large-scale structure with regular spatial oscillations\nstemming from the limited number of basis states eﬀec-\ntively involved into dynamics. The other pattern of spin\ndensity in Fig. 3 has an irregular and spatially chaotic\ncharacter, mostly on the small and medium scales with\nformation of peaks with variable height. The amplitude\nof small scale irregular contribution to the spin density\ngrowswith increasingdriving strength. This is an indica-\ntion of the chaotic regime which is induced in our system\nat strong driving ﬁeld. We have observed the formation\nof similar two-scale density distributions for both charge\nand spin also at longer times compared to the snapshot\nin Fig.3. The formation of this stable picture can be at-\ntributed to the onset ofthe quasi-stationaryproﬁle of the5\nFIG. 3: (Color online) Probability density distributions f or\n(a)-(c) the charge and (d)-(f) the Szcomponent of spin den-\nsity in the billiard (a),(d) for the initial state taken as an\neigenstate of the billiard with high number of spatial har-\nmonics, and after NT= 500 periods of driving ﬁeld with am-\nplitude (b),(e) E0= 0.14 V/cm, and (c),(f) E0= 0.70 V/cm.\nThe spatial distributions for the Sz(and other spin compo-\nnents) have the regular component on the medium scale and\nthe irregular peaked contribution on small scale for both in i-\ntial state (a),(d) and after the driven evolution (b,c,e,f) .\ndynamic regime for our driven evolution after the period\nof initial quasiclassical chaotic-like regime. This can be\nobserved, for example, in the Hilbert space dynamics of\nlevel number shown in Fig.1(a),(b). After the starting\nperiod of diﬀusion in the Hilbert space described by the\ngrowing number of levels involved into dynamics the evo-\nlution is transformedintothe quasi-periodicpattern with\nthestablebehaviorofalltheobservablesatarbitrarylong\ntimes, including the spin textures shown in Fig.3. Thus,\nwe believe that these stable and predictable two-scale\ndensity patterns with both regular and irregular contri-\nbutions can be observed in scanning probe experiments\nperformed in various time frames.\nIn fact, such complicated spatial proﬁles of proba-\nbility density are known in quantum systems like bil-\nliards or waveguides demonstrating chaotic behavior\n[4, 6, 20, 21, 25–27]. Finally, from Fig. 3 one may no-\ntice a developing with the time diﬀerence between the\ndistribution of spin and charge, possibly indicating the\nspin-charge separation in this system. Since the billiard\nis a simple rectangular, we do not see any speciﬁc “scars”\nin the density usually occurring in the chaotic billiards\nwhere chaos appears due to their shape [20, 21, 26, 28].051015202530\nt  [T]00.20.40.60.81spin-charge correlators0.14 V/cmelectric field\nFIG. 4: (Color online) The magnitude of the time-dependent\ncorrelator between the charge density and the Sxcomponent\nof spin density for E0= 0.14 V/cm; αR= 0.5 meVnm (solid\nline),αR= 1.5 meVnm (dashed line), and αR= 5.0 meVnm\n(dashed-dottedline). Thecorrelators fall below0 .2whenSOC\nstrength (5) is high, and the spin-charge separation occurs .\nIn our billiard the chaos is generated by the SOC rather\nthan by the geometry [22]. After turning on the driv-\ning this initially chaotic structure of the spectrum and\nthe eigenstates determines the dynamical variables and\ndensity distributions such as the spin textures in Fig.3.\nOne of the tools for checking the degree of spin-charge\nseparation is the correlators between the charge- and the\nspin density components taken on the spatial grid with\nNpoints and treated as statistical variables. We then\ncalculate the correlators of quantities qandvasrq,v=\nKq,v/wqwv, where\nKq,v=/summationdisplay\niqivi\nN−mqmv (11)\nis the correlation coeﬃcient, mq,vis the mean value,\nandwq,vis the corresponding variance. An example of\nsuch a time-dependence is shown in Fig.4. The ampli-\ntudes of correlators are below 0 .2 when SOC is strong\n(5) mixing several energy levels, and the spin-charge sep-\naration occurs. This result can be viewed as another\nconsequence of the strong SOC and driving producing\nthe time-dependent entanglement of the charge and spin\ndegrees of freedom.\nVI. CONCLUSIONS\nWe have studied the electron dynamics in a quan-\ntum billiard with spin-orbit coupling and driven by a\nmonochromatic electric ﬁeld. It was found that the spa-\ntial and time resolved patterns for probability and spin\ndensities demonstrate the onset of chaotic dynamics with\nqualitativemodiﬁcationsofthe powerspectraandspatial\npatterns. In particular, we have identiﬁed new regimesof\nquantum chaosin this system describedby two-scalespa-\ntial charge and spin density distributions. The onset of6\nspin-chargeseparationeﬀectispredictedbythedynamics\nof the spin and charge density correlators. Our predic-\ntions can be important for the understanding of the cou-\npled spin-charge transport through mesoscopic billiards\ndriven by a uniform electric ﬁeld, where chaos can arise\nfor both spin and charge current observables. The sta-\ntionary density distributions seen in the absence of the\ndriving can be veriﬁed in the tunneling experiments sim-\nilar to those presented in Ref.[24].\nAcknowledgements\nThe authors are grateful to A.F. Sadreev for stim-\nulating discussions. D.V.K. and A.I.M. are supportedby the RFBR Grants No. 13-02-00717a, 13-02-00784a.\nE.S. is supported by of the University of Basque Coun-\ntry UPV/EHU under program UFI 11/55, Spanish MEC\nGrant No. FIS2012-36673-C03-01, and “Grupos Consol-\nidados UPV/EHU del Gobierno Vasco” grant IT-472-10.\nM.D. acknowledges support from DOE grant DE-FG02-\n05ER46204.\n[1] M.C. Gutzwiller, Chaos in Classical and Quantum Me-\nchanics, Springer-Verlag, New York, 1990.\n[2] L.E. Reichl, The Transition to Chaos. Conservative Clas-\nsical Systems and Quantum Manifestations , 2nd Ed.,\nSpringer-Verlag, New York, 2004.\n[3] F. Haake, Quantum Signatures of Chaos , Springer-Verlag\nBerlin Heidelberg, 3rd Ed., 2010.\n[4] H-J. St¨ ockmann, Quantum Chaos: An Introduction ,\nCambridge University Press, 1999.\n[5] G. Casati, B.V. Chirikov, F.M. Izrailev, and J. Ford,\nLect. Notes Phys. 93, 334 (1979)\n[6] K. Nakamura and T. Harayama, Quantum Chaos and\nQuantum Dots , Oxford University Press, New York,\n2004.\n[7] N. Bushong, Y.V. Pershin, and M. Di Ventra, Phys. Rev.\nLett.99, 226802 (2007).\n[8] M. Di Ventra, Electrical transport in nanoscale systems ,\n(Cambridge University Press, Cambridge, 2008).\n[9] T. Dietl, D.D. Awschalom, M. Kaminska, and H. Ohno,\nSpintronics , (Elsevier, Amsterdam), 2008.\n[10]Spin physics in semiconductors , ed. by M.I. Dyakonov,\nSpringer-Verlag Berlin Heidelberg, 2008.\n[11] I. Zˇ uti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[12] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91,\n126405 (2003).\n[13] K.C. Nowack, F.H.L. Koppens, Yu.V. Nazarov, and\nL.M.K. Vandersypen, Science 318, 1430 (2007).\n[14] M. Pioro-Ladriere, T. Obata, Y. Tokura, Y.-S. Shin,\nT.Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Na-\nture Physics 4, 776 (2008).\n[15] V.N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B\n74, 165319 (2006).\n[16] L.S. Levitov and E.I. Rashba, Phys. Rev. B 67, 115324\n(2003).\n[17] D. V. Khomitsky and E. Ya. Sherman, Phys. Rev. B 79,\n245321 (2009).\n[18] D. V. Khomitsky, L.V. Gulyaev, and E. Ya. Sherman,Phys. Rev. B 85, 125312 (2012).\n[19] L. Chotorlishvili, Z. Toklikishvili, A. Komnik, and J. Be-\nrakdar, Phys. Lett. A 377, 69 (2012).\n[20] K.-F. Berggren, A.F. Sadreev, and A.A. Starikov, Phys.\nRev. E66, 016218 (2002).\n[21] K.-F. Berggren, D.N. Maksimov, A.F. Sadreev, R.\nH¨ ohmann, U. Kuhl, and H.-J. St¨ ockmann, Phys. Rev.\nE77, 066209 (2008).\n[22] K.-F. Berggren and T. Ouchterlony, Found. Phys. 31,\n233 (2001). For a detailed analysis of semiclassical eigen-\nstates in Rashba billiards, see A. Csord´ as, J. Cserti, A\nP´ alyi, and U. Z¨ ulicke, Eur. Phys. J. B 54, 189 (2006). For\nthe analysis of spin-orbit coupling eﬀects on level statis-\ntics and electromagnetic response of metal nanoparticles\nsee, for example, L.P. Gor’kov and G.M. ´Eliashberg, Sov.\nPhys. JETP 21, 940 (1965).\n[23] V.Ya. Demikhovskii, F.M. Izrailev, and A.I. Malyshev,\nPhys. Rev. Lett. 88, 154101 (2002); Phys. Rev. E 66,\n036211 (2002); A.I. Malyshev and L.A. Chizhova, J. Exp.\nTheor. Phys. 110, 837 (2010).\n[24] R. Crook, C.G. Smith, A.C. Graham, I. Farrer, H.E.\nBeere, and D.A. Ritchie, Phys. Rev. Lett. 91, 246803\n(2003).\n[25] E.N. Bulgakov and A.F. Sadreev, Phys. Rev. E 70,\n056211 (2004).\n[26] A.T. Ngo, E.H. Kim, and S.E. Ulloa, Phys. Rev. B 84,\n155457 (2011).\n[27] J.D. Urbina, M. Wimmer, D. Bauernfeind, D. Espitia,\n˙I Adagideli, and K. Richter, Phys. Rev. E 87, 042115\n(2013).\n[28] A.F. Sadreev, E.N. Bulgakov, and I. Rotter, Phys. Rev.\nB73, 235342 (2006); E.N. Bulgakov and I. Rotter, Phys.\nRev. E73, 066222 (2006); ˙I. Adagideli, Ph. Jackoud, M.\nScheid, M. Duckheim, D. Loss, and K. Richter, Phys.\nRev. Lett. 105, 246807 (2010); D. Waltner, J. Kuipers,\nPh. Jacqoud, and K. Richter, Phys. Rev. B 85, 024302\n(2012)."    },    {        "title": "2211.02490v1.Derivation_of_Interacting_Two_Qubit_Dynamics_from_Spin_Boson_Model.pdf",        "content": "arXiv:2211.02490v1  [quant-ph]  4 Nov 2022Derivation of Interacting Two-Qubit Dynamics from\nSpin-Boson Model\nHiroaki Matsueda1,2, Yukiya Ide1, and Sadamichi M aekawa3,4\n1Department of Applied Physics, Graduate School of Engineee ring, Tohoku University, Sendai\n980-8579, Japan\n2Center for Science and Innovation in Spintronics, Tohoku Un iversity, Sendai 980-8577, Japan\n3RIKEN Center for Emergent Matter Science, Wako, Saitama 351 -0198, Japan\n4Kavli Institute for Theoretical Physics, University of Chi nese Sciences, Beijing 100190, China\nE-mail: hiroaki.matsueda.c8@tohoku.ac.jp\n(Received July 31, 2022)\nWe derive damping equations of motion for interacting two-s pin states from a spin-boson model\nin order to examine qubit dynamics in quantum computers. On t he basis of the composite operator\nmethod, we develop the Caldeira-Leggett approach for open q uantum systems so that the entangle-\nment dynamics originated from the two-spin correlation can be taken. We demonstrate numerical\nresults for time dependence on the two-spin dynamics. We ﬁnd that the relaxation of the total spin is\ndescribed by a quantum version of the Landau-Lifshitz-Gilb ert equation for magnetic materials. We\nalso ﬁnd that a two-spin composite mode keeps oscillation ev en after the total spin has been fully\nrelaxed. We thus conclude that the two-spin correlation due to the presence of the composite mode\nis stable against dissipation. We consider the mechanism of why the correlation is maintained.\nKEYWORDS: qubit, spin-boson model, relaxation, composite operator, equation of motion,\nCaldeira-Leggett approach\n1. Introduction\nNowadays, exploring quantum technologies such as quantum c omputation, quantum cryptogra-\nphy, and quantum sensing is turning into a realistic goal for current engineering. Ten years have\npassed already after the D-wave machine, a kind of quantum an nealers, was commercially provided,\nand the machine is getting used for various optimization pro blems. Recent ﬂagship research projects\nassociated with quantum technologies aim to construct larg e-scale fault-tolerant universal quantum\ncomputers in the middle of this century. For this purpose, de velopment of noisy intermediate-scale\nquantum (NISQ) computers is an important milestone at the pr esent stage. We believe that future\nsophisticated society will be highly supported from these t echnologies, and thus it is necessary to\npromote basic science behind the technologies. There are ma inly two directions for the advanced re-\nsearch. One is direct treatment of fault tolerance algorith m and implementation as long-term research,\nand the other is deep examination of NISQ itself as short or me dium-term plan. The latter is closely\nrelated to non-equilibrium physics in which the dynamics of our qubit system is highly disturbed\nby the environmental noise and interaction among qubits the mselves when the qubits are massively\nintegrated on the substrate. Since a qubit can be identiﬁed w ith a quantum spin, our target model is\nthe so-called spin-boson model in which the interacting qua ntum spins couple with bosonic degrees\nof freedom. Thus, our interest is to understand operational stability of single spin and entanglement\namong multiple spins in this model. This is because quantum c omputation is realized by sequential\nchange of qubit states with external perturbation as unitar y gates.\nMotivated by the abovementioned consideration, we theoret ically examine the spin dynamics in\n1the spin-boson model. Here, the entanglement control of qub its is a key for various quantum tech-\nnologies, and thus we particularly focus on whether dynamic al behavior of non-local correlation or\nentanglement is stably controlled against dissipation due to the presence of the environment. This\ntype of works was recently done as a toy model for the D-wave ma chine [1]. However, this is a very\nspecial case in which we can successfully integrate out boso nic degrees of freedom in terms of the\nSuzuki-Trotter decomposition. We would like to get versati le techniques for more general cases.\nFor this purpose, we ﬁrst derive damping equations of motion for the total spin from our spin-\nboson model in which two spins interact with each other. We de veloped old approaches such as Feyn-\nman’s inﬂuential functional and Caldeira-Leggett model. W e assume Ohmic spectral distribution of\nenvironmental degrees of freedom and Markovian approximat ion to get a closed equation of motion.\nWe then ﬁnd that the result is equivalent to a quantum version of the Landau-Lifshitz-Gilbert (LLG)\nequation for macroscopic spin precession in magnetic mater ials [2, 3]. Here, the Gilbert constant is\nproportional to the coe ﬃcient of the distribution function. On the basis of the total -spin dynamics, we\nnext focus on the internal dynamics of two spins in order to un derstand the stability of their nonlocal\ncorrelation. The abovementioned theoretical method is als o applied to a composite operator associ-\nated with the correlation. We ﬁnd that the two-spin correlat ion is maintained even after the total spin\nhas been relaxed and the stability of two-spin correlation i s diﬀerent from that of the total-spin dy-\nnamics [3]. In the usual spectroscopy in quantum many-body s ystems, low-lying states are dominated\nby composite spins [4]. We expect that the stability of the dy namics strongly depends on the spatial\nsize of correlated spin cloud. We demonstrate numerical res ults and mention why the stability of the\nnonlocal correlation appears.\n2. Model and Equations of Motion for Interacting Two-Spin Dy namics\nWe consider interacting two qubits (quantum spins) coupled to the bosonic environment. We start\nwith the following Hamiltonian\nH=/summationdisplay\nk,αωkbα†\nkbα\nk+/summationdisplay\nαBα\n0Sα+J/summationdisplay\nαSα\n1Sα\n2+/summationdisplay\nk,α/parenleftBig\nνkbα\nk+ν∗\nkbα†\nk/parenrightBig\nSα, (1)\nwhere we consider two S=1/2 spins, Sα=Sα\n1+Sα\n2(α=1,2,3),bα†\nkandbα\nkare boson operators\nwith mode kand index of angular momentum α,Bα\n0is associated with the energy di ﬀerence of qubit\nstates,ǫ, and transverse ﬁeld, h,/vectorB0=(h,0,ǫ), and Jis antiferromagnetic coupling between spins.\nHere we assume a special form of boson operators with index αso that we can ﬁnd simple damping\nequations for spins. Furthermore, the coupling between qub its is assumed to be of Heisenberg type\nfor simplicity, but this is an ideal situation. We should not e that the coupling actually depends on the\ntype of qubit design. For simplicity we introduce\nAα=Bα\n0+Lα,Lα=/summationdisplay\nk/parenleftBig\nνkbα\nk+ν∗\nkbα†\nk/parenrightBig\n, (2)\nand then the Hamiltonian is simply represented as H=/summationtext\nkωk/vectorb†\nk·/vectorbk+J/vectorS1·/vectorS2+/vectorA·/vectorS.\nThe equations of motion for single spin operators, /vectorS1and/vectorS2, are represented as\n∂\n∂t/vectorS1=J/vectorm−/vectorS1×/vectorA,∂\n∂t/vectorS2=−J/vectorm−/vectorS2×/vectorA, (3)\nwhere the composite spin operator /vectormis deﬁned by\nmα=ǫαβγSγ\n1Sβ\n2, /vectorm=/vectorS2×/vectorS1, (4)\n2and this operator characterizes entanglement between two s pins. To understand the relation between\n/vectormand entanglement, it is useful to remember the deﬁnition of t he single spin /vectorS1. The states,|↑∝angbracketright\nand|↓∝angbracketright, are the eigenstates of Sz\n1in the single-spin case. The transition between these state s are\nrepresented by Sx\n1andSy\n1. Here, the eigenstates of mzare the product and entangled states ( |↑↑∝angbracketright,\n|↓↓∝angbracketright, and|↑↓∝angbracketright± i|↓↑∝angbracketright). Then, mxandmycorrespond to the transition from the product state ( |↑↑∝angbracketrightor\n|↓↓∝angbracketright) to the singlet state, and these transition operators play a crucial role on generating or keeping\nentanglement. Thus, we particularly focus on the relaxatio n dynamics of /vectorm. Note that the equation of\nmotion for the total spin /vectorSdoes not include /vectormbecause/vectorScommutes with the Heisenberg coupling:\n∂\n∂t/vectorS=−/vectorS×/vectorA=−1\n2/parenleftBig/vectorS×/vectorA−/vectorA×/vectorS/parenrightBig\n. (5)\nThus, the alternative treatment of /vectormis important for the examination of internal dynamics betwe en\nspins. The equation of motion for /vectormis given by\n∂\n∂t/vectorm=1\n2J/parenleftBig/vectorS2−/vectorS1/parenrightBig\n−/vectorm×/vectorA=1\n2J/parenleftBig/vectorS2−/vectorS1/parenrightBig\n−1\n2/parenleftBig\n/vectorm×/vectorA−/vectorA×/vectorm/parenrightBig\n. (6)\nNote that we have taken symmetrized procedure in Eqs. (5) and (6) in order to avoid technical di ﬃ-\nculty associated with noncommutativity of quantum operato rs.\nThese equations still contain bosonic operators through /vectorA. Let us remove the bosonic degrees of\nfreedom. For this purpose, the Heisenberg equation of motio n for environmental boson is given by\ni∂\n∂tbα\nk=ωkbα\nk+ν∗\nkSα, (7)\nand the formal solution can be obtained as\nbα\nk(t)=e−iωktbα\nk(0)−iν∗\nk/integraldisplayt\n0dt′e−iωk(t−t′)Sα/parenleftbigt′/parenrightbig. (8)\nWe would like to obtain a closed form of equations of motion fo r/vectorSand/vectorm, and for this purpose we\nsubstitute the bosonic solution into the equations. We assu me the bosonic spectrum as\nJ(ω)=/summationdisplay\nk|νk|2δ(ω−ωk)=ηω, (9)\nwhere this assumption represents the Ohmic process and the c oeﬃcientηplays a central role on the\nrelaxation of spin dynamics.\nBy combining these equations with use of Markovian approxim ation (ωcis the cut-oﬀfrequency\nforJ(ω), and we take it as a large constant), the ﬁnal form of total sp in dynamics is given by\n∂\n∂t/vectorS(t)=/vectorB(t)×/vectorS(t)−ηsinωct\nt/parenleftBig/vectorS(t)×/vectorS(0)−/vectorS(0)×/vectorS(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorS(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorS(t)/parenrightBigg\n, (10)\nwhere the coeﬃcientα(t) is deﬁned by\nα(t)=η/integraldisplayωct\n0dτsinτ\nτ,lim\nωct→∞α(t)=πη\n2, (11)\nand/vectorB(t) is deﬁned by\nBα(t)=Bα\n0+/summationdisplay\nk/parenleftBig\nνke−iωktbα\nk(0)+ν∗\nkeiωktbα†\nk(0)/parenrightBig\n. (12)\n3The result is essentially a quantum version of the LLG equati on with the damping coe ﬃcientα(t). In\nthe right hand side of Eq. (10), the second term represents co rrelation between the initial state and the\nstate at time t. For largeωcvalues, the second term becomes negligible with time. The la st term in\nEq. (10) shows damping, and also produces quantum e ﬀects that do not contain in the classical LLG\ndynamics. The quantum e ﬀects are originated in non-commutativity between /vectorSand∂/vectorS/∂t, and then\nthe magnitude of the expectation value of /vectorSmay not be a conserved quantity. We will brieﬂy discuss\nthis point later.\nOn the other hand, /vectormshows the following dynamics\n∂\n∂t/vectorm(t)=/vectorB(t)×/vectorm(t)+1\n2(J−2ηωc)/parenleftBig/vectorS2(t)−/vectorS1(t)/parenrightBig\n−ηsinωct\nt/parenleftBig\n/vectorm(t)×/vectorS(0)−/vectorS(0)×/vectorm(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorm(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorm(t)/parenrightBigg\n. (13)\nBy substituting /vectorm×/vectorS=/parenleftBig/vectorS1−/vectorS2/parenrightBig\n/2+i/vectormand/vectorS×/vectorm=/parenleftBig/vectorS2−/vectorS1/parenrightBig\n/2+i/vectorminto Eq. (13), we obtain\n∂\n∂t/vectorm(t)=/vectorB(t)×/vectorm(t)−1\n2(J−2ηωc)/parenleftBig\n/vectorm(t)×/vectorS(t)−/vectorS(t)×/vectorm(t)/parenrightBig\n−ηsinωct\nt/parenleftBig\n/vectorm(t)×/vectorS(0)−/vectorS(0)×/vectorm(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorm(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorm(t)/parenrightBigg\n. (14)\nA striking feature of this equation is that the relaxation te rm (the last term in the right hand side)\nturns oﬀwhen/vectorSis in a stationary condition ∂/vectorS/∂t=0. In this case, the operator /vectorSin the second\nterm behaves as a static ﬁeld. The third term represents quan tum correlation between initial state and\nthe state at time t, but this term is negligible as we have already pointed out. F urthermore, decoupling\nof the nature of /vectorSand/vectormis facilitated by taking J=2ηωc. Therefore, the dynamics of /vectormis very\nstable against the relaxation of /vectorS. More precisely, the stationary condition must be represen ted by\n∝angbracketleftψ|∂/vectorS/∂t|ψ∝angbracketright=∂∝angbracketleft/vectorS∝angbracketright/∂t=0 with the initial quantum state |ψ∝angbracketright. Thus, the abovementioed statement\nwould be too strong. We expect slow damping of the composite s pin/vectorm, and the damping behavior of\n/vectormwould be represented by sophisticated treatment of higher- order equation of motion. We thus think\nthat the energy relaxation time T1is determined by the time scale of relaxation of the total spi n∝angbracketleft/vectorS∝angbracketright,\nand the decoherence time T2between two spins is determined by the time scale of relaxati on of the\ncomposite spin∝angbracketleft/vectorm∝angbracketright. In the next section, we analyze these coupled equations of m otion to examine\nthe feature of the dynamics of ∝angbracketleft/vectorS∝angbracketrightand∝angbracketleft/vectorm∝angbracketright.\nIn the previos works associated with inertial spin dynamics in ferromagnets, higher-order terms\nof the LLG equation have been considered [5, 6]. It is an inter esting future work to examine their\nrelationship with the present result.\n3. Numerical Results\nBefore going into numerical details, we brieﬂy examine the q ubit system without bosons in order\nto ﬁnd a guideline for determining the magnitudes of Jandǫ(h=0). In this case, the Hamiltonian\nwithout bosons, H0, is transformed into\nH0=J/vectorS1·/vectorS2+ǫSz=J\n2S(S+1)−3J\n4+ǫSz, (15)\nwhere/vectorS·/vectorS=S(S+1)=2/vectorS1·/vectorS2+3/2 and thus H0can be represented by using the total spin\nS. The singlet is characterized by S=0 and Sz=0, while the triplet is characterized by S=1\nandSz=−1,0,+1. Here, we compare the energy of singlet E(S=0,Sz=0)=−3J/4 with\nthe energy of one of triplets E(S=1,Sz=−1)=J/4−ǫ. Then, E(S=1,Sz=−1) becomes\n4lower than E(S=0,Sz=0) forǫ > J. When we take a parameter range in which the triplet\n(disentangled product state) is stabilized, a viewpoint of classical LLG dynamics would be reasonable\nfor describing the dynamics of /vectorS. In this proceeding, we would like to start with such a simple case,\nand then consider the coupling with bosonic environment. We are interested in a parameter region in\nwhich singlet and triplet states are strongly competing wit h each other, but this is a future work. For\ncomparison, we show that the operator /vectormsatisﬁes the following relation\n/vectorm·/vectorm=3\n8−1\n2/vectorS1·/vectorS2=3\n4−1\n4S(S+1). (16)\nThus the magnitude of this quantity is also characterized by the total spin S. This value for the singlet\nstate with ﬁnite amont of entanglement is larger than that fo r the triplet state. This result also supports\nthat/vectormcharacterizes entanglement between two spins.\nFor numerical simulation, we take J=1,ωc=200, and/vectorB0=(h,0,ǫ)=(0,0,2). We intro-\nduce the initial quantum state as a product (disentangled) s tate, and then consider how the two-spin\ncorrelation or entanglement is generated by the time evolut ion:\n|ψ∝angbracketright∝(a|↑∝angbracketright1+(1−a)|↓∝angbracketright1)⊗(b|↑∝angbracketright2+(1−b)|↓∝angbracketright2)⊗|ϕ∝angbracketright, (17)\nwhere we take a=0.7 and b=0.3, and|ϕ∝angbracketrightis a bosonic part. For this initial state, the expectation\nvalues of/vectorSand/vectormare, respectively, given by\n∝angbracketleft/vectorS∝angbracketright=∝angbracketleftψ|/vectorS|ψ∝angbracketright=(0.72,0,0),∝angbracketleft/vectorm∝angbracketright=∝angbracketleftψ|/vectorm|ψ∝angbracketright=(0,−0.25,0). (18)\nNote that the magnitudes of these vectors depend on the selec tion of the initial quantum state |ψ∝angbracketright.\nWe do not take a full polarized state |↓↓∝angbracketright(or the maximally-entangled singlet state) at t=0, since\n∝angbracketleft/vectorm∝angbracketright(or∝angbracketleft/vectorS∝angbracketright) is zero in this case. Our equations of motion, Eqs. (10) and ( 14), are operator relations,\nnot classical vector equations. Thus, we must take expectat ion values by the state |ψ∝angbracketrightin order to\nintroduce graphical representation. In this process, /vectorS×∂/vectorS/∂tin Eq. (10) and /vectorm×∂/vectorS/∂tin Eq. (14)\nare respectively decomposed into two independent terms:\n∝angbracketleft/vectorS×∂\n∂t/vectorS∝angbracketright∼∝angbracketleft/vectorS∝angbracketright×∂\n∂t∝angbracketleft/vectorS∝angbracketright,∝angbracketleft/vectorm×∂\n∂t/vectorS∝angbracketright∼∝angbracketleft/vectorm∝angbracketright×∂\n∂t∝angbracketleft/vectorS∝angbracketright. (19)\nThe dynamics of∝angbracketleft/vectorS∝angbracketrightafter this approximation becomes equivalent to the classic al LLG equation,\nexcept that∝angbracketleft/vectorB∝angbracketrightstill contains information of bosons. Here we neglect time d ependence on /vectorB(t) that\nappears as a result of the second term in Eq. (12). In this case , we can solve the equation of motion,\nand we ﬁnd that the relaxation time scale T1is proportional to (1 +π2η2|∝angbracketleft/vectorS∝angbracketright|2)/2πηǫ|∝angbracketleft/vectorS∝angbracketright|. Unfor-\ntunately, quantum e ﬀects originated from non-commutativity between /vectorSand∂/vectorS/∂tand polaronic\neﬀects are lost in this approximation, and then |∝angbracketleft/vectorS∝angbracketright|is kept. Thus, we suppose that the realistic relax-\nation time scale may change. In the present approximation, t he decoherence time T2becomes inﬁnity\ndue to the stability of the dynamics of /vectorm. The precise estimation of T2is an important future work,\nbut we can say T2>T1even within the present simple analysis.\nWe demonstrate time evolution of ∝angbracketleft/vectorS∝angbracketrightand∝angbracketleft/vectorm∝angbracketrightforη=0.008 (2ηωc=3.2>J=1) in Fig. 1.\nWe ﬁnd that∝angbracketleft/vectorS∝angbracketrightdecays into the direction of −∝angbracketleft/vectorB∝angbracketright. This feature is consistent with the classical LLG\ndynamics. As we have already discussed, the length of ∝angbracketleft/vectorS∝angbracketright, 0.72, is conserved in the present approx-\nimation, and∝angbracketleft/vectorS∝angbracketrightdoes not become (0 ,0,−1) even after the long time. We particularly focus on the\nentanglement dynamics represented by ∝angbracketleft/vectorm∝angbracketright. In contrast to∝angbracketleft/vectorS∝angbracketright, the coherent oscillation of ∝angbracketleft/vectorm∝angbracketrightis main-\ntained even after∝angbracketleft/vectorS∝angbracketrighthas been relaxed to the stationary point. The coherent oscil lation corresponds to\ncontinuous spin ﬂip ( |↑↓∝angbracketright↔|↓↑∝angbracketright ) between two spins. We ﬁnd that the phase di ﬀerence between∝angbracketleftmx∝angbracketright\nand∝angbracketleftmy∝angbracketrightisπ/2. As we have already mentioned, the equation of motion for /vectormdoes not contain the\ndamping term if ∂S/∂tbecomes zero. This is the origin of the stable oscillation of ∝angbracketleft/vectorm∝angbracketright. Therefore, the\ntwo-spin dynamics is essentially di ﬀerent from total-spin dynamics.\n5Fig. 1. Spin dynamics for η=0.008. (a)∝angbracketleft/vectorS∝angbracketright, (b)/angbracketleftbig/vectorm/angbracketrightbig, and (c) Graphical representation of (a) and (b). In\nﬁgure (c), the blue curve represents ∝angbracketleft/vectorS∝angbracketright, and the red curve represents/angbracketleftbig/vectorm/angbracketrightbig.\n4. Concluding Remarks\nWe derived the spin dynamics in the spin-boson model in order to examine the entanglement\ncontrol of qubits against dissipation due to the presence of the environment. For the total spin, we\nfound that the result is consistent with the LLG equation for macroscopic spin precession in magnetic\nmaterials although the result also contains some quantum e ﬀects. Here, the Gilbert constant is propor-\ntional to the coeﬃcient of the distribution function of the bosonic degrees of freedom. However, the\nentanglement dynamics originated from a composite spin sho ws diﬀerent behavior. We numerically\nshowed the relaxation dynamics of the total spin and the stab ility of the entanglement dynamics. In\nthe present approximation, the decoherence time T2is inﬁnity, and more precise treatment based on\nthe higher-order equation of motion is an interesting futur e work.\nH.M. is supported by JPSJ KAKENHI (Nos. 21K03380, 21H04446, 21H03455) from MEXT\nJapan and CSIS, Tohoku University, Japan. S.M is supported b y JST CREST Grant (Nos. JPMJCR19J4,\nJPMJCR1874, and JPMJCR20C1) and JSPJ KAKENHI (Nos. 17H0292 7 and 20H01865) from MEXT,\nJapan.\nReferences\n[1] Y . Bando and H. Nishimori, Phys. Rev. A 104, 022607 (2021).\n[2] J. Anders, C. R. J. Sait, and S. A. R. Horsley, New. J. Phys. 24, 033020 (2022).\n[3] H. Matsueda, Y . Ide, and S. Maekawa, in preparation.\n[4] T. Otaki, Y . Yahagi, and H. Matsueda, J. Phys. Soc. Jpn. 86, 084709 (2017).\n[5] K. Neeraj et al., Nat. Phys. 17, 245 (2021).\n[6] M.-C. Ciornei, J. M. Rubi, and J.-E. Wegrowe, Phys. Rev. B 83, 020410(R) (2011).\n6"    },    {        "title": "1611.07964v2.Dynamical_Time_Reversal_Symmetry_Breaking_and_Photo_Induced_Chiral_Spin_Liquids_in_Frustrated_Mott_Insulators.pdf",        "content": "Dynamical Time-Reversal Symmetry Breaking and\nPhoto-Induced Chiral Spin Liquids in Frustrated Mott Insulators\nMartin Claassen,1,2Hong-Chen Jiang,2Brian Moritz,2and Thomas P. Devereaux2,3,∗\n1Department of Applied Physics, Stanford University, CA 94305, USA\n2Stanford Institute for Materials and Energy Sciences,\nSLAC & Stanford University, CA 94025, USA\n3Geballe Laboratory for Advanced Materials,\nStanford University, Stanford, CA 94305, USA\n(Dated: October 13, 2017)\n1arXiv:1611.07964v2  [cond-mat.str-el]  11 Oct 2017Abstract\nThe search for quantum spin liquids in frustrated quantum magnets recently has enjoyed a surge\nof interest, with various candidate materials under intense scrutiny. However, an experimental\nconﬁrmation of a gapped topological spin liquid remains an open question. Here, we show that\ncircularly-polarized light can provide a novel knob to drive frustrated Mott insulators into a chiral\nspin liquid (CSL), realizing an elusive quantum spin liquid with topological order. We ﬁnd that\nthe dynamics of a driven Kagome Mott insulator is well-captured by an eﬀective Floquet spin\nmodel, with heating strongly suppressed, inducing a scalar spin chirality Si·(Sj×Sk)term which\ndynamically breaks time-reversal while preserving SU(2) spin symmetry. We ﬁngerprint the tran-\nsient phase diagram and ﬁnd a stable photo-induced CSL near the equilibrium state. The results\npresented suggest employing dynamical symmetry breaking to engineer quantum spin liquids and\naccess elusive phase transitions that are not readily accessible in equilibrium.\nControl of quantum materials out of equilibrium represents one of the grand challenges of\nmodern condensed matter physics. While an understanding of general non-equilibrium set-\ntingsbeyondheatingandthermalizationisstillinitsinfancy, aloopholeconcernsconsidering\ninstead the transient quantum states of quasi-periodic perturbations such as wide-envelope\nlaser pulses. Here, much of the intuition and language of equilibrium survives in a distinctly\nnon-equilibrium setting within the framework of Floquet theory. While recently enjoying\nmuch attention and experimental success in the manipulation of single-particle spectra [1–4]\nand band topology or short-range entangled topological states [5–7], a natural extension\nregards pumping of strongly-correlated systems. Here, the essence of Floquet physics lies\nnot merely in imbuing one- [1] and two-particle [2] responses with the pump frequency as\nan additional energy scale, but in reshaping the underlying Hamiltonian to stabilize novel\nphases of matter that might be inaccessible in equilibrium.\nIndeed, initial investigations suggest that the notion of eﬀective low-energy physics per-\nsists in certain high-frequency regimes of time-periodic perturbations, leading for instance\nto enhancement of correlated hopping [8, 9], strong-ﬁeld sign reversal of nearest-neighbor\nHeisenberg exchange in a 1D magnet [10, 11], or enhancement of Cooper-pair formation\n[12–14]. Similar ideas are being pursued in the ﬁeld of ultracold atoms to simulate artiﬁcial\n∗Author to whom correspondence should be addressed to: M. C. (mclaassen@stanford.edu) or T. P. D.\n(tpd@stanford.edu)\n2gauge ﬁelds, to dynamically realize topological band structures [15] or even propose a frac-\ntional quantum Hall eﬀect in optical lattices [16, 17]. At the same time, recent advances\nin Floquet thermodynamics indicate that, while driven non-integrable closed systems are in\nprinciple expected to heat up to inﬁnite temperature [18, 19], heating can be exponentially\nslow on pre-thermalized time scales [20–25] or altogether avoided via many-body localiza-\ntion [26–28] or dissipation [29–31]. An ideal condensed-matter realization hence entails a\ncharge gap to limit absorption, as well as a delicate balance of competing phases, such that\ntime-dependent perturbations and dynamical symmetry breaking can be expected to have\nan outsized eﬀect and phase boundaries can be reached on pre-thermalized time scales with\nmoderate eﬀort.\nFrustrated quantum magnets [32] are prime candidates for such ideas. Strong local\nCoulomb repulsion between electrons freezes out the charge degrees of freedom, whereas\nthe spin degrees of freedom are geometrically obstructed from ordering, hosting a delicate\ncompetition of conventionally-ordered phases as well as quantum spin liquids (QSLs) with\nlong-range entanglement and exotic excitations [33–40]. The chiral spin liquid (CSL) con-\nstitutes one of the earliest proposals of a topologically-ordered QSL; it breaks time-reversal\nsymmetry (TRS) and parity, while preserving SU(2) spin symmetry, and can be regarded\nas a bosonic ν=½fractional quantum Hall state of spins with zero net magnetization and\ngapped semion excitations [41–44]. While an unlikely ground state in unperturbed micro-\nscopic models, recently the CSL was found to be a competing state [45–52], in particular\nafter explicit breaking of TRS and parity [45–49]. However, TRS breaking in experiment is\nrealized canonically via external magnetic ﬁelds, necessarily entailing a Zeeman shift as the\ndominant contribution, which breaks SU(2) symmetry and disfavors CSLs [45].\nHere, we show that pumping a Mott insulator with circularly-polarized light below the\nMott gap can dynamically break TRS without breaking of SU(2) or translation symmetry,\nproviding a new knob to drive a frustrated quantum magnet into a CSL. Starting from a\nprototypical Hubbard model, the key questions posed by this work are three-fold: First, how\ndoes optically-induced time-reversal symmetry breaking manifest itself in a Mott insulator;\nsecond, can the ensuing eﬀective Floquet spin model support a transient CSL and what are\nits signatures; and ﬁnally, does such an eﬀective Floquet steady state description capture\nthe many-body time evolution of an optically-driven Hubbard model? In the following, we\nanswer all three questions aﬃrmatively.\n3RESULTS\nFloquet-Hubbard Model.— Our focus lies on Kagome antiferromagnets, which have re-\ncently garnered much attention due to novel candidate materials herbertsmithite, kapellasite\nand others [33], with putative spin-liquid behavior at low temperatures. Experiments [53]\nand ﬁrst-principles calculations [54,55] indicate that the ground state and low-energy excita-\ntion spectra of these materials are well-captured by antiferromagnetic Heisenberg exchange\nbetweend9spins localized on Cu [33]. However, as photons couple to charge, a microscopic\nmodellingofthelight-matterinteractioninprinciplemustaccountforthemulti-orbitalstruc-\nture at higher energies [56], above the ∼2 eV charge gap [57]. Here, we take a phenomeno-\nlogical approach, and, as an eﬀective starting point that captures the essential physics but\nwithout pretense of a direct materials connection start from a driven single-orbital Hubbard\nmodel at half ﬁlling\nˆH(t) =−th/summationdisplay\n/angbracketleftij/angbracketrightσeie\n~rij·A(t)ˆc†\niσˆcjσ+U/summationdisplay\niˆni↑ˆni↓ (1)\nHere,th,U,edenote nearest-neighbor hopping, Coulomb interaction and electron charge, rij\ndenotes vectors between sites i,j, and A(t) =A(t)[cos(Ωt),sin(Ωt)]Tmodels a circularly-\npolarized pump beam with wide pulse envelope A(t), coupling to electrons via Peierls substi-\ntution. Comparisonofnearest-neighborexchange J≈4t2\nh/Uwithﬁrst-principlespredictions\nfor herbertsmithite [55] suggests U/thof up to 40 due to the exceedingly narrow width of\nCud-orbital derived bands.\nIfA(t)varies slowly with respect to the pump period, then the Hamiltonian becomes\napproximately periodic under a translation ˆH(t+ 2π/Ω) = ˆH(t). Floquet theory then\ndictates that the behavior near the pump plateau is completely determined via many-body\neigenstates of the form |Ψn(t)/angbracketright=e−i/epsilon1nt/summationtext\nmeimΩt|Φm/angbracketrightwith/epsilon1nthe Floquet quasi-energy,\nwherethe|Φm/angbracketrightconvenientlyfollowaseigenstatesofthestaticFloquet-HubbardHamiltonian\nˆH=−th/summationdisplay\n/angbracketleftij/angbracketrightσ\nmm/primeJm−m/prime(A)ei(m−m/prime) argrijˆc†\niσˆcjσ⊗|m/angbracketright/angbracketleftm/prime|\n+U/summationdisplay\niˆni↑ˆni↓⊗1−/summationdisplay\nmmΩ1⊗|m/angbracketright/angbracketleftm| (2)\n4whereAdenotes the dimensionless ﬁeld strength at the pump plateau, such that A(t)≈\nA~/(ea0)witha0the nearest-neighbor distance, m∈Zis the Floquet index, and Jm(·)\ndenotes the Bessel function of the ﬁrst kind [see Methods]. Note that the apparent Hilbert\nspace expansion is merely a gauge redundancy of Floquet theory, as eigenstates with energy\n/epsilon1n+mΩidentify with the same physical state ∀m.\nFloquet Chiral Spin Model.— Physically, Eq. (2) describes photon-assisted hopping in\nthe presence of interactions, where electrons can enlist mphotons to hop at a reduced energy\ncostU−mΩof doubly-occupying a site. Deep in the Mott phase the formation of local\nmoments persists out of equilibrium as long as the pump remains oﬀ resonance and red-\ndetuned from the charge gap. However, photon-assisted hopping reduces the energy cost\nof virtual exchange, pushing the system closer to the Mott transition and enlarging the\nrange of virtual hopping paths that provide non-negligible contributions to longer-ranged\nexchange or multi-spin processes. Second, electrons acquire gauge-invariant phases when\nhopping around loops on the lattice for circular polarization. Crucially, and in contrast to\nan external magnetic ﬁeld, an optical pump precludes a Zeeman shift, retaining the SU(2)\nsymmetry that is shared by CSL ground states. Symmetry considerations dictate that a\nmanifestation of TRS breaking must to lowest-order necessarily involve a photo-induced\nscalar spin chirality χijkterm, with:\nˆHspin=/summationdisplay\nijJijSi·Sj+/summationdisplay\nijkχijkSi·(Sj×Sk) (3)\nThis Floquet Chiral Spin Hamiltonian is the central focus of the paper; to derive it\nmicroscopicallyfromthedrivenKagome-Hubbardmodel(1), itisinstructivetoﬁrstconsider\nthe high-frequency limit Ω/greatermuchU,th. Here, circularly-polarized pumping induces complex\nnearest-neighbor hoppings ˜t=th(1−A2/4) +i(√\n3/4)t2\nhA2/Ωas well as purely-complex\nnext-nearest-neighbor hoppings ˜t/prime=−i(√\n3/4)t2\nhA2/Ω, analogous to a staggered magnetic\nﬂux pattern in the unit cell [see Supplementary Note 1]. To third order in ˜t,˜t/prime, a spin\ndescription then includes scalar spin chirality contributions, with χ= 9√\n3t4\nhA2/2U2Ωof\nequal handedness for both equilateral triangles per unit cell, as depicted in Fig. 1(a), and\nsix isosceles triangles of opposite handedness with χ/prime=χ/3, such that the total chiral\ncouplings in the unit cell sum to zero.\nNowconsidersub-gappumping Ω<U. StartingfromEq. (2), amicroscopicderivationof\n5the Floquet spin Hamiltonian proceeds via quasi-degenerate perturbation theory, where care\nmust be taken to simultaneously integrate out m,0Floquet states and many-body states\nwith doubly-occupied sites [see Supplementary Note 2]. Fig. 1(b) and (c) depict relevant\nvirtual processes, involving simultaneous hopping of electrons and absorption of mphotons\nwith intermediate energy cost U−mΩ. To second order in virtual hopping, two-site ex-\nchange processes [Fig. 1(b)] are phase-agnostic and solely renormalize the nearest-neighbor\nHeisenberg exchange ˜J= 4/summationtext\nm|Jm(A)|2t2\nh/(U−mΩ)[8, 10]. While every process con-\ntributes to Heisenberg exchange, a scalar spin chirality contribution appears for multi-hop\nprocesses that enclose an area. Naïvely, to third order, an electron could simply circum-\nnavigate the elementary triangles of the Kagome lattice; however, these processes interfere\ndestructively and cancel exactly to all orders in Aeven though time-reversal symmetry is\nbroken, and in contrast to an external magnetic ﬁeld [see Supplementary Note 2]. This is\nconsistent with results on the resonant A 2gRaman response of Mott insulators [58], that\nconnect to the A→0,m= 1limit. Instead, time-reversal symmetry breaking ﬁrst manifests\nitself to fourth order in virtual hopping. Here, processes [Fig. 1(c)] can either encompass\nan elementary triangle, or virtually move an electron back and forth two legs of a hexagon,\ninducing scalar spin chirality contributions as shown in Fig. 1(a), with\nχ= 3/summationdisplay\nm/bracketleftBig\nsin/bracketleftBig2π(m1−m2+m3)\n3/bracketrightBig\nΛ(1)\nm−sin/bracketleftBig\n2πm2\n3/bracketrightBig\nΛ(2)\nm/bracketrightBig\n(4)\nχ/prime=/summationdisplay\nm/bracketleftBig\nsin/bracketleftBigπ(m1−m2+m3)\n3/bracketrightBig\nΛ(1)\nm−sin/bracketleftBig\nπm2\n3/bracketrightBig\nΛ(2)\nm/bracketrightBig\n(5)\nwhere m={m1,m2,m3}are Floquet indices, and\nΛ(1)\nm=8t4\nhJm1(A)Jm2−m1(A)Jm2−m3(A)Jm3(A)\n/producttext3\ni=1(U−miΩ)(6)\nΛ(2)\nm= 2(1−δm2)U−m2Ω\nm2Ω(−1)m1−m3cos2/bracketleftBig\nm2π\n2/bracketrightBig\nΛ(1)\nm (7)\nHere, Λ(1)\nmandΛ(2)\nmparameterize fourth-order virtual hopping processes for which the second\nintermediate virtual state retains a single double-occupied site or returns to local half ﬁll-\ning (albeit with non-zero Floquet index), respectively. Furthermore, next-nearest-neighbor\n6Heisenberg exchange\nJ/prime=/summationdisplay\nm/bracketleftBigg\nΓm−cos/bracketleftBigπ(m1−m2+m3)\n3/bracketrightBigΛ(1)\nm\n2+ cos/bracketleftBig\nπm2\n3/bracketrightBigΛ(2)\nm\n2/bracketrightBigg\nJ3=/summationdisplay\nm/bracketleftBigg\nΓm−Λ(1)\nm\n2+Λ(2)\nm\n2/bracketrightBigg\n(8)\nand corrections to nearest-neighbor Heisenberg exchange\nJ=˜J+/summationdisplay\nm/braceleftBig/bracketleftBig\n2 + cos/bracketleftBigπ(m1−m3)\n3/bracketrightBig\n+ cos/bracketleftBig2π(m1−m3)\n3/bracketrightBig\n+\n+ cos/bracketleftBigπ(m1−m2+m3)\n3/bracketrightBig\n+1\n2cos/bracketleftBig2π(m1−m2+m3)\n3/bracketrightBig/bracketrightBig\nΛ(1)\nm\n−/bracketleftBig\n1 + 2 cos [πm2] + cos/bracketleftBig\nπm2\n3/bracketrightBig\n+1\n2cos/bracketleftBig\n2πm2\n3/bracketrightBig/bracketrightBig\nΛ(2)\nm−7Γm/bracerightBig\n(9)\nappear at the same order, with\nΓm=/bracketleftBiggδm3,0\nU−m1Ω+δm3,0\nU−m2Ω/bracketrightBigg2/productdisplay\ni=12t2\nh[Jmi(A)]2\n(U−miΩ)(10)\nparameterizing a two-fold virtual nearest-neighbor exchange process.\nSteady-State Phase Diagram.— Having established the eﬀective steady-state physics\nfor the duration of the pump pulse, the next question concerns whether the photo-induced\nFloquet spin model [Eq. 3] can indeed stabilize a CSL. Consider its parameter space as a\nfunction of A,Ω, depicted in Fig. 2 for U= 20th. Adiabatic ramping up of the circularly-\npolarized pump then corresponds to horizontal trajectories with ﬁxed Ω. Fig. 2 (a) and (b)\nshowthatTRS-breakingscalarspinchiralitiesdevelopwithincreasingﬁeldstrength,whereas\nthe eﬀect on longer-ranged Heisenberg exchange [(c) and (d)] is comparatively weak. This\nimmediately suggests that circularly-polarized pumping grants a handle to change the un-\nderlying many-body state. Close to the one-photon resonance ( Ω =U), chiral contributions\nare staggered between elementary and isosceles triangles [Fig. 1(a)], whereas χ/primechanges sign\nwhen Ωapproaches a two-photon resonance ( Ω =U/2).\nWe analyze the steady state phase diagram using exact diagonalization of the Floquet\nspin model [Eq. 3], parameterized by pump strength and frequency. In equilibrium ( A= 0),\nthe ground state of Eq. (3) is gapped and TRS invariant. Absence of conventional spin order\nis evidenced by a rapid decay of spin-spin and chiral-chiral correlation functions on a 36-site\n7cluster [Fig. 3(a)], consistent with density-matrix renormalization group simulations that\nﬁnd a gapped Z2QSL [59–62]. We adopt this view for the thermodynamic limit, but note\nthat the ground state degeneracy of a Z2QSL remains inaccessible in exact diagonalization\nof ﬁnite-size clusters [see Methods]. Upon pumping ( A,0), the spin correlator displays\nno propensity for ordering; however, chiral correlations develop smoothly [Fig. 3(a)]. Im-\nportantly, a two-fold ground state quasi-degeneracy develops continuously, with a gap to\nmany-body excitations, indicative of a CSL. To track the phase boundary as a function of\nA,Ω, we determine the parameter space region within which the ground state degeneracy\nas well as the gap ∆CSLto the many-body excitation manifold above the CSL survives in-\nsertion of a ﬂux quantum through the torus [see Methods]. As shown in Fig. 3(b), a robust\nphoto-induced CSL develops already for weak A, with excited states well-separated in en-\nergy. Finally, a proper veriﬁcation of the CSL necessitates characterizing its ground state\ntopological order. We therefore ﬁngerprint the photo-induced phase by determining a basis\nof minimally-entangled states from combinations of the two degenerate ground states |ψ1,2/angbracketright,\nminimizing the Renyi entropy for their reduced density matrices in two distinct bipartitions\n[Fig. 3(d)] on a 36-site torus [see Methods]. C6symmetry allows extraction of both modular\nU,Smatrices [63–65] that encode self- and mutual-braiding statistics of the elementary\nexcitations. We ﬁnd that the photo-induced CSL corresponds uniquely to a ν=1\n2bosonic\nfractional quantum Hall state, with matching modular matrices [39]\nU=e−i2π\n24×1.02\n1 0\n0 0.99i\n,S=1√\n2\n1.00 1.00\n1.00−1.02\n (11)\nTime Evolution.— Having established a photo-induced CSL for the Floquet spin model,\nthe ﬁnal question concerns whether this eﬀective spin model qualitatively captures the time\nevolution of the driven Hubbard model [Eq. 1].\nTo this end, we consider a circularly-polarized optical pump pulse with a slow sinu-\nsoidal ramp-up and a wide pump plateau [Fig. 4(a)], and simulate the exact many-body\ndynamics of driven 12-site U= 30Kagome Hubbard clusters for long times t≤1000t−1\nh.\nConceptually, the transient state can then be thought of as dynamically following the in-\nstantaneous Floquet eigenstate/vextendsingle/vextendsingle/vextendsingleΨ(τ,¯Tslow)/angbracketrightBig\n≈e−i/epsilon1(¯Tslow)τ/summationtext\nmeimΩτ/vextendsingle/vextendsingle/vextendsingleΦ(¯Tslow)/angbracketrightBig\n, with the time\nvariablet“separating”intofast( τ)andslow( ¯Tslow)movingcomponents. Reachingthepump\n8plateau, thetime-evolvedstatewillneverthelessretainaﬁnitequasi-energyspread ρ(/epsilon1α)with\n|Ψ(t)/angbracketright=/summationtext\nαρ(/epsilon1α)e−i/epsilon1αt/summationtext\nmeimΩt|Φα(t)/angbracketright(withαindexing the Floquet eigenstates). While\ndephasing of these constituent Floquet eigenstates should ultimately thermalize the system\nto inﬁnite temperature, the system nevertheless matches the eﬀective spin dynamics de-\nscribed by Eq. 3, and barely absorbs energy on the broad “pre-thermalized” time scales of\ninterest, as we show below.\nFirst, to compare to the Floquet spin description, we focus on time-dependent scalar\nspin chirality expectation values χijk(t) =/angbracketleftSi·(Sj×Sk)/angbracketright(Si= ˆc†\niσ/vectorsσσ/primeˆciσ/prime) on elementary\ntriangles of the Kagome cluster [Fig. 4(b)]. Vanishing in equilibrium due to TRS, the\npump-period average of χijk(t)should saturate to its Floquet expectation value at the pump\nplateau. Fig. 4(c) compares χijk(t), time-averaged over the pump plateau, to corresponding\nstaticχijkexpectation values of the Floquet spin model (3) ground state. The latter follows\nfrom choosing χ,χ/prime,J,J/prime,J3via Eqns. (4)-(10), with A,Ωthe pump parameters of the\nHubbard time evolution. Intriguingly, the electronic time evolution is in excellent qualitative\nagreement with predictions for the Floquet spin model, even when driven close to the Mott\ntransition.\nQuantitative discrepancies predominantly originate from deviations of the local moment\n/angbracketleft(Sz)2/angbracketright<1/4; additionally, the 12-site cluster with periodic boundary conditions permits\nweak ring exchange contributions from loops of virtual hopping around the cluster. Impor-\ntantly, the transient increase in double occupancies is not an indication of heating – instead,\nthis follows from a reduction of the eﬀective Uof the transient Floquet-Hubbard Hamilto-\nnian, a consequence of the photo-assisted hopping processes depicted in Fig. 1 (b) and (c).\nQuantitative diﬀerences between spin and fermionic observables are therefore analogous to\ndiﬀerences between canonical spin and fermionic descriptions of equilibrium quantum mag-\nnets for a ﬁnite Hubbard- U.\nTo analyze this in detail, we focus on pumping the system across the charge resonance\nwith the upper Hubbard band, where the photo-induced scalar spin chirality contribution is\nexpected to be largest. Figs. 5(a)-(c) show the period-averaged double occupancy /angbracketleftˆn↑ˆn↓/angbracketrightas\na function of pump strength and detuning from the charge resonance ≈U−5.5th. Upon res-\nonant charge excitation, the system heats up rapidly and the double occupancy approaches\nits inﬁnite-temperature limit /angbracketleftˆn↑ˆn↓/angbracketright→1/4. Importantly, this entails that thermalization at\nlong times is independent of the pump strength A.\n9Conversely, in the oﬀ-resonant regime, one observes a pump-strength dependent satura-\ntion of the double occupancy. Here, proper heating is strongly suppressed and the system\ninstead realizes the eﬀective Floquet chiral quantum magnet with a transient reduction of\nthe Hubbard interaction U. To verify that the driven steady state indeed follows the ground\nstate of the eﬀective Floquet Hamiltonian adiabatically, consider a period-shifted Floquet\n“ﬁdelity measure” F=|/angbracketleftΨ(t+T)|Ψ(t)/angbracketright|, whereT= 2π/Ωis the pump period. At the\npump plateau with discrete time translation symmetry, Fis time-independent and quan-\ntiﬁes the Floquet quasi-energy spread of the transient steady state [Fig. 5(d)]. For a pure\nFloquet state, 1−F→ 0, suggesting that the driven state below resonance adiabatically\nfollows a Floquet eigenstate, whereas adiabaticity is lost when crossing the absorption edge.\nTo distinguish residual heating on these pre-thermalized time scales of interest [24, 25]\nfrom a transient increase in energy in the chiral quantum magnet due to modulation of the\nHamiltonian, consider the period-averaged stroboscopic energy operatorˆ/angbracketleftE/angbracketright[see Methods].\nOn the pump plateau, both the double occupancy andˆ/angbracketleftE/angbracketrightsaturate to their pre-thermalized\nsteady state expectation values; however, a minuscule residual gradient over thousands of\npump cycles remains. To good approximation for the time scales considered here, we can\nlinearize the energy on the pump plateauˆ/angbracketleftE/angbracketright(t)≈E0+t∆E, and extract the heating rate\n∆Efrom simulations. Fig. 5(e) depicts the absorbed energy per pump cycle on the pump\nplateau, asafunctionofpumpstrengthanddetuningfromtheabsorptionedge. Remarkably,\nresidual heating is largely suppressed close to resonance, with an absorbed energy on the\norder of 10−6thper pump cycle. Naïvely, this extraordinary meta-stability suggests that it\ncould take on the order of tens of thousands of pump cycles for heating to dominate the\ndynamics, absorbing a total energy ∼Jthe exchange coupling.\nA more microscopic analysis of photo-excitation for realistic materials will likely lead\nto a less optimistic upper bound on the time scales of interest. First, a materials-speciﬁc\nmodelling of electron-photon coupling and multi-band eﬀects will modify the eﬀective photo-\ninduced spin Hamiltonian, albeit necessarily retaining the salient symmetry properties and\nscalar spin chirality contributions that stabilize the CSL. Second, an intriguing follow-up\nquestion regards the role of coupling to - and heating of - the lattice. While magneto-elastic\ncoupling to phonons is weak in most materials and the optical frequencies under consider-\nation are far from resonance with infrared-active phonon modes, electron phonon coupling\nwill nevertheless indirectly heat the lattice due to Raman-assisted hopping processes. Con-\n10versely, theseparationoftimescalesforelectronsandphononssuggeststhatthephononbath\ncould similarly play the role of a dissipative channel, eﬀectively “cooling” the electronic sys-\ntem. While initial investigations have already studied the case of free or weakly-interacting\nelectrons [29–31], a proper understanding of the conﬂuence of strong interactions, external\ndrive and dissipation remains an interesting topic for future study.\nCONCLUSIONS\nIn summary, we have shown that pumping a frustrated Mott insulator with circularly-\npolarized light can dynamically break TRS while preserving SU(2) symmetry of the under-\nlying spin system, by augmenting its eﬀective dynamics with a transient scalar spin chirality\nterm. Remarkably, on the Kagome lattice this eﬀective Floquet spin model was found to\nstabilize a transient CSL in a broad parameter regime. Our results suggest that wide-pulse\noptical perturbations can provide an intriguing knob to tune the low-energy physics of frus-\ntrated quantum magnets, shedding light on novel regions of their phase diagram hithero\nunexplored.\nACKNOWLEDGMENTS\nWe acknowledge support from the U. S. Department of Energy, Oﬃce of Basic Energy\nSciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-\n76SF00515. Computational resources were provided by the National Energy Research Sci-\nentiﬁc Computing Center supported by the Department of Energy, Oﬃce of Science, under\nContract No. DE- AC02-05CH11231.\nAUTHOR CONTRIBUTIONS\nM. C. conceived the project, developed the theoretical description and performed the\nnumerical simulations. M. C., H.-C. J. and B. M. analyzed the numerical results. The\nmanuscript was written by M. C. with input from all authors. T. P. D. supervised the\nproject.\n11COMPETING FINANCIAL INTERESTS\nThe authors declare no competing ﬁnancial interests.\nMETHODS\nFloquet Theory\nConsideragenerictime-dependentmany-bodyHamiltonianwithdiscretetime-translation\ninvariance ˆH(t) = ˆH(t+ 2π/Ω). Instead of solving the many-body time evolution, one\ncan reexpress the time-dependent Schrödinger equation in a Floquet eigenbasis |Φα(t)/angbracketright=\ne−i/epsilon1αt/summationtext\nmeimΩt|uα,m/angbracketright, whereαindexes the basis wave function and /epsilon1αis its respective Flo-\nquet quasi-energy. Then, determination of the time-dependent eigenstates of the driven\nsystem reduces to ﬁnding the time-independent eigenstates of the Floquet Hamiltonian\nˆHﬂoquet =/summationdisplay\nmm/prime/bracketleftBigˆHm−m/prime−mΩδm,m/prime/bracketrightBig\n|m/angbracketright/angbracketleftm/prime| (12)\nwhere ˆHm−m/primeare the Fourier expansion coeﬃcients of ˆH(t). Taking ˆH(t)as the driven\nKagome-Hubbard model of Eq. (1) in the main text straightforwardly recovers the Floquet-\nHubbard Hamiltonian [Eq. (2)]. Here, the dimensionless pump strength Athat enters\nvia Peierls substitution, relates to the electric ﬁeld EasE= Ω/(ea0), withethe electron\ncharge and a0the nearest-neighbor bond distance. While realistic estimates for materials\nwould necessarily entail signiﬁcant contributions from multi-orbital eﬀects and local dipole\ntransitions, all of which are not captured within the single-orbital Hubbard model, a naïve\nestimatefromsolelyPeierlssubstitutionforasingle-orbitalapproximationofherbertsmithite\nyieldsA= 0.25...0.5forE...100...200meVÅ−1for a 900nm near-infrared pump.\nNumerical Simulations\nTo characterize the photo-induced chiral spin liquid state, we performed exact diagonal-\nization calculations of the Floquet chiral spin Hamiltonian, described in Eq. (3). Due to\nthree-spin interactions and longer-ranged exchange interactions, the sparsity of the resulting\nHamiltonian matrix is over an order of magnitude lower than for a nearest-neighbor Heisen-\n12berg antiferromagnet. Spin and chiral correlation functions as well as minimally-entangled\nstates were calculated for a 36-site cluster with periodic boundary conditions, spanned by\nvectors R1= 4a1−2a2,R2=−2a1+ 4a2, where a1,a2are the lattice vectors. This choice\nretains the rotational symmetry of the Kagome lattice, facilitating extraction of the mod-\nular matrices. Previous exact diagonalization studies have simulated the 36-site cluster for\na purely nearest-neighbor Heisenberg model [66, 67]. While density-matrix renormalization\ngroup studies indicate that such a model is likely to stabilize a Z2QSL in the thermody-\nnamic limit, it is well-known that its ground state degeneracy remains inaccessible in exact\ndiagonalization of ﬁnite-size clusters [66, 67], and the 36-site cluster is likely to host a QSL\nclose to a phase boundary with a valence bond crystal [68]. However, the nature of the equi-\nlibrium ground state and its extrapolation to the thermodynamic limit does not aﬀect the\nconclusions regarding transient state; crucially, the chiral spin liquid state is well-stabilized\nalready on the ﬁnite-size systems under consideration, with a robust many-body excitation\ngap.\nThe ground state degeneracy and minimal many-body excitation gap under ﬂux inser-\ntion was calculated by imposing a spin-dependent twist of boundary conditions along one\ndirection and tracking the many-body spectral ﬂow as a function of twist angle, as depicted\nin the inset of Fig. 3(b). A ﬁne sampling of ﬂux insertion, as depicted in Fig. 3(b), was\nperformed for 30-site clusters, spanned by vectors R1= 2a1+a2,R2=−2a1+ 4a2, and\nchecked against the 36-site cluster. The winding of the quasi-degenerate ground states upon\nﬂux insertion is a signature of CSLs, with the two quasi-degenerate ground states exchanging\nonce under ﬂux insertion, or remaining separated, depending on whether they lie in diﬀerent\n(30-site cluster) or the same (36-site cluster) momentum sectors.\nWe furthermore consider two bipartitions A,Bof the 36-site cluster with periodic bound-\nary conditions, as depicted in the inset in Fig. 3(c), and calculate the Rényi entropies\nSα(θ,φ) =−logtr/braceleftBig\nρ2\nα(θ,φ)/bracerightBig\n, α=A,B (13)\nwhereρα(θ,φ) =trα{|Ψ(θ,φ)/angbracketright/angbracketleftΨ(θ,φ)|}is the reduced density matrix on bipartition α=\nA,Bfor superpositions |Ψ(θ,φ)/angbracketright= cos(θ)|ψ1/angbracketright+sin(θ)eiφ|ψ2/angbracketrightof the quasi-degenerate ground\nstates|ψ1/angbracketright,|ψ2/angbracketright, as a function of θ,φ. Fig. 3(c) depicts SAandSBcalculated from two-fold\nquasi-degenerate ground states of the Floquet chiral spin model. For a CSL, Sα(θ,φ)is\n13expected to display two entanglement minima; the two corresponding minimally-entangled\nstates|Ψ(θ,φ)/angbracketrightpermit extraction of the modular matrices [63], which match expectations\nfor a Kalmeyer-Laughlin CSL and are quoted in the main text.\nTime Evolution\nThe electronic many-body time evolution was simulated for a 12-site Kagome-Hubbard\ncluster (4 unit cells) with periodic boundary conditions, spanned by vectors R1= 2a1,R2=\n2a2, and with the time propagation employing adaptive step size control. We note this is\nthe minimum cluster size to faithfully host all permutations of virtual hopping processes\nthat give rise to the eﬀective Floquet chiral spin Hamiltonian discussed in the main text\n[Eq. (3)].\nTo model broad circularly-polarized pump pulses, we consider a pulsed ﬁeld\nA(t) =A(t) [cos(Ωt),sin(Ωt)]/latticetop(14)\nwith a smooth sinusoidal pump envelope\nA(t) =\n\n0, t ≤0\nsAsin2/parenleftBig\nπ\n2t\ntplateau/parenrightBig\n,0<t<t plateau\nA, t ≥tplateau(15)\nwheretplateau = 700t−1\nhfortheresultsofthemaintext. Detailsonpumpenvelopedependence\ncan be found in Supplementary Note 3.\nFinally, to quantify energy absorption in the driven system, we compute the period-\naveraged energy operator\nˆE=1\nT/integraldisplayt+T\ntdt/primeˆH(t/prime)\n=Uˆn↑ˆn↓−J 0(A)th/summationdisplay\n/angbracketleftij/angbracketrightσˆc†\niσˆcjσ (16)\nwhereJ0(A)denotes the zeroth Bessel function of the ﬁrst kind. Note thatˆ/angbracketleftE/angbracketrightis time-\nindependent for a pure Floquet state, in theory. Instead, the ﬁniteness of the pump envelope\n14entailsaresidualquasi-energyspread,withtheresultingdephasingofthedrivenstateleading\nto residual heating on the pump plateau. 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Sørensen, Ground-state energy and spin gap of spin- 1/2\nKagomé-Heisenberg antiferromagnetic clusters: Large-scale exact diagonalization results , Phys.\nRev. B83, 212401 (2011).\n[67] H. Nakano, and T. Sakai, Numerical-Diagonalization Study of Spin Gap Issue of the Kagome\nLattice Heisenberg Antiferromagnet , J. Phys. Soc. Jpn. 80, 053704 (2011).\n[68] T. Li, The spin-1\n2Heisenberg model on Kagome lattice as a quantum critical system , Preprint\nat http://arxiv.org/abs/1106.6134 (2011).\n20(b)\n(c)E=0 E=U−Ω\nE=0\nE=U−ΩE=0\nE=0E=U−ΩE=U−2Ω(a)FIG. 1.Photo-induced Kagome chiral antiferromagnet. (a) Starting from a Kagome Mott\nantiferromagnet, pumping with circularly-polarized light dynamically breaks time-reversal and\nparity while preserving SU(2) spin symmetry, photo-inducing scalar spin chirality Si·(Sj×Sk)\ncontributions on elementary equilateral ( χ) and isosceles ( χ/prime) triangles. Pump strength and fre-\nquency provide knobs to tune χ,χ/primeas well as Heisenberg exchange J,J/prime,J3, as described in the\nmain text. Examples of (b) nearest-neighbor and (c) three-site Floquet virtual hopping processes\nincluding absorption of photons, in the Mott-insulating regime. Boxes graphically depict the ex-\nample initial, virtual intermediate and ﬁnal states for second- and fourth-order virtual hopping\nprocesses, in terms of the product space of electronic degrees of freedom and the Floquet index.\nTRS is broken in (c), inducing scalar spin chiralities χon triangles of the lattice, whereas (b) solely\ninduces nearest-neighbor Heisenberg exchange.\n21(a)\nA A(b)\n(c) (d)\neffective parameters [in units of J]FIG. 2.Pump dependence of chiral spin model parameters. Photo-induced time reversal\nsymmetry (TRS) breaking scalar spin chirality χ,χ/prime[(a), (b)] and TRS preserving Heisenberg\nexchangeJ/prime,J3[(c), (d)] terms for a Kagome Mott insulator pumped by circularly-polarized\nlight, as a function of pump strength Aand pump frequency [from Eqns. (4-10)]. Parameters\nare depicted in units of nearest-neighbor exchange coupling J. Dashed lines indicate one- and\ntwo-photon resonances with respect to the Hubbard repulsion U. Equal signs of χ,χ/primesignify a\nstaggering of Si·(Sj×Sk)contributions between elementary triangles and isosceles triangles inside\nthe hexagons.\n22 \n(b) (c)(a)\nCSL\nZ2/VBCΔCSL/J\nA=0A=0.5A=0.75\nΔCSL\nSA\nSB\nAFIG. 3.Identiﬁcation of a photo-induced chiral spin liquid. (a) Evolution of spin /angbracketleftSi·Sj/angbracketright\n(circles on vertices) and chiral /angbracketleft[Si·(Sj×Sk)] [Sl·(Sm×Sn)]/angbracketright(triangles) correlation functions\nas a function of pump strength A, from 36-site exact diagonalization of Eq. (3). (b) Minimum\nchiral spin liquid (CSL) excitation gap ∆CSLabove two-fold degenerate ground states, under ﬂux\ninsertion through the torus. Inset depicts the two-fold quasi-degenerate ground states (blue and\ndashed-red lines) and gap to excited states (brown lines), as a function of ﬂux φthreaded through\nthe torus [see Methods section]. The equilibrium ground state (a putative Z2spin liquid or valence\nbond crystal) transitions into a photo-induced CSL at ﬁnite pump strength. (c) Rényi entropies\nand entanglement minima, of CSL ground state superpositions cos(θ)|ψ1/angbracketright+ sin(θ)eiφ|ψ2/angbracketrightfor two\nbipartitions of the 36-site torus [inset], discussed in the main text.\n23A=0A=0(b) (a)\n(c)\n|χijk|Ω [units of th]\nx\ntime [th-1]Ω [units of th](d)\ntime evolution steady stateFIG. 4.Time evolution for broad-pulse irradiation. Comparison of the Floquet chiral spin\nmodel [Eq. (3)] and the exact time evolution of the 12-site U= 30Kagome Hubbard model [Eq.\n(1)] driven by circularly-polarized light with ultra-slow adiabatic pump envelopes (a). The scalar\nspin chiralities χijk=/angbracketleftSi·(Sj×Sk)/angbracketrighton elementary triangles of the unit cell (b) are depicted\nin (c) and (d). (c) depicts χijk(t)measured from exact time evolution and averaged over the\npump plateau; (d) depicts corresponding static expectation values of the Floquet spin model with\nHeisenberg and chiral couplings chosen as a function of A,Ω.\n24resonantresonant\nresonantdouble occupancy 〈n ↑n↓〉double occupancy 〈n↑n↓〉\ndouble occupancy 〈n ↑n↓〉 absorbed energy per period [t h]detuning from charge resonance [t h]\ndetuning from charge resonance [t h] detuning from charge resonance [t h]pump cyclespump cycles\na\nb(a)\n(b)(c)\n(d) (e) off resonance\n(scaled)〈Ψ(t+T)│Ψ(t)〉 Floquet fidelity  1−│                    │FIG. 5.Pre-thermalization and transient Floquet steady states. (a), (b) depict the time\nevolution of the period-averaged local double occupancy of the 12-site driven Hubbard model as a\nfunction of cycles under the pump, on and oﬀ the charge resonance with the upper Hubbard band,\nrespectively. Dashed lines indicate onset of the pump plateau. (c) Extracted transient expectation\nvaluesofthedoubleoccupancyatthepumpplateau, asafunctionofpumpstrength Aanddetuning\nfrom the upper Hubbard band. Red arrows indicate values of thused for (a), (b). Lines are guides\nto the eye. On resonance (gray region), the system heats rapidly, with /angbracketleftˆn↑ˆn↓/angbracketrightthermalizating inde-\npendent of the pump strength Aand approaching its inﬁnite-temperature expectation value 1/4.\nBelow resonance (white region), the system transiently realizes the TRS-breaking chiral quantum\nmagnet, with a tunable Hubbard interaction U(as well as correspondingly tunable magnetic J,χ,\nsee Fig. 4) as a function of pump strength. (d) Floquet ﬁdelity F(T) =|/angbracketleftΨ(t+T)|Ψ(t)/angbracketright|on the\npump plateau – below charge resonance, (1−F)→0, indicating the controlled preparation of a\nFloquet eigenstate. (e) Extracted stroboscopic heating rates per pump cycle and below resonance.\nRemarkably, heating is strongly suppressed close to the charge resonance, with the driven system\nrequiring many thousands of pump cycles to ﬁnally absorb energy on the order of the Heisenberg\nexchangeJ.\n25Appendix A: Perturbation Theory for the Non-Interacting Model\nBefore discussing sub-gap pumping in the interacting Mott insulator, it is instructive to\nﬁrst consider electrons on the Kagome lattice in the absence of interactions, described by a\nBloch Hamiltonian\nh(k) =−2th\n0 cos/parenleftBig\na1·k\n2/parenrightBig\ncos/parenleftBig\na2·k\n2/parenrightBig\ncos/parenleftBig\na1·k\n2/parenrightBig\n0 cos/parenleftBig\na3·k\n2/parenrightBig\ncos/parenleftBig\na2·k\n2/parenrightBig\ncos/parenleftBig\na3·k\n2/parenrightBig\n0\n(A1)\nIn equilibrium, the band structure mirrors graphene, with two Dirac points at the corners\nK,K/primeof the Brillouin zone, and a third ﬂat band with a quadratic band touching at Γ.\nA circularly-polarized pump ﬁeld now couples to electrons via Peierls substitution k→\nk+A(t), with units ~=e= 1. As there is no charge gap in the absence of interactions, we\nconsider here the oﬀ-resonant high-frequency regime with Ωmuch larger than the electronic\nband width. At weak pump strengths, the eﬀective Hamiltonian now follows as\nheﬀ(k) =h0(k)−1\nΩ[h1(k), h−1(k)]\n=−2\n0 ˜tcos/parenleftBig\na1·k\n2/parenrightBig˜t⋆cos/parenleftBig\na2·k\n2/parenrightBig\n˜t⋆cos/parenleftBig\na1·k\n2/parenrightBig\n0 ˜tcos/parenleftBig\na3·k\n2/parenrightBig\n˜tcos/parenleftBig\na2·k\n2/parenrightBig˜t⋆cos/parenleftBig\na3·k\n2/parenrightBig\n0\n\n−2i˜t/prime\n0 cos/parenleftBig(a2−a3)·k\n2/parenrightBig\n−cos/parenleftBig(a1+a3)·k\n2/parenrightBig\n−cos/parenleftBig(a2−a3)·k\n2/parenrightBig\n0 cos/parenleftBig(a1+a2)·k\n2/parenrightBig\ncos/parenleftBig(a1+a3)·k\n2/parenrightBig\n−cos/parenleftBig(a1+a2)·k\n2/parenrightBig\n0\n(A2)\nwith lattice vectors a1= (1,0)T,a2= (1/2,√\n3/2)T, andhm(k) =Ω\n2π/integraltext2π/Ω\n0dt eimΩth(k+\nA(t)). The eﬀective Hamiltonian is now parameterized by time-reversal symmetry breaking\nnearest and next-nearest neighbor hoppings\n˜t=th/parenleftBigg\n1−A2\n4/parenrightBigg\n−i√\n3t2\nhA2\n4Ω(A3)\n˜t/prime=√\n3t2\nhA2\n4Ω(A4)\n26whereAis the dimensionless ﬁeld strength as deﬁned in the main text. Time-reversal\nsymmetry breaking entails that a gap opens up at the two Dirac points at K,K/prime, however\nthe quadratic band touching at Γpersists to lowest order in A.\nAppendix B: Quasi-Degenerate Perturbation Theory for the Driven Kagome Hub-\nbard Model\nConsider now a strongly-interacting Mott-Hubbard insulator on the Kagome lattice,\ndriven by circularly-polarized light. To simplify notation, we consider a generic driven\nHubbard model\nˆH(τ) =−/summationdisplay\nijσtij(τ) ˆc†\niσˆcjσ+U/summationdisplay\niˆni↑ˆni↓ (B1)\nwhere the external drive enters in the time-dependent hopping amplitudes, and we denote\ntime byτ. The case of external drive via Peierls substitution corresponds to tij(τ) =\ntheiA(τ)·rijas discussed in the main text. In Floquet language, one can similarly write:\nˆH=−/summationdisplay\nijσ\nmm/primet(m−n)\nij ˆc†\niσˆcjσ⊗|m/angbracketright/angbracketleftm/prime|+U/summationdisplay\niˆni↑ˆni↓⊗1−/summationdisplay\nmmΩ1⊗|m/angbracketright/angbracketleftm|(B2)\nwheret(m−n)\nij =/integraltext2π/Ω\n0dτ ei(m−n)τtij(τ)are Floquet hopping amplitudes that obey/parenleftBig\nt(−m)\nji/parenrightBig⋆=\nt(m)\nij.\nSo far, this description is exact. As described in the main text, the Hamiltonian retains a\nlocal moment description of interacting spins as long as Uis much larger than hopping, and\nthe pump is oﬀ-resonant with charge excitations. A spin description for pumping below the\ncharge gap then follows from simultaneously integrating out doubly-occupied many-body\nstates and photon (Floquet) side bands, treating strong-coupling and Floquet energy scales\non equal footing.\nBefore discussing the derivation in detail, it is instructive to compare our methodology to\nthe Schrieﬀer-Wolﬀ transformation for driven systems [4]. The methods are a priori equiv-\nalent for pumping oﬀ the charge resonance (the regime of interest in this work), which is\nto be expected, as the perturbation expansion should give the same results independent of\nchoice of basis. Importantly however, as discussed in detail below, the lowest-order contri-\n27bution in virtual hopping merely serves to “renormalize” the strength of nearest-neighbor\nspin exchange interactions [4, 5], which cannot change the transient many-body state as long\nas total spin is conserved. Instead, as discussed in the main text, the subleading photon-\nassisted contributions – to fourth order in virtual hopping – are essential to manipulate\nthe equilibrium state, break time-reversal symmetry, and ultimately stabilize a chiral spin\nliquid.\nSecond Order in Virtual Hopping\nIn Floquet language, simultaneously integrating out charge and Floquet degrees of free-\ndom as discussed above amounts to applying standard quasi-degenerate perturbation theory\nto the time-independent Floquet Hamiltonian in frequency formulation. We constrain our-\nselves to the half-ﬁlled Mott insulator. To second order in virtual hopping processes, the\nperturbation theory entails a renormalization of nearest-neighbor Heisenberg exchange in-\nteractions\nˆH(2nd order )=/summationdisplay\nijσσ/primet(−m)\nijt(m)\nji\nU+mΩˆc†\niσˆcjσˆc†\njσ/primeˆciσ/prime\n= 2/summationdisplay\n/angbracketleftij/angbracketrightσσ/prime/vextendsingle/vextendsingle/vextendsinglet(m)\nji/vextendsingle/vextendsingle/vextendsingle2\nU+mΩˆc†\niσˆciσ/primeˆcjσˆc†\njσ/prime\n= 4/summationdisplay\n/angbracketleftij/angbracketright/vextendsingle/vextendsingle/vextendsinglet(m)\nji/vextendsingle/vextendsingle/vextendsingle2\nU+mΩSi·Sj (B3)\nwhere the last line follows from a canonical change to spin-1/2 operators and the local\nmomentconstraint. ChoosingFloquethoppingamplitudes t(m)\nijthatcorrespondtocircularly-\npolarized light ﬁnally leads to the result quoted in the main text.\nThird Order in Virtual Hopping\nAs a short digression, a more interesting scenario appears to third-order in virtual hop-\nping. In equilibrium, it is well-known that the third-order expansion vanishes in the presence\nof time-reversal symmetry. Conversely, if time-reversal symmetry is broken via an external\nmagnetic ﬁeld, a scalar spin chirality term appears already to this order, and is proportional\n28to the phase acquired by electrons hopping around a triangle. Naïvely, a similar scenario\nshould then arise already to third order, in the presence of a circularly-polarized ﬁeld, with\nelectrons acquiring a phase when absorbing or emitting photons during virtual hopping.\nHere, we show that this is not the case.\nTo third order in virtual hopping, and without any assumptions on symmetries of the\nproblem, the generic fermionic starting point of the expansion reads\nˆH(3)=/summationdisplay\nijk\nσσ/primeσ/prime/prime\nm1m21\n(U+m1Ω)(U+m2Ω)/bracketleftBig\nˆc†\niσˆcjσˆc†\njσ/primeˆckσ/primeˆc†\nkσ/prime/primeˆciσ/prime/primet(−m2)\nijt(m2−m1)\njkt(m1)\nki+\n+ ˆc†\njσ/primeˆckσ/primeˆc†\niσˆcjσˆc†\nkσ/prime/primeˆciσ/prime/primet(−m2)\njkt(m2−m1)\nijt(m1)\nki/bracketrightBig\n(B4)\nwherei,j,ksumoverall sites. Afterappropriateexpansionof allpermutations, reorderingof\nfermionicoperators and recasting in terms of spin-1/2operators, one arrives ata Heisenberg-\nchiral spin Hamiltonian\nˆH3rd order=/summationdisplay\n/angbracketleftij/angbracketrightJ(3rd order)Si·Sj+/summationdisplay\n/triangle\nijkχ(3rd order)Si·(Sj×Sk) (B5)\nHere, the Heisenberg exchange coupling and scalar spin chirality coupling can be expressed\nas\nχ3rd order=/summationdisplay\nm1m22\n(U+m1Ω)(U+m2Ω)Im/bracketleftBig\nt(m2)\nijt(m1−m2)\njkt(−m1)\nki +t(m1−m2)\nijt(m2)\njkt(−m1)\nki +\n+t(m2−m1)\nijt(−m2)\njkt(m1)\nki+t(−m2)\nijt(m2−m1)\njkt(m1)\nki+t(m2)\nijt(−m1)\njkt(m1−m2)\nki +t(−m1)\nijt(m2)\njkt(m1−m2)\nki +\n+t(m2−m1)\nijt(m1)\njkt(−m2)\nki +t(m1)\nijt(m2−m1)\njkt(−m2)\nki +t(m1−m2)\nijt(−m1)\njkt(m2)\nki+t(−m1)\nijt(m1−m2)\njkt(m2)\nki+\n+t(−m2)\nijt(m1)\njkt(m2−m1)\nki +t(m1)\nijt(−m2)\njkt(m2−m1)\nki/bracketrightBig\n(B6)\nJ3rd order=/summationdisplay\nm1m21\n(U+m1Ω)(U+m2Ω)Re/bracketleftBig\n−(t(m2)\nijt(m1−m2)\njkt(−m1)\nki )−t(m1−m2)\nijt(m2)\njkt(−m1)\nki +\n+t(m2−m1)\nijt(−m2)\njkt(m1)\nki+t(−m2)\nijt(m2−m1)\njkt(m1)\nki+t(m2)\nijt(−m1)\njkt(m1−m2)\nki−t(−m1)\nijt(m2)\njkt(m1−m2)\nki−\n−t(m2−m1)\nijt(m1)\njkt(−m2)\nki−t(m1)\nijt(m2−m1)\njkt(−m2)\nki +t(m1−m2)\nijt(−m1)\njkt(m2)\nki+t(−m1)\nijt(m1−m2)\njkt(m2)\nki−\nt(−m2)\nijt(m1)\njkt(m2−m1)\nki +t(m1)\nijt(−m2)\njkt(m2−m1)\nki ] (B7)\n29To proceed, consider a generic parameterization of the Floquet hoppings\nt(m)\nij=tijJm(Aij)eimψij(B8)\nImportantly, one then ﬁnds that the terms in the bracket in Supplementary Eq. (B6) are\nreal and the chiral contribution χ3rd ordervanishes exactly even if time-reversal symmetry is\nbrokenis by the external ﬁeld. In other words, the photon-mediated phases acquired by\nelectrons hopping around a triangle remarkably cancel exactly and to all orders in pump\nstrengthA. The third-order Heisenberg contribution J3rd ordervanishes analogously, with\nthe terms in the bracket in Supplementary Eq. (B7) purely imaginary.\nInstead, in analogy to the equilibrium case, solely a staticmagnetic ﬁeld will induce a\nscalar spin chirality term to third order. As discussed in the main text however, in this case\nthe Zeeman shift will generically dominate for an external magnetic ﬁeld and preclude the\nformation of a chiral spin liquid.\nFor completeness, we could now consider the joint eﬀect of a staticmagnetic ﬁeld and\ntime-dependent circularly-polarized pump. Here, the nearest-neighbor Floquet hopping am-\nplitudes\nt(m)\n/angbracketleftij/angbracketright=eiφmagthJm(Aij)eimψij(B9)\nwould acquire an additional phase φmagdue to the static magnetic ﬂux through the triangle.\nUnlike the equilibrium problem, the third-order contribution to the eﬀective Hamiltonian\nin this case entails not only a scalar spin chirality term but further corrections to nearest-\nneighbor Heisenberg exchange\nJ3rd order=/summationdisplay\nm1m24t3\nhJm1(A)Jm2−m1(A)J−m2(A)\n(U+m1Ω)(U+m2Ω)sin/bracketleftBig2π(2m1−m2)\n3/bracketrightBig\nsin(3φmag)(B10)\nχ3rd order=/summationdisplay\nm1m224t3\nhJm1(A)Jm2−m1(A)J−m2(A)\n(U+m1Ω)(U+m2Ω)cos/bracketleftBig2π(2m1−m2)\n3/bracketrightBig\nsin(3φmag)(B11)\nThe well-known equilibrium result [1–3] for solely a static magnetic ﬁeld can be readily\nrecovered setting Ato zero, which yields J3rd order= 0andχ3rd order= 24t3\nh/U2sin(3φmag).\n30Fourth Order in Virtual Hopping\nConstitutingthecentralresultofthemaintext, dynamicaltime-reversalsymmetrybreak-\ning due to circularly-polarized light ﬁrst enters the eﬀective spin description to fourth order\nin photon-assisted virtual hopping processes. The presence and relevance of this eﬀect in\ndetermining the transient steady state is remarkable – whereas fourth-order contributions\nto spin interactions will commonly be negligible in equilibrium Mott insulators, the out-of-\nequilibrium problem provides a powerful knob to tune the spin system into a regime where\nsuch contributions emerge, in fact, as the determining mechanism to stabilize the transient\nphase.\nThe quasi-degenerate perturbation theory again proceeds as usual, however entails a large\nnumber of combinatoric contributions due to the various permutations of hopping virtually\nthrough diﬀerent Floquet side bands. We therefore developed a computer algebra program\nto perform the perturbation expansion and recasting into spin language. Here, we solely\nquote the fermionic starting point of the fourth-order expansion\n31ˆH(4)=−/summationdisplay\nijkl\nσσ/primeσ/prime/primeσ/prime/prime/prime\nm1m2m3\nˆc†\niσˆcjσˆc†\njσ/primeˆckσ/primeˆc†\nkσ/prime/primeˆclσ/prime/primeˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/prime(1−δikδm2)t(−m3)\nijt(m3−m2)\njkt(m2−m1)\nklt(m1)\nli\n(U+m1Ω)(U(1−δik) +m2Ω)(U+m3Ω)+\n+ ˆc†\njσ/primeˆckσ/primeˆc†\niσˆcjσˆc†\nkσ/prime/primeˆclσ/prime/primeˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/prime(1−δikδm2)t(−m3)\njkt(m3−m2)\nijt(m2−m1)\nklt(m1)\nli\n(U+m1Ω)(U(1−δik) +m2Ω)(U+m3Ω)+\n+ ˆc†\njσ/primeˆckσ/primeˆc†\nkσ/prime/primeˆclσ/prime/primeˆc†\niσˆcjσˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/prime(1−δjlδm2)t(−m3)\njkt(m3−m2)\nklt(m2−m1)\nijt(m1)\nli\n(U+m1Ω)(U(1−δjl) +m2Ω)(U+m3Ω)+\n+ ˆc†\nkσ/prime/primeˆclσ/prime/primeˆc†\njσ/primeˆckσ/primeˆc†\niσˆcjσˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/prime(1−δjlδm2)t(−m3)\nklt(m3−m2)\njkt(m2−m1)\nijt(m1)\nli\n(U+m1Ω)(U(1−δjl) +m2Ω)(U+m3Ω)\n+\n−/summationdisplay\nijkl\nσσ/primeσ/prime/primeσ/prime/prime/prime\nm1m2m31\n(U+m1Ω)(2U+m2Ω)(U+m3Ω)/bracketleftbigg\nˆc†\niσˆclσˆc†\njσ/primeˆckσ/primeˆc†\nkσ/prime/primeˆcjσ/prime/primeˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/primet(−m3)\nilt(m3−m2)\njkt(m2−m1)\nkjt(m1)\nli+\n+ ˆc†\njσˆclσˆc†\niσ/primeˆckσ/primeˆc†\nkσ/prime/primeˆcjσ/prime/primeˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/primet(−m3)\njlt(m3−m2)\nikt(m2−m1)\nkjt(m1)\nli\n+ ˆc†\njσˆckσˆc†\niσ/primeˆclσ/primeˆc†\nkσ/prime/primeˆcjσ/prime/primeˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/primet(−m3)\njkt(m3−m2)\nilt(m2−m1)\nkjt(m1)\nli\n+ ˆc†\niσˆckσˆc†\njσ/primeˆclσ/primeˆc†\nkσ/prime/primeˆcjσ/prime/primeˆc†\nlσ/prime/prime/primeˆciσ/prime/prime/primet(−m3)\nikt(m3−m2)\njlt(m2−m1)\nkjt(m1)\nli/bracketrightBig\n+1\n2/summationdisplay\nijkl\nσσ/primeσ/prime/primeσ/prime/prime/prime\nm1m2ˆc†\nkσˆclσˆc†\nlσ/primeˆckσ/primeˆc†\niσ/prime/primeˆcjσ/prime/primeˆc†\njσ/prime/prime/primeˆciσ/prime/prime/primet(−m2)\nklt(m2)\nlkt(−m1)\nijt(m1)\nji/bracketleftBigg1\n(U+m1Ω)(U+m2Ω)2+\n+1\n(U+m1Ω)2(U+m2Ω)/bracketrightBigg\n(B12)\nwhere we imply within the terms of the ﬁrst sum that(1−δikδm2)\nU(1−δik)+m2Ω= 0ifi=kandm2= 0.\nThe resulting spin Hamiltonian for circularly-polarized pumping can now be derived from\ntedious but straightforward algebraic operations, and is presented in the main text.\nWe note that similar perturbative expansions must be used to recast electronic experi-\nmental observables in terms of the eﬀective low-energy spin degrees of freedom. A detailed\ndiscussion of optical and Raman probes (in particular, the A 2gchannel and its analogue\nfor the Kagome lattice) to fourth order in virtual hopping processes will be discussed in an\nupcoming publication.\n32Appendix C: Pump-Envelope Dependence and Many-Body Dynamics of the Driven\nKagome-Hubbard Model\nThe main text investigates the many-body dynamics of a Kagome-Hubbard model\npumped with circularly-polarized light, and compares the driven steady state at the pump\nplateau to the eﬀective Floquet chiral spin model that is the main focus of this work. To\nextract scalar spin chirality expectation values from the driven steady state at the pump\nplateau, we consider pump ﬁelds of the form\nA(t) =A(t)\ncos(Ωt)\nsin(Ωt)\n (C1)\nwith a smooth sinusoidal pump envelope\nA(t) =\n\n0, t ≤0\nAsin2/parenleftBig\nπ\n2t\ntplateau/parenrightBig\n,0<t<t plateau\nA, t ≥tplateau(C2)\nwithtplateau = 500t−1\nh. SupplementaryFig. 6depictstheperiod-averagedscalarspinchirality\nexpectation values /angbracketleftχijk/angbracketrightfor three distinct triangles of the Kagome unit cell, as well as a the\nraw dynamical data (including micro-motion within the pump periods), close to the charge\nresonance for a 12-site cluster. Fig. 4(c) of the main text now follows straight-forwardly via\nextracting the average expectation values /angbracketleftχijk/angbracketrightat the pump plateau.\nTo analyze the inﬂuence of the pump envelope on controlled preparation of the steady\nstate, we calculate the scalar spin chirality expectation values as well as double occupancies\nand period-averaged energy, as a function of tplateau, i.e. as a function of ramp-up speed.\nSupplementary Fig. 7 depicts the envelope dependence for scalar spin chirality expectation\nvalues, showing excellent control of the transient steady state for the pump envelope widths\nused in the main text. To consider residual eﬀects of heating and energy absorption, Sup-\nplementary Fig. 8 depicts transient double occupancies and period-averaged energy on the\npump plateau, as a function of pump envelope width, for intermediate and strong pumping.\nAnalogously, the steady state is remarkably robust, with heating strongly suppressed. To\nquantify the latter, residual energy absorption rates can be extracted from the gradient of\n33the period-averaged energy, and are depicted in Fig. 5(e) of the main text.\n0.000.050.100.150 tplateau<χijk>elementaryΩ=220 tplateauΩ=22.50 tplateauΩ=230 tplateauΩ=23.50 tplateau\n0.000.050.100.15Ω=24\nA=0.125\nA=0.25\nA=0.375\nA=0.5A=0.625\nA=0.75\nA=0.875\nA=1\n0.000.050.100.15<χijk>isosceles0.000.050.100.15\n0 tplateau0.000.050.100.15<χijk>in hexagon0 tplateau0 tplateau0 tplateau0 tplateau0.000.050.100.15\n(a) Period-averaged scalar spin chirality expectation values.\n(b) Raw dynamical data.\nFIG. 6.Time evolution of scalar spin chirality expectation values. Panels depict time-\ndependent expectation values /angbracketleftSi·(Sj×Sk) (t)/angbracketrightfor circularly-polarized pumping close to the one-\nphoton charge resonance, as a function of pump strength and frequency. (a) depicts a triplet of\nrows corresponding to measuring the period-averaged scalar spin chirality on three triangles in the\nunit cell: elementary triangles of the Kagome lattice, isosceles triangles inside the hexagon, and\nequilateral triangles inside the hexagon, as depicted graphically in Fig. 4 (b) of the main text.\n(b) depicts the corresponding raw dynamical data. The broadening of lines stems from micro-\nmotion within individual pump periods, shown in detail in the inset. Floquet expectation values\nare recovered by averaged out the micro-motion, as shown in (a).\n34T\nT\nT\nT\nTFIG. 7.Pump envelope dependence of scalar spin chirality expectation values . The\ntime evolution of period-averaged scalar spin chirality expectation values is shown for elementary\ntriangles of the Kagome lattice, for Ω = 23th. Here,tplateau/Tquantiﬁes the number of pump cycles\nunder the ramp-up, before reaching the pump plateau at time t=tplateau(time axis is normalized\naccordingly). Pump envelopes used for Fig. 4 and Fig. 5 of the main text are depicted in green\n(tplateau/T= 915).\nT\nT\nT\nT\nTT\nT\nTT\nFIG. 8.Pump envelope dependence of period-averaged double occupancies and energy.\nTow(bottom)rowscorrespondtointermedite(strong)pumping, for Ω = 23th, onthepumpplateau\n(pump cycle 0≡tplateau). Energy depicted in units of hopping strength. The time derivative of\ndouble occupancies and energy yields eﬀective heating rates, demonstrating negligible heating both\nduring the pump ramp-up and on the pump plateau, for pumping below the one-photon charge\nresonance.\n35SUPPLEMENTARY REFERENCES\n[1] A. H. MacDonald, S. M. Girvin, and D. Yoshioka, t/Uexpansion for the Hubbard model , Phys.\nRev. B37, 9753 (1988).\n[2] D. Sen, and R. Chitra, Large-Ulimit of a Hubbard model in a magnetic ﬁeld: Chiral spin\ninteractions and paramagnetism , Phys. Rev. B 51, 1922 (1995).\n[3] O. I. Motrunich, Orbital magnetic ﬁeld eﬀects in spin liquid with spinon Fermi sea: Possible\napplication to κ-(ET) 2Cu2(CN) 3, Phys. Rev. B 73, 155155 (2006).\n[4] M. Bukov, M. Kolodrubetz, and A. Polkovnikov, Schrieﬀer-Wolﬀ Transformation for Period-\nically Driven Systems: Strongly Correlated Systems with Artiﬁcial Gauge Fields , Phys. Rev.\nLett.116, 125301 (2016).\n[5] J. H. Mentink, K. Balzer, and M. Eckstein, Ultrafast and reversible control of the exchange\ninteraction in Mott insulators , Nature Comm. 6, 6708 (2015).\n36"    },    {        "title": "2109.05628v2.Spin_excitation_spectra_in_helimagnetic_states__proper_screw__cycloid__vortex_crystal__and_hedgehog_lattice.pdf",        "content": "Spin excitation spectra in helimagnetic states: proper-screw, cycloid, vortex crystal,\nand hedgehog lattice\nYasuyuki Kato, Satoru Hayami, and Yukitoshi Motome\nDepartment of Applied Physics, the University of Tokyo, Tokyo 113-8656, Japan\nWe investigate the spin excitation spectra in chiral and polar magnets by the linear spin-wave\ntheory for an e\u000bective spin model with symmetric and antisymmetric long-range interactions. In one\ndimension, we obtain the analytic form of the dynamical spin structure factor for proper-screw and\ncycloidal helical spin states with uniform twists, which shows a gapless mode with strong intensity\nat the helical wave number. When introducing spin anisotropy in the symmetric interactions, we\nnumerically show that the stable spin spirals become elliptically anisotropic with nonuniform twists\nand the spin excitation is gapped. In higher dimensions, we \fnd that similar anisotropy stabilizes\nmultiple-Qspin states, such as vortex crystals and hedgehog lattices. We show that the anisotropy\nin these states manifests itself in the dynamical spin structure factor: a strong intensity in the\ntransverse components to the wave number appears only when the helical wave vector and the\ncorresponding easy axis are perpendicular to each other. Our \fndings could be useful not only\nto identify the spin structure but also to deduce the stabilization mechanism by inelastic neutron\nscattering measurements.\nI. INTRODUCTION\nThe helimagnetic orders are periodic spin states found\nin a wide range of materials, from metals to insula-\ntors, where the magnetic moments form twisting and\nswirling textures, such as spin spirals and vortex crystals\n(VCs) [1]. Of particular interest is the cases where the\nspin textures de\fne topologically nontrivial objects [2{\n5]. There are many examples of such helimagnetic orders,\ne.g., one-dimensional (1D) proper-screw helical spin (HS)\nstates [Fig. 1(a)] [6], 1D cycloidal HS states [Fig. 1(b)] [7],\n1D chiral soliton lattice [8{14], two-dimensional (2D)\nskyrmion crystals (SkXs) [15{18], 2D vortex crystal\n(VC) [19], and three-dimensional (3D) hedgehog lattices\n(HLs) [20{25]. These helimagnetic states have been at-\ntracting a lot of attention since they induce intriguing\nelectronic and transport properties, such as the magne-\ntoelectric e\u000bect [26] and the topological Hall e\u000bect [27],\nwhich would lay the cornerstone of future technology.\nFIG. 1. Helimagnetic orders: (a) proper-screw and (b) cy-\ncloidal helical spin states. The latter is obtained by \u0019=2 spin\nrotation of the former about the zaxis.\nSeveral mechanisms have been proposed for the sta-\nbility of these helimagnetic spin textures, including the\nDzyaloshinskii-Moriya antisymmetric exchange interac-tions [28{30], frustration among the competing exchange\ninteractions [31{33], four spin interactions [34{39], long-\nrange interactions via itinerant electrons [40{48], long-\nrange dipole interactions [49{51], and bond-dependent\nanisotropic interaction [52{54]. To elucidate the relevant\nmechanism, it is desired to clarify the microscopic infor-\nmation of the magnetic interactions. Inelastic neutron\nscattering is a useful experimental tool to obtain such\nmicroscopic information from the analysis of the spin ex-\ncitation spectrum. It is, however, not always an easy\ntask, especially for the complex spin textures. For ex-\nample, while the SkXs and the HLs are stably obtained\nfor the models with either short-range [28, 31, 55, 56] or\nlong-range interactions [44{46, 57, 58], it remains yet to\nbe clari\fed which is the most relevant mechanism in each\nsubstance. This is mainly due to less available informa-\ntion on the spin excitations for the detailed comparison\nbetween theory and experiment.\nIn this paper, we systematically study the spin ex-\ncitation spectra for spin models which stabilize various\ntypes of helimagnetic spin textures, by tuning the range\nof magnetic interactions in real space. Speci\fcally, start-\ning from the e\u000bective spin model for spin-charge coupled\nsystems, which has in\fnite-range interactions [45, 57], we\nextend it by including both symmetric and antisymmet-\nric exchange interactions with spatial decay, and obtain\nthe ground states and spin excitation spectra by varia-\ntional calculations and the linear spin-wave theory, re-\nspectively. We \fnd that our models stabilize 2D VCs\nand 3D HLs in addition to 1D HS states, by introduc-\ning spin anisotropy in the symmetric interaction. In the\n1D case, we show that the dynamical spin structure fac-\ntor for both proper-screw and cycloidal HS states has a\ngapless mode with strong intensity at the helical wave\nnumber in the isotropic case, but they are gapped in the\npresence of the anisotropy which modulates the stable\nspin spirals into elliptically anisotropic ones and makes\nthe twists nonuniform. We also clarify that the lowest-\nenergy excitation mode with the strongest intensity canarXiv:2109.05628v2  [cond-mat.str-el]  2 Dec 20212\nbe regarded as a phase shift of the spin helix. In higher\ndimensions, we \fnd that while the system exhibits a HS\nstate in the isotropic case, the anisotropy can stabilize\nmultiple-Qspin states which are composed of superpo-\nsitions of multiple spin helices; we obtain four di\u000berent\ntypes of double- Q(2Q) VCs in two dimensions and three\ndi\u000berent types of triple- Q(3Q) HLs in three dimensions.\nWe \fnd that the dynamical spin structure factor for the\nmultiple-Qspin states exhibits a strong intensity in the\nlowest-energy excitation mode when the helical wave vec-\ntor is perpendicular to the easy axis of the corresponding\ninteraction. This means that the experimental identi\f-\ncation of such strong intensity by the inelastic neutron\nscattering would provide the information of not only the\npropagating direction and magnetic period of the helices\nbut also the anisotropy in the e\u000bective magnetic inter-\nactions. In addition to the experimental relevance, our\npresent scheme provides a versatile theoretical framework\nto investigate spin-wave excitations in a wide variety of\nmultiple-Qspin states, even beyond those treated in this\npaper, such as SkXs and other multiple- QHLs.\nThe structure of this paper is as follows. In Sec. II, we\n\frst introduce the e\u000bective spin model for chiral magnets\nwith in\fnite-range interactions. Several types of the sym-\nmetric and antisymmetric interactions are introduced for\nthe 1D, 2D, and 3D cases. Then, we extend the model by\nintroducing spatial decay in the interactions. In Sec. III,\nwe describe the methods used in the present study: the\nvariational method for the ground state and the linear\nspin-wave theory for the spin excitations. In Sec. IV, the\nresults for the 1D, 2D, and 3D cases are shown. For the\n1D case, we present the results of the analytical calcula-\ntions for the HS states with spatially uniform spin twist\nin the isotropic case, and the results of the numerical cal-\nculations for the e\u000bect of the anisotropy. For the 2D and\n3D cases, we show the ground-state phase diagrams while\nchanging the anisotropy in the symmetric interaction and\nthe strength of the antisymmetric interaction. Then, we\ndiscuss the details of the stabilized spin states, the spin-\nwave dispersion, and the dynamical spin structure factor,\nwhich is relevant to the inelastic neutron scattering ex-\nperiments, for di\u000berent types of VCs and HLs. Section V\nis devoted to the summary and discussion.\nII. MODEL\nA. E\u000bective spin model\nWe begin with a generic spin model for chiral magnets\nwhose Hamiltonian is de\fned in momentum space as\nH=X\nq21BZHq; (1)\nwith\nHq=\u0000X\n\u000b;\fJ\u000b\f\nqS\u000b\nqS\f\n\u0000q\u0000iDq\u0001(Sq\u0002S\u0000q);(2)where\u000b,\f=x,y,z;Sq= (Sx\nq;Sy\nq;Sz\nq) is de\fned by the\nFourier transform of the spin in real space, Sr, as\nSq=L\u0000d\n2X\nrSre\u0000iq\u0001r: (3)\nHere we de\fne this model on a d-dimensional hypercu-\nbic lattice with linear dimension Lunder the periodic\nboundary condition; the lattice site ris denoted as\nr=8\n><\n>:x(\u0011`);(d= 1)\n(x;y); (d= 2)\n(x;y;z );(d= 3); (4)\nwith integers x,y, andzin [0;L). The sumP\nq21BZ\nin Eq. (1) runs over all the wave numbers in the \frst\nBrillouin zone (1BZ):\nq=8\n><\n>:qx\u0011q=2\u0019\nLnx; (d= 1)\n(qx;qy) =2\u0019\nL(nx;ny); (d= 2)\n(qx;qy;qz) =2\u0019\nL(nx;ny;nz);(d= 3); (5)\nwith integers n\u000bin [\u0000L=2;L=2), that is,\u0000\u0019\u0014q\u000b< \u0019.\nThe \frst term of Hqin Eq. (2) represents the symmetric\nexchange interaction ( J\u000b\f\nq=J\f\u000b\nq), while the second term\nrepresents the antisymmetric one of the Dzyaloshinskii-\nMoriya type [7, 59]. For the former, we include only the\ndiagonal elements, namely, J\u000b\f\nq=J\u000b\u000b\nq\u000e\u000b;\f, for simplicity\n(\u000e\u000b;\fis the Kronecker delta). Then, Hqis expressed as\nHq=\u0000X\n\u000b;\fS\u000b\nqJ\u000b\f\nqS\f\n\u0000q; (6)\nwith\nJq=2\n4Jxx\nqiDz\nq\u0000iDy\nq\n\u0000iDz\nqJyy\nqiDx\nq\niDy\nq\u0000iDx\nqJzz\nq3\n5=J\u0003\n\u0000q: (7)\nB. In\fnite-range limit\nA particular case of the model in Eq. (1) was studied\nfor 2D VCs and SkXs [57], where J\u000b\u000b\nqandDqin Eq. (7)\nare taken as\nJ\u000b\u000b\nq=X\n\u0011J\u000b\u000b\nQ\u0011(\u000eq;Q\u0011+\u000eq;\u0000Q\u0011); (8)\nDq=X\n\u0011DQ\u0011(\u000eq;Q\u0011\u0000\u000eq;\u0000Q\u0011): (9)\nThis corresponds to the model in the limit of in\fnite-\nrange interactions in real space. The summations in\nEqs. (8) and (9) are taken for a particular set of the\nwave vectors Q\u0011, which correspond to the nesting vec-\ntors of the Fermi surfaces when the model is constructed\nas an e\u000bective model for itinerant electron systems of the\nKondo lattice type [45, 57]. This in\fnite-range model3\nTABLE I. Theoretical models in the present study: dimension d, target spin states, spin con\fgurations, equations and\nschematics of the symmetric and antisymmetric interactions, crystallographic point groups, and corresponding sections for the\nresults.\nd target spin state spin con\fguration J\u000b\u000b\nQ\u0011DQ\u0011 schematic crystallographic point group results\n1Dproper-screw HS state Figs. 11(a){11(c) Eq. (15) Eq. (14) Fig. 2(a) orthorhombic D2(222)\nSec. IV A cycloid(I) HS state Eq. (16) Eq. (17) Fig. 2(b) orthorhombic C2v(mm2)\ncycloid(II) HS state Eq. (18) Eq. (17) Fig. 2(c) orthorhombic C2v(mm2)\n2Dproper-screw(I) VC Fig. 15(a) Eq. (22) Eq. (21) Fig. 3(a) tetragonal D4(422)\nSec. IV Bcycloid(I) VC Fig. 15(b) Eq. (23) Eq. (24) Fig. 3(b) tetragonal C4v(4mm)\nproper-screw(II) VC Fig. 15(c) Eq. (25) Eq. (26) Fig. 3(c) tetragonal D2d(\u001642m)\ncycloid(II) VC Fig. 15(d) Eq. (27) Eq. (28) Fig. 3(d) tetragonal D2d(\u00164m2)\n3Dproper-screw HL Fig. 19(a) Eq. (30) Eq. (29) Fig. 4(a) cubic T(23)\nSec. IV C cycloid(I) HL Fig. 19(b) Eq. (31) Eq. (32) Fig. 4(b) trigonal C3(3)\ncycloid(II) HL Fig. 19(c) Eq. (33) Eq. (34) Fig. 4(c) trigonal C3(3)\nwas shown to stabilize VCs in two dimensions, which\nturn into SkXs in an applied magnetic \feld [57]. It was\nalso shown that the models with an additional in\fnite-\nrange biquadratic interaction stabilize SkXs and HLs in\ntwo and three dimensions, respectively [58, 60, 61].\nC. Helical wave number and anisotropy\nIn the present study, starting from the in\fnite-range\nmodel, we consider its extension by introducing expo-\nnential decay in the long-range interactions. Before go-\ning into the extension, we de\fne the characteristic wave\nnumbers Q\u0011and the anisotropy in the magnetic interac-\ntion in this section. With regard to Q\u0011, for simplicity,\nwe take them being parallel to the principal axes of the\nhypercube andjQ\u0011j=Q:\nQ\u0011=8\n><\n>:Q^x; (d= 1)\nQ^x;Q^y; (d= 2)\nQ^x;Q^y;Q^z;(d= 3); (10)\nwhere ^x,^y, and ^zare the unit vectors along the x,\ny, andzaxes, respectively. Meanwhile, regarding the\nanisotropy, we introduce it in the symmetric part of the\ninteraction J\u000b\u000b\nq, following Ref. [57]. In the following, we\ndescribe the speci\fc forms of the anisotropic interactions\nin each spatial dimension.\n1. One-dimensional case\nIn the 1D case ( d= 1), we choose\nJ\u000b\u000b\nq=J\u000b\u000b\nQ(\u000eq;Q+\u000eq;\u0000Q); (11)\nDq=DQ(\u000eq;Q\u0000\u000eq;\u0000Q); (12)\nFIG. 2. Pictorial representations of the coupling constants\nfor the symmetric and antisymmetric interactions in the 1D\nmodels for (a) proper-screw [Eqs. (14) and (15)], (b) cycloid(I)\n[Eqs. (16) and (17)], and (c) cycloid(II) [Eqs. (17) and (18)]\nHS states. The blue ellipsoids represent J\u000b\u000b\n\u0006Q: the lengths\nalong the principal axes [100], [010], and [001] denote the\namplitudes of Jxx\n\u0006Q,Jyy\n\u0006Q, andJzz\n\u0006Q, respectively. The red\narrows represent D\u0006Q. The axes for the spin space are shown\nin (a).\nwith\nQ=2\u0019\n\u0003; (13)\nwhere \u0003 gives the period of the HS states. We consider\nthree sets of the coupling constants with di\u000berent spin\nanisotropy in J\u000b\u000b\nQand the direction of DQas described\nbelow. They are summarized in Table I, including the\ncrystallographic point groups of the resultant models.\nThe \frst is the one which stabilizes a proper-screw HS\nstate shown in Fig. 1(a). In this case, to align the helical\nplane perpendicular to the propagating direction, we set\nDQas\nDQ=D^x: (14)\nIn addition, we introduce an anisotropy \u0001 in J\u000b\u000b\nQas\nJxx\nQ=Jzz\nQ=J(1\u0000\u0001); Jyy\nQ=J(1 + 2\u0001): (15)\nAs we will discuss later, this anisotropy modulates the\nspin helix from circular to elliptical and makes the\ntwist angle between neighboring spins nonuniform, which4\nopens a gap in the magnetic excitation spectrum. The\npictorial representations of J\u000b\u000b\nqandDqare shown in\nFig. 2(a).\nThe second one is for realizing a cycloidal HS state,\nwhose spin structure is obtained by \u0019=2 spin rotation\nof the proper-screw one about the zaxis, as shown in\nFig. 1(b). To stabilize this, we rotate the spin axis in the\ncoupling constants as\nJyy\nQ=Jzz\nQ=J(1\u0000\u0001); Jxx\nQ=J(1 + 2\u0001); (16)\nDQ=D^y; (17)\nas shown in Fig. 2(b). We call the spin state realized by\nthis model the cycloid(I) HS state.\nThe last one is for a di\u000berent type of the cycloidal HS\nstate, which we call cycloid(II), obtained by additional\n\u0019=2 spin rotation about the yaxis. In this case, we set\nJxx\nQ=Jyy\nQ=J(1\u0000\u0001); Jzz\nQ=J(1 + 2\u0001); (18)\nwith the same DQas Eq. (17). This case is shown in\nFig. 2(c).\n2. Two-dimensional case\nFIG. 3. Similar pictorial representations to Fig. 2 for the 2D\nmodels realizing the VCs of (a) proper-screw(I) [Eqs. (21) and\n(22)], (b) cycloid(I) [Eqs. (23) and (24)], (c) proper-screw(II)\n[Eqs. (25) and (26)], and (d) cycloid(II) [Eqs. (27) and (28)]\ntypes. The notations are common to those in Fig. 2.\nIn the 2D case ( d= 2), we choose\nJ\u000b\u000b\nq=X\n\u0011J\u000b\u000b\nQ\u0011(\u000eq;Q\u0011+\u000eq;\u0000Q\u0011); (19)\nDq=X\n\u0011DQ\u0011(\u000eq;Q\u0011\u0000\u000eq;\u0000Q\u0011); (20)withQ1=Q^xandQ2=Q^y[see Eq. (10)]. We consider\nfour sets of J\u000b\u000b\nQ\u0011andDQ\u0011. The \frst is the one which can\nstabilize a superposition of two proper-screw spirals. In\nthis case, to align each helical plane perpendicular to the\ncorresponding helical direction, we set DQ\u0011as\nDQ\u0011=(\nD^x;(\u0011= 1)\nD^y;(\u0011= 2): (21)\nFor the symmetric part, we introduce the anisotropy\ncompatible with C4rotational or S4rotore\rection sym-\nmetry about the zaxis, that is,\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n(\n[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 1)\n[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 2):(22)\nThe pictorial representations of J\u000b\u000b\nqandDqare shown\nin Fig. 3(a). We call the 2 Qspin state stabilized in this\nsetting the proper-screw(I) VC.\nThe second one is for realizing a superposition of two\ncycloidal spirals. Similar to the 1D case, we apply \u0000\u0019=2\nspin rotation to the \frst case about the zaxis and set\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n(\n[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 1)\n[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 2);(23)\nDQ\u0011=(\n\u0000D^y;(\u0011= 1)\nD^x;(\u0011= 2): (24)\nSee Fig. 3(b). We call the 2 Qspin state stabilized in this\nsetting the cycloid(I) VC.\nThe third one is for realizing a superposition of two\nproper-screw spirals which are di\u000berent from the \frst\ncase. This is obtained by additional \u0019spin rotation about\nthe [1 \u001610] axis, and hence, we set\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n(\n[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 1)\n[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 2);(25)\nDQ\u0011=(\nD^x;(\u0011= 1)\n\u0000D^y;(\u0011= 2); (26)\nas shown in Fig. 3(c). We call the 2 Qspin state stabilized\nin this setting the proper-screw(II) VC.\nThe last one is for realizing a di\u000berent superposition\nof two cycloidal spirals from the second case. This is\nobtained by additional \u0019=2 spin rotation to the third case\nabout thezaxis, and hence, we set\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n(\n[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 1)\n[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 2);(27)\nDQ\u0011=(\nD^y;(\u0011= 1)\nD^x;(\u0011= 2); (28)5\nas shown in Fig. 3(d). We call the 2 Qspin state stabilized\nin this setting the cycloid(II) VC.\nThe four sets of the coupling constants are summarized\nin Table I, including the crystallographic point groups of\nthe resultant models. See also Fig. 15 for the spin con\fg-\nurations of each 2 Qspin state. We note that the proper-\nscrew(I) VC is categorized into the so-called Bloch-type\nVCs, while the cycloid(I) VC is the so-called N\u0013 eel-type.\nIn general, the Bloch- and N\u0013 eel-type multiple- Qspin\nstates are realized under the Rashba- and Dresselhaus-\ntype spin-orbit couplings, respectively [57, 62]. Note that\nmultiple-Qstates in the presence of both types of the\nspin-orbit coupling were studied for a model which ex-\nplicitly includes itinerant electrons [63].\n3. Three-dimensional case\nFIG. 4. Similar pictorial representations to Fig. 2 for the 3D\nmodels realizing the HLs of (a) proper-screw [Eqs. (29) and\n(30)], (b) cycloid(I) [Eqs. (31) and (32)], and (c) cycloid(II)\n[Eqs. (33) and (34)] types. The notations are common to\nthose in Fig. 2.\nIn the 3D case ( d= 3), we choose the same forms of\nJ\u000b\u000b\nqandDqas Eqs. (19) and (20), but with Q1=Q^x,\nQ2=Q^y, andQ3=Q^z[see Eq. (10)]. We consider three\nsets ofJ\u000b\u000b\nQ\u0011andDQ\u0011as described below; see Table I. See\nalso Fig. 19 for the spin con\fgurations stabilized in each\nmodel.\nThe \frst is the one which can stabilize a superposition\nof three proper-screw spirals. In this case, to align the\nhelical plane perpendicular to the corresponding helical\ndirection, we set DQas\nDQ\u0011=8\n><\n>:D^x;(\u0011= 1)\nD^y;(\u0011= 2)\nD^z;(\u0011= 3): (29)\nFor the symmetric part, we introduce the anisotropy\ncompatible with C3rotational symmetry about the [111]\naxis, that is,\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n8\n><\n>:[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 1)\n[J(1\u0000\u0001);J(1\u0000\u0001);J(1 + 2\u0001)];(\u0011= 2)\n[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 3):(30)The pictorial representations of J\u000b\u000b\nqandDqare shown\nin Fig. 4(a). We call the 3 Qspin state realized in this\nsetting the proper-screw HL.\nThe second one is for realizing a superposition of three\ncycloidal spirals. This is obtained by \u00002\u0019=3 spin rotation\nof the proper-screw HL about the [111] axis, and hence,\nwe set\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n8\n><\n>:[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 1)\n[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 2)\n[J(1\u0000\u0001);J(1\u0000\u0001);J(1 + 2\u0001)];(\u0011= 3);(31)\nDQ\u0011=8\n><\n>:D^z;(\u0011= 1)\nD^x;(\u0011= 2)\nD^y;(\u0011= 3): (32)\nSee Fig. 4(b). We call the 3 Qspin state stabilized in this\nsetting the cycloid(I) HL.\nThe last one is for realizing a di\u000berent superposition\nof three cycloidal spirals. This is obtained by additional\n\u00002\u0019=3 spin rotation about the [111] axis, and hence, we\nset\n(Jxx\nQ\u0011;Jyy\nQ\u0011;Jzz\nQ\u0011) =\n8\n><\n>:[J(1\u0000\u0001);J(1\u0000\u0001);J(1 + 2\u0001)];(\u0011= 1)\n[J(1 + 2\u0001);J(1\u0000\u0001);J(1\u0000\u0001)];(\u0011= 2)\n[J(1\u0000\u0001);J(1 + 2\u0001);J(1\u0000\u0001)];(\u0011= 3);(33)\nDQ\u0011=8\n><\n>:D^y;(\u0011= 1)\nD^z;(\u0011= 2)\nD^x;(\u0011= 3); (34)\nas shown in Fig. 4(c). We call the 3 Qspin state stabilized\nin this setting the cycloid(II) HL.\nD. Finite-range model\nAs introduced in Sec. II B, the model in Eq. (1) has\nbeen studied in the limit of the in\fnite-range interac-\ntions in Eqs. (8) and (9). In the following, we extend\nthe model by introducing spatial decay in the interac-\ntions. After explaining the extension in detail for the\n1D case in Sec. II D 1, we describe the 2D and 3D cases\nin Secs. II D 2 and II D 3, respectively. Throughout this\nsection, we assume the set of the coupling constants for\nthe proper-screw states \frstly introduced for each dimen-\nsional case in Sec. II C; the extensions to the other sets are\nstraightforward by using the spin rotations introduced\nabove.6\n1. One-dimensional case\nJq/JDq/D-\u0001-\u0001\u0001\u0001\u0001\u0001-\u0001\u0002\u0003-\u0003\u0002\u0004\u0003\u0002\u0004\u0001\u0002\u0003<latexit sha1_base64=\"VeKlLpA9elKfa1zVxjn7PbUwt7g=\">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomIeix68diC/YA2lM120q7dbOLuRiihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6nfqtJ1Sax/LejBP0IzqQPOSMGivVH3ulsltxZyDLxMtJGXLUeqWvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYslTRC7WezQyfk1Cp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDaz7hMUoOSzReFqSAmJtOvSZ8rZEaMLaFMcXsrYUOqKDM2m6INwVt8eZk0zyveZcWtX5SrN3kcBTiGEzgDD66gCndQgwYwQHiGV3hzHpwX5935mLeuOPnMEfyB8/kD3UOM+g==</latexit>q\n<latexit sha1_base64=\"BChiLIkIeKtMnz+DOJz908oNMLo=\">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEtMeCF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1Fip2RyUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasKan3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1W9m6rbvK7Ua3kcRTiDc7gED26hDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBqcGM0A==</latexit>Q\n<latexit sha1_base64=\"iaH4NBgCY0OLb3otPrN/X4jkA10=\">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyUR0R4LXjy2Ym2hDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjRx2nimGLxSJWnYBqFFxiy3AjsJMopFEgsB2Mb2d++wmV5rF8MJME/YgOJQ85o8ZK9xfNfrniVt05yCrxclKBHI1++as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXUsljVD72fzSKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP66YmrPkZl0lqULLFojAVxMRk9jYZcIXMiIkllClubyVsRBVlxoZTsiF4yy+vksfLqndddZtXlXotj6MIJ3AK5+DBDdThDhrQAgYhPMMrvDlj58V5dz4WrQUnnzmGP3A+fwATFY0H</latexit>\u0000Q<latexit sha1_base64=\"K7u0QFDKHbcLJKWNfjieRuaTHYc=\">AAAB/nicbVDLSgMxFM3UV62vUXHlJliEKqXMqKjLohtx1YJ9QDsOmTRtY5PMmGSEMhT8FTcuFHHrd7jzb0zbWWj1wIXDOfdy7z1BxKjSjvNlZebmFxaXssu5ldW19Q17c6uuwlhiUsMhC2UzQIowKkhNU81IM5IE8YCRRjC4HPuNByIVDcWNHkbE46gnaJdipI3k2zt3ftLuIc5RsXp76F8Xq6PC/YFv552SMwH8S9yU5EGKim9/tjshjjkRGjOkVMt1Iu0lSGqKGRnl2rEiEcID1CMtQwXiRHnJ5PwR3DdKB3ZDaUpoOFF/TiSIKzXkgenkSPfVrDcW//Nase6eewkVUayJwNNF3ZhBHcJxFrBDJcGaDQ1BWFJzK8R9JBHWJrGcCcGdffkvqR+V3NPScfUkX75I48iCXbAHCsAFZ6AMrkAF1AAGCXgCL+DVerSerTfrfdqasdKZbfAL1sc3i9CUkg==</latexit>j\u0000,Q⇤J,Q(q)\n<latexit sha1_base64=\"blafztd2zm5sbjFdocI4UYXSrV0=\">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1iEKqUkKuqyqAuXLdgHtDFMJpN26EwSZyZCCQV/xY0LRdz6He78G6dtFtp64MLhnHu59x4vZlQqy/o2cguLS8sr+dXC2vrG5pa5vdOUUSIwaeCIRaLtIUkYDUlDUcVIOxYEcY+Rlje4HvutRyIkjcI7NYyJw1EvpAHFSGnJNfd8N+32EOeoXL8/dm/K9VHp4cg1i1bFmgDOEzsjRZCh5ppfXT/CCSehwgxJ2bGtWDkpEopiRkaFbiJJjPAA9UhH0xBxIp10cv4IHmrFh0EkdIUKTtTfEyniUg65pzs5Un05643F/7xOooJLJ6VhnCgS4umiIGFQRXCcBfSpIFixoSYIC6pvhbiPBMJKJ1bQIdizL8+T5knFPq+c1s+K1assjjzYBwegBGxwAargFtRAA2CQgmfwCt6MJ+PFeDc+pq05I5vZBX9gfP4AeQSUhg==</latexit>d\u0000,Q⇤D,Q(q)00.0\nFIG. 5. qdependences of j\r;Q\u0003\nJ;Q(q) [Eq. (44)] and\nd\r;Q\u0003\nD;Q(q) [Eq. (45)] in the 1D \fnite-range model. We take\n\u0003 = 16 and \r= 0:2.Q\u0003\nJandQ\u0003\nDare set toQ\u0003\nJ'0:396\nandQ\u0003\nD'0:390 so that J\u000b\u000b\nqandjDqjtake their maxima at\nq=\u0006Q=\u0006\u0019=8.\nLet us begin with the real-space representation of the\n1D in\fnite-range model with Eqs. (11) and (12). By the\nFourier transformation, the Hamiltonian reads\nH=\u00002\nLX\n`;`1X\n\u000b;\fJ\u000b\f\nQS\u000b\n`S\f\n`\u0000`1e\u0000iQ`1; (35)\nwhere the sum of `1runs over all the integers in the\nrange [\u0000L=2;L=2). Here and hereafter, we assume that\nthe helix has a commensurate period to the lattice for\nsimplicity; namely, L=\u0003 is an integer. To introduce spa-\ntial decay in the in\fnite-range interactions, we multiply\nan exponential dumping factor as\nH! ~H=\u00002\nLX\n`;`1X\n\u000b;\fJ\u000b\f\nQS\u000b\n`S\f\n`\u0000`1e\u0000iQ`1e\u0000\rj`1j:(36)\nFor su\u000eciently large L, the modi\fed Hamiltonian ~Hcan\nbe expressed as\n~H=\u00002\nLX\nq21BZX\n\u000b;\fJ\u000b\f\nQS\u000b\nqS\f\n\u0000qf\r;Q(q); (37)\nwhere\nf\r;Q(q) =sinh\r\ncosh\r\u0000cos(Q\u0000q): (38)\nThis function f\r;Q(q) for\r\u001c1 is well approximated\nnearq=Qby the Lorentzian function as\nf\r;Q(q)\u00192\r\n\r2+ (Q\u0000q)2: (39)\nBy symmetrizing the terms of \u0006q, we end up with the\nHamiltonian in the form\n~H=\u0000X\nq21BZX\n\u000b;\f~J\u000b\f\nqS\u000b\nqS\f\n\u0000q; (40)where\n~J\u000b\f\nq=(J\u000b\u000b\nQ\nL[f\r;Q(q) +f\r;Q(\u0000q)];(\u000b=\f)\nJ\u000b\f\nQ\nL[f\r;Q(q)\u0000f\r;Q(\u0000q)];(\u000b6=\f):(41)\nThe model in Eq. (40) stabilizes a spin helix whose pe-\nriod deviates from \u0003 = 2 \u0019=Q because the peaks of j~J\u000b\f\nqj\nare shifted due to the factors of f\r;Q(\u0006q). To facilitate\nthe following analyses, we adjust the form of the interac-\ntions so thatj~J\u000b\f\nqjhave peaks exactly at q=\u0006Qand the\nperiod of the stable spin helix becomes \u0003 = 2 \u0019=Q. This is\nachieved by replacing f\r;Q(\u0006q) byf\r;Q\u0003(\u0006q), whereQ\u0003\nis determined so that the derivative of the correspond-\ning coupling constant with respect to qbecomes zero at\nq=\u0006Q. In addition, we rescale all the elements of ~J\u000b\f\nq\nindividually so that they take the same values with the\nin\fnite-range model at q=\u0006Q, namely, ~J\u000b\f\nQ=J\u000b\f\nQ.\nThen, \fnally we obtain the Hamiltonian with the \fnite-\nrange interactions in the same form of Eqs. (1) and (2)\nwith\nJ\u000b\u000b\nq=J\u000b\u000b\nQj\r;Q\u0003\nJ;Q(q); (42)\nDq=DQd\r;Q\u0003\nD;Q(q); (43)\nwhereJ\u000b\u000b\nQandDQare given in Eqs. (15) and (14),\nrespectively, for the proper-screw HS case; j\r;Q\u0003\nJ;Qand\nd\r;Q\u0003\nJ;Qare de\fned as\nj\r;Q\u0003\nJ;Q(q) =f\r;Q\u0003\nJ(q) +f\r;Q\u0003\nJ(\u0000q)\nf\r;Q\u0003\nJ(Q) +f\r;Q\u0003\nJ(\u0000Q); (44)\nd\r;Q\u0003\nD;Q(q) =f\r;Q\u0003\nD(q)\u0000f\r;Q\u0003\nD(\u0000q)\nf\r;Q\u0003\nD(Q)\u0000f\r;Q\u0003\nD(\u0000Q): (45)\nHere,Q\u0003\nJandQ\u0003\nDare determined by solving\n@j\r;Q\u0003\nJ;Q(q)\n@q\f\f\f\f\nq=Q= 0; (46)\n@d\r;Q\u0003\nD;Q(q)\n@q\f\f\f\f\nq=Q= 0; (47)\nrespectively. Figure 5 exempli\fes j\r;Q\u0003\nJ;Q(q) and\nd\r;Q\u0003\nD;Q(q) for \u0003 = 16 and \r= 0:2. The other cases\nfor the cycloidal HS states are obtained by the spin ro-\ntations in Sec. II C 1.7\n2. Two-dimensional case\nFIG. 6. q= (qx;qy) dependences of (a) Jxx\nq, (b)Jyy\nq, (c)Jzz\nq,\nand (d)jDqjin the 2D \fnite-range model. The color bar is\ncommon to (a{d), while jDqjin (d) is plotted by multiplying\na factor of 10 for better visibility. (e) Pro\fle of (a{d) along\nthe path (\u0019;0){(0;0){(0;\u0019) [white dotted lines in (a{d)]. We\ntake \u0003 = 16, \r= 0:2,J= 1,D= 0:2, and \u0001 = 0 :3. The\nvalues forQ\u0003\nJandQ\u0003\nDare the same as those in Fig. 5.\nFollowing the 1D case, we can construct the \fnite-\nrange model in two dimensions. The Hamiltonian also\nhas the same form of Eqs. (1) and (2). Using the func-\ntionsj\r;Q\u0003\nJ;Q(q) andd\r;Q\u0003\nD;Q(q) in Eqs. (44) and (45),\nrespectively, the coupling constants for the symmetric\ninteractions are given as\nJ\u000b\u000b\nq=J\u000b\u000b\nQ1j\r;Q;Q\u0003\nJ(qx)j\r;0;0(qy)\n+J\u000b\u000b\nQ2j\r;0;0(qx)j\r;Q;Q\u0003\nJ(qy); (48)\nwithJ\u000b\u000b\nQ\u0011in Eq. (22), and those for the antisymmetric\ninteractions are given as\nDq=Dqq\njqj; (49)\nwhere\nDq=D\u0002\f\fd\r;Q;Q\u0003\nD(qx)j\r;0;0(qy)\f\f\n+\f\fj\r;0;0(qx)d\r;Q;Q\u0003\nD(qy)\f\f\u0003\n; (50)\nfor the case of the proper-screw(I) VC. Figure 6 exem-\npli\fesJ\u000b\u000b\nqandDqfor \u0003 = 16, \r= 0:2,J= 1,D= 0:2,\nand \u0001 = 0:3. The other cases are obtained by the proper\nspin rotations in Sec. II C 2.\nFIG. 7. q= (qx;qy;qz) dependences of [(a){(c)] Jxx\nq, [(d){\n(f)]Jyy\nq, [(g){(i)]Jzz\nq, and [(j){(l)]jDqjin the 3D \fnite-range\nmodel. (a,d,g,j), (b,e,h,k), and (c,f,i,l) show the qz= 0,qx=\n0, andqy= 0 planes, respectively. The color bar is common to\n(a{l), whilejDqjin (j,k,l) are plotted by multiplying a factor\nof 5 for better visibility. (m) Pro\fle along the white dotted\nlines in (a{l). We take \u0003 = 12, \r= 0:2,J= 1,D= 0:3, and\n\u0001 = 0:3.Q\u0003\nJandQ\u0003\nDare set toQ\u0003\nJ= 0:525 andQ\u0003\nD= 0:522\nso thatJ\u000b\u000b\nqandjDqjtake their maxima at q=\u0006Q=\u0006\u0019=6.\n3. Three-dimensional case\nIn a similar manner, we can obtain the forms of the\n\fnite-range interactions for the 3D case as\nJ\u000b\u000b\nq=J\u000b\u000b\nQ1j\r;Q;Q\u0003\nJ(qx)j\r;0;0(qy)j\r;0;0(qz)\n+J\u000b\u000b\nQ2j\r;0;0(qx)j\r;Q;Q\u0003\nJ(qy)j\r;0;0(qz)\n+J\u000b\u000b\nQ3j\r;0;0(qx)j\r;0;0(qy)j\r;Q;Q\u0003\nJ(qz); (51)\nDq=D\u0002\f\fd\r;Q;Q\u0003\nD(qx)j\r;0;0(qy)j\r;0;0(qz)\f\f\n+\f\fj\r;0;0(qx)d\r;Q;Q\u0003\nD(qy)j\r;0;0(qz)\f\f\n+\f\fj\r;0;0(qx)j\r;0;0(qy)d\r;Q;Q\u0003\nD(qz)\f\f\u0003\n; (52)\nwhere we use Eqs. (30) and (49) for J\u000b\u000b\nQ\u0011andDq, re-\nspectively, in the case of the proper-screw HL. Figure 7\nexempli\fes J\u000b\u000b\nqandDqfor \u0003 = 12, \r= 0:2,J= 1,\nD= 0:3, and \u0001 = 0 :3. The other cases are obtained by\nthe proper spin rotations in Sec. II C 3.8\nIII. METHODS\nA. Variational method\nIn this study, we investigate the spin excitation spec-\ntrum of the stable ground state for each model introduced\nin the previous section. For this purpose, we \frst deter-\nmine the ground state by using variational calculations\nin the classical limit where Sris regarded as a 3D vector\nwith \fxed length of jSrj= 1. In the case of the isotropic\nsymmetric interactions (\u0001 = 0), we perform the varia-\ntional calculation analytically by assuming a 1 QHS state\nwith a uniform twist in all dimensions, as we do not \fnd\nany other lower-energy state in the numerical variational\ncalculation described below. Meanwhile, in the presence\nof the spin anisotropy with \u0001 6= 0, we employ the numer-\nical variational calculation, as the 1 QHS state is mod-\nulated and other multiple- Qspin states may have lower\nenergy. In the numerical calculation, starting from sev-\neral di\u000berent initial spin con\fgurations (see below), we\ndetermine the lowest-energy state by optimization of the\nindividual spin orientation taking into account the inter-\nnal magnetic \feld from the other spins and the single-ion\nanisotropy ( S\u000b\nr)2appearing in the real-space form of the\nHamiltonian. As the initial spin con\fgurations, we take\ninto account a 1 QHS state with a uniform twist for the\n1D case, the 1 Qstate and a 2 QVC [57] for the 2D case,\nand the 1Qand 2Qstates and a 3 QHL [58] for the 3D\ncase; in each state, we set an appropriate helical plane\ndepending on the type of Dq, namely, the proper-screw\ntype ( Dqkq) or cycloid type ( Dq?q). We \frst perform\nthe numerical calculations for the in\fnite-range model\nwith\r= 0, and then, study the \fnite-range model with\n\r > 0 starting from the solution for the in\fnite-range\nmodel as the initial state.\nB. Linear spin-wave theory\nFor the stable spin con\fguration obtained by the vari-\national method, we study the spin excitation by using\nthe linear spin-wave theory. For the 1D 1 QHS states\nwith uniform twists, we obtain the analytic form of\nthe excitation spectra regardless of the range of interac-\ntions (Sec. IV A 1). Meanwhile, for the anisotropic cases\n(\u0001>0) as well as the 2D and 3D cases, we perform the\nspin-wave calculations numerically as follows. For each\nstable spin con\fguration, we introduce new local spin\naxes at each site so that all the spins point to the zdi-\nrection. We denote the spins in the new spin frame as ~Sr.\nThen, the stable spin con\fguration is regarded as a fer-\nromagnetic state, namely, ~Sr=^zfor all r. We apply the\nHolstein-Primako\u000b transformation to the Hamiltonian in\nthe new spin frame, leaving the lowest order of bosonicoperators:\n2\n64~Sx\nr\n~Sy\nr\n~Sz\nr3\n75!2\n664q\nS\n2(ar+ay\nr)q\nS\n21\ni(ar\u0000ay\nr)\nS\u0000ay\nrar3\n775; (53)\nwherearanday\nrrepresent the annihilation and creation\noperators of magnon at site r, respectively; Sis the spin\nquantum number of Sr.\nWe denote the spatial coordinate rasr=R+r0,\nwhere Randr0are the position vectors of each mag-\nnetic unit cell and the sublattice site within the unit\ncell, respectively: fRjR\u0016= \u0003N\u0016;N\u00162[0;L=\u0003)gand\nfr0jr\u0016\n02[0;\u0003)g, whereN\u0016andr\u0016\n0are integers. The\nBrillouin zone is folded from fqj\u0000\u0019\u0014q\u0016< \u0019gto\nfKj\u0000\u0019=\u0003\u0014K\u0016<\u0019= \u0003gunder the magnetic order with\nperiod of \u0003. Using the Fourier transformation\naK;r0=\u0012\u0003\nL\u0013dX\nRaR+r0e+iK\u0001R; (54)\nwe obtain the linear spin-wave Hamiltonian expressed as\nHSW=S\n2\u0003dX\nK0\n\u000by\nKAK\u000bK; (55)\nwhere\n\u000by\nK= [aK+;a\u0000K+;aK\u0000;a\u0000K\u0000]; (56)\nwith\naK+=h\nay\nK;(0;\u0001\u0001\u0001);\u0001\u0001\u0001;ay\nK;r;\u0001\u0001\u0001;ay\nK;(\u0003\u00001;\u0001\u0001\u0001)i\n;(57)\naK\u0000=h\naK;(0;\u0001\u0001\u0001);\u0001\u0001\u0001;aK;r;\u0001\u0001\u0001;aK;(\u0003\u00001;\u0001\u0001\u0001)i\n:(58)\nIn Eq. (55), AKis a 4\u0003d\u00024\u0003dmatrix for generic K,\nwhile it becomes a 2\u0003d\u00022\u0003dmatrix for K= 0 or on the\nzone boundary; each term in the sum of Kincludes all\nthe contributions from a\u0006K;r0anday\n\u0006K;r0, and the sumP0\nKruns over a half of the folded Brillouin zone (e.g.,\nKx\u00150). By the Bogoliubov transformation [64], the\nHamiltonian is diagonalized as\nHSW=X\nK0X\np\"Kpby\nKpbKp+ const:; (59)\nwhere\"Kp>0 represents the pth spin-wave dispersion\n(p= 1;\u0001\u0001\u0001;dim[AK]=2), andbKpandby\nKprepresent the\nannihilation and creation operators of a bosonic quasipar-\nticle, respectively, which are given by linear combinations\nofa\u0006K;r0anday\n\u0006K;r0.\nBy using the linear spin-wave theory, we evaluate the\ndynamical spin structure factor given by\nS\u0016\u0017(q;!) =\u000f\n\u0019X\nK0X\nphvacjS\u0016\n\u0000qjKpihKpjS\u0017\nqjvaci\n(!\u0000\"Kp)2+\u000f2;(60)9\nwherejvaciis the vacuum of the quasiparticles b,jKpi=\nby\nKpjvaci, and\u000fcorresponds to the relaxation rate. In\ninelastic neutron scattering experiments, the transverse\ncomponents to the incident wave number qare ob-\nserved [65]. Thus, we study the transverse component\nof the dynamical spin structure factor de\fned as\nS?(q;!) =S\u00161\u00161(q;!) +S\u00162\u00162(q;!); (61)\nwhere\u00161and\u00162are the two orthogonal directions per-\npendicular to q, e.g.,\u00161=yand\u00162=zforqk^x.\nFurthermore, using a polarized neutron beam, two trans-\nverse components can be decomposed by measuring the\nspin-\rip and non-spin-\rip cross sections.\nIV. RESULTS\nA. One-dimensional magnetic helices\nFirst, we present the results for the 1D HS states. In\nSec. IV A 1, we discuss the case of the isotropic symmetric\ninteraction, where the stable state has a uniform twist.\nIn this case, we can derive the analytic forms of the spin-\nwave dispersion and the dynamical spin structure factor.\nWe discuss their dependences on the interaction range\n\r, including the limit of D!0. In Sec. IV A 2, we\nnumerically show that the anisotropy \u0001 makes the twist\nof the HS state nonuniform, accordingly, modulates the\nexcitation spectra. Finally, in Sec. IV A 3, we study the\nlowest-energy excitation mode.\n1. Uniform helical spin state in the isotropic case\n\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0007\u0002\u0001\u0001\u0007\u0003\b\u0004<latexit sha1_base64=\"y6d1t3DDt8sJS4WZVB5osJ05DNE=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKe6KosegHsRTRPOAZAmzk04yZHZ2mZkVwpJP8OJBEa9+kTf/xkmyB00saCiquunuCmLBtXHdbye3tLyyupZfL2xsbm3vFHf36jpKFMMai0SkmgHVKLjEmuFGYDNWSMNAYCMYXk/8xhMqzSP5aEYx+iHtS97jjBorPdyc3HWKJbfsTkEWiZeREmSodopf7W7EkhClYYJq3fLc2PgpVYYzgeNCO9EYUzakfWxZKmmI2k+np47JkVW6pBcpW9KQqfp7IqWh1qMwsJ0hNQM9703E/7xWYnqXfsplnBiUbLaolwhiIjL5m3S5QmbEyBLKFLe3EjagijJj0ynYELz5lxdJ/bTsnZfd+7NS5SqLIw8HcAjH4MEFVOAWqlADBn14hld4c4Tz4rw7H7PWnJPN7MMfOJ8/mq+NWQ==</latexit>D/J\n<latexit sha1_base64=\"4cRCdYHHu1DaaLluXXPLd6Zad/s=\">AAAB+3icbVDLTsMwENzwLOUVypGLRYVULiVBIDhWcEGciqAPqY0qx3Vaq44TbKeiivorXDiAEFd+hBt/g9vmAC0jrTSe2ZV3x485U9pxvq2l5ZXVtfXcRn5za3tn194r1FWUSEJrJOKRbPpYUc4ErWmmOW3GkuLQ57ThD64nfmNIpWKReNCjmHoh7gkWMIK1kTp2oT3EksaKcfN6PCnd3h937KJTdqZAi8TNSBEyVDv2V7sbkSSkQhOOlWq5Tqy9FEvNCKfjfDtRNMZkgHu0ZajAIVVeOt19jI6M0kVBJE0Jjabq74kUh0qNQt90hlj31bw3Ef/zWokOLr2UiTjRVJDZR0HCkY7QJAjUZZISzUeGYCKZ2RWRPpaYaBNX3oTgzp+8SOqnZfe87NydFStXWRw5OIBDKIELF1CBG6hCDQg8wTO8wps1tl6sd+tj1rpkZTP78AfW5w9Ih5Pv</latexit>\"q/(JS)<latexit sha1_base64=\"tIV8jVDPdQc95usbwHAKt/YE8VM=\">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ9OKxov2ANpTNdtIu3Wzi7kYooT/BiwdFvPqLvPlv3LY5aPXBwOO9GWbmBYng2rjul1NYWl5ZXSuulzY2t7Z3yrt7TR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMrqd+6xGV5rG8N+ME/YgOJA85o8ZKdw+Xbq9ccavuDOQv8XJSgRz1Xvmz249ZGqE0TFCtO56bGD+jynAmcFLqphoTykZ0gB1LJY1Q+9ns1Ak5skqfhLGyJQ2ZqT8nMhppPY4C2xlRM9SL3lT8z+ukJrzwMy6T1KBk80VhKoiJyfRv0ucKmRFjSyhT3N5K2JAqyoxNp2RD8BZf/kuaJ1XvrOrenlZqV3kcRTiAQzgGD86hBjdQhwYwGMATvMCrI5xn5815n7cWnHxmH37B+fgGzW+New==</latexit>q=0\n<latexit sha1_base64=\"9aNzNTd+JoEPniZXZ73Ns5ft870=\">AAAB8nicbVDLSgNBEJyNrxhfUY9eBoPgKeyKqMegF48RzAM2S5idzCZD5rHM9AphyWd48aCIV7/Gm3/jJNmDJhY0FFXddHfFqeAWfP/bK62tb2xulbcrO7t7+wfVw6O21ZmhrEW10KYbE8sEV6wFHATrpoYRGQvWicd3M7/zxIzlWj3CJGWRJEPFE04JOCnMe0ZiDSNmpv1qza/7c+BVEhSkhgo0+9Wv3kDTTDIFVBBrw8BPIcqJAU4Fm1Z6mWUpoWMyZKGjikhmo3x+8hSfOWWAE21cKcBz9fdETqS1Exm7TklgZJe9mfifF2aQ3EQ5V2kGTNHFoiQTGDSe/Y8H3DAKYuIIoYa7WzEdEUMouJQqLoRg+eVV0r6oB1d1/+Gy1rgt4iijE3SKzlGArlED3aMmaiGKNHpGr+jNA+/Fe/c+Fq0lr5g5Rn/gff4AetGRYg==</latexit>other\n<latexit sha1_base64=\"3/BaNWSnq+Bll3RRMhbIOKARzt8=\">AAAB7nicbVBNSwMxEJ31s9avqkcvwSJ4Krsi6kUoevHYgv2AdinZNNuGJtmYZIWy9Ed48aCIV3+PN/+NabsHbX0w8Hhvhpl5keLMWN//9lZW19Y3Ngtbxe2d3b390sFh0ySpJrRBEp7odoQN5UzShmWW07bSFIuI01Y0upv6rSeqDUvkgx0rGgo8kCxmBFsntR5vukqgeq9U9iv+DGiZBDkpQ45ar/TV7SckFVRawrExncBXNsywtoxwOil2U0MVJiM8oB1HJRbUhNns3Ak6dUofxYl2JS2aqb8nMiyMGYvIdQpsh2bRm4r/eZ3UxtdhxqRKLZVkvihOObIJmv6O+kxTYvnYEUw0c7ciMsQaE+sSKroQgsWXl0nzvBJcVvz6Rbl6m8dRgGM4gTMI4AqqcA81aACBETzDK7x5ynvx3r2PeeuKl88cwR94nz+koo8d</latexit>q=±Q\n<latexit sha1_base64=\"mMcT4cVOE17Yt++87nUM5LQDuao=\">AAAB73icbVBNSwMxEJ2tX7V+VT16CRbBU9ktUr0IRS8eW7Af0C4lm2bb0CS7TbJCWfonvHhQxKt/x5v/xrTdg7Y+GHi8N8PMvCDmTBvX/XZyG5tb2zv53cLe/sHhUfH4pKWjRBHaJBGPVCfAmnImadMww2knVhSLgNN2ML6f++0nqjSL5KOZxtQXeChZyAg2VupMbnuxqKBGv1hyy+4CaJ14GSlBhnq/+NUbRCQRVBrCsdZdz42Nn2JlGOF0VuglmsaYjPGQdi2VWFDtp4t7Z+jCKgMURsqWNGih/p5IsdB6KgLbKbAZ6VVvLv7ndRMT3vgpk3FiqCTLRWHCkYnQ/Hk0YIoSw6eWYKKYvRWREVaYGBtRwYbgrb68TlqVslctu42rUu0uiyMPZ3AOl+DBNdTgAerQBAIcnuEV3pyJ8+K8Ox/L1pyTzZzCHzifPxe0j1k=</latexit>q=±2Q0.0\nFIG. 8.Ddependence of the spin excitation energy \"qof\nthe in\fnite-range model ( \r= 0) with the isotropic symmetric\ninteractions (\u0001 = 0) in one dimension for q= 0 (blue), q=\n\u0006Q(orange),q=\u00062Q(green), and the other generic q(red)\n[Eq. (74)].\nJ↵↵q/J\n<latexit sha1_base64=\"4pmMcy2fn+Jg+hZgg/9M9CgTyJU=\">AAAB/XicbVDLSgMxFL3js9bX+Ni5CRbBVZ2Rii6LbqSrCvYB7Thk0rQNzWTGJCPUofgrblwo4tb/cOffmLaz0NYD93I4515yc4KYM6Ud59taWFxaXlnNreXXNza3tu2d3bqKEklojUQ8ks0AK8qZoDXNNKfNWFIcBpw2gsHV2G88UKlYJG71MKZeiHuCdRnB2ki+vV+5S9uYx3087SP//qTi2wWn6EyA5ombkQJkqPr2V7sTkSSkQhOOlWq5Tqy9FEvNCKejfDtRNMZkgHu0ZajAIVVeOrl+hI6M0kHdSJoSGk3U3xspDpUahoGZDLHuq1lvLP7ntRLdvfBSJuJEU0GmD3UTjnSExlGgDpOUaD40BBPJzK2I9LHERJvA8iYEd/bL86R+WnRLxbObUqF8mcWRgwM4hGNw4RzKcA1VqAGBR3iGV3iznqwX6936mI4uWNnOHvyB9fkDVteVKA==</latexit><latexit 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sha1_base64=\"4cRCdYHHu1DaaLluXXPLd6Zad/s=\">AAAB+3icbVDLTsMwENzwLOUVypGLRYVULiVBIDhWcEGciqAPqY0qx3Vaq44TbKeiivorXDiAEFd+hBt/g9vmAC0jrTSe2ZV3x485U9pxvq2l5ZXVtfXcRn5za3tn194r1FWUSEJrJOKRbPpYUc4ErWmmOW3GkuLQ57ThD64nfmNIpWKReNCjmHoh7gkWMIK1kTp2oT3EksaKcfN6PCnd3h937KJTdqZAi8TNSBEyVDv2V7sbkSSkQhOOlWq5Tqy9FEvNCKfjfDtRNMZkgHu0ZajAIVVeOt19jI6M0kVBJE0Jjabq74kUh0qNQt90hlj31bw3Ef/zWokOLr2UiTjRVJDZR0HCkY7QJAjUZZISzUeGYCKZ2RWRPpaYaBNX3oTgzp+8SOqnZfe87NydFStXWRw5OIBDKIELF1CBG6hCDQg8wTO8wps1tl6sd+tj1rpkZTP78AfW5w9Ih5Pv</latexit>\"q/(JS)\n<latexit sha1_base64=\"ombNEeML1r4BDikZ0Rg+dFmsmjs=\">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRbBU90VUS9CUQ/iqYLbFtqlZNNsG5pklyQrlKW/wYsHRbz6g7z5b0zbPWjrg4HHezPMzAsTzrRx3W+nsLS8srpWXC9tbG5t75R39xo6ThWhPol5rFoh1pQzSX3DDKetRFEsQk6b4fBm4jefqNIslo9mlNBA4L5kESPYWMm/Pbm/crvlilt1p0CLxMtJBXLUu+WvTi8mqaDSEI61bntuYoIMK8MIp+NSJ9U0wWSI+7RtqcSC6iCbHjtGR1bpoShWtqRBU/X3RIaF1iMR2k6BzUDPexPxP6+dmugyyJhMUkMlmS2KUo5MjCafox5TlBg+sgQTxeytiAywwsTYfEo2BG/+5UXSOK1651X34axSu87jKMIBHMIxeHABNbiDOvhAgMEzvMKbI50X5935mLUWnHxmH/7A+fwBi8+N2w==</latexit>D/J=000.00\n0\n00.0\n0.0\n0.0FIG. 9. Interaction range \rdependences of (a) the sym-\nmetric interaction Jq=J\u000b\u000b\nq, (b) the antisymmetric interac-\ntionDq, and [(c) and (d)] the spin-wave dispersion \"qfor the\nisotropic case (\u0001 = 0) with \u0003 = 16 in one dimension. The\nantisymmetric interaction is set to (c) D=J = 0:2 and (d)\nD=J = 0. The black dots and lines in (c) and (d) represent\nthe results in the in\fnite-range limit of \r!0; see also Fig. 8.\nWe begin with a HS state in one dimension which\nis stable when the symmetric interactions are isotropic,\nnamely, \u0001 = 0 in Eq. (15). We here consider a proper-\nscrew spin state given by\nS`= [0;sin(Q`);cos(Q`)]t; (62)\nand derive the analytic forms of the dispersion of spin\nexcitation and the dynamical spin structure factor.\nThe Hamiltonian in Eq. (1) reads\nHq=\u0000\u0002\nJqSq\u0001S\u0000q+iDq\u0000\nSy\nqSz\n\u0000q\u0000Sz\nqSy\n\u0000q\u0001\u0003\n=\u00001\nLX\n`;`0[JqS`\u0001S`0\n+iDq(Sy\n`Sz\n`0\u0000Sz\n`Sy\n`0)]e\u0000iq(`\u0000`0);(63)\nwithJq\u0011J\u000b\u000b\nqandDqin Eqs. (42) and (43), respectively.\nBy substituting Eq. (62) with the rotation of the local\nspin axes, namely,\nS`=0\nB@0 1 0\n\u0000cos(Q`) 0 sin(Q`)\nsin(Q`) 0 cos(Q`)1\nCA~S`; (64)10\nFIG. 10. Anisotropy dependence of the dynamical spin structure factor S\u0016\u0016(q;!)=Sfor the proper-screw HS state in one\ndimension. The anisotropy \u0001 is set to [(a){(c)] \u0001 = 0 (the isotropic case), [(d){(f)] \u0001 = 0 :1, [(g){(i)] \u0001 = 0 :2, and [(j){(l)]\n\u0001 = 0:4. (a,d,g,j), (b,e,h,k), and (c,f,i,l) show the \u0016=x,y, andzcomponents, respectively. The color bar is common to all\nplots whereas the data for \u0001 = 0 :4 in (j{l) are plotted by multiplying a factor of 5 for better visibility. The parameters are\ntaken as \u0003 = 16, D=J = 0:2,\r= 0:3, and\u000f= 0:1.\nand applying the Holstein-Primako\u000b transformation in\nEq. (53), we obtain the linear spin-wave Hamiltonian as\nHSW=SX\nq21BZ\u001aJq\n2(a\u0000q\u0000ay\nq)(aq\u0000ay\n\u0000q)+2(J+D)ay\nqaq\n\u0000Jq+Dq\n4[a\u0000(q\u0000Q)+ay\nq\u0000Q][aq\u0000Q+ay\n\u0000(q\u0000Q)]\n\u0000Jq\u0000Dq\n4[a\u0000(q+Q)+ay\nq+Q][aq+Q+ay\n\u0000(q+Q)]\u001b\n=X\nq0\u0002\nay\nqa\u0000q\u0003\nAq\"\naq\nay\n\u0000q#\n+ const:; (65)\nwhere\nay\nq=1p\nLX\n`ay\n`e\u0000iq`; (66)Aq=S\"\nJq\"\n\u00001 1\n1\u00001#\n+ 2(J+D)\"\n1 0\n0 1#\n\u0000Jq+Q+Dq+Q+Jq\u0000Q\u0000Dq\u0000Q\n2\"\n1 1\n1 1##\n;(67)\nand the sumP0\nqin Eq. (65) runs over a half of the \frst\nBrillouin zone (e.g., q2[0;\u0019])1. Note that no term lin-\near to the bosonic operators appears as long as the HS\nstate in Eq. (62) is energetically stable. Using the Bo-\ngoliubov transformation, the Hamiltonian in Eq. (65) is\n1Strictly speaking, special treatment is required when q= 0 or\nq=\u0019, but in reality, the same result is obtained by considering\nthe limit of q!+0 orq!\u0019\u00000.11\ndiagonalized as\nUqAqUq=\"\n\"q0\n0\"q#\n; (68)\nwith\nUq=\"\ncosh\u0018qsinh\u0018q\nsinh\u0018qcosh\u0018q#\n; (69)\nwhere\n\u0018q=1\n4[lnBq\u0000lnCq]; (70)\nBq=J+D\u0000Jq; (71)\nCq=J+D\u0000Jq+Q+Dq+Q+Jq\u0000Q\u0000Dq\u0000Q\n2:(72)\nThe excitation spectrum is obtained as\n\"q= 2Sp\nBqCq: (73)\nLet us \frst consider the in\fnite-range limit ( \r= 0)\n[Eqs. (11) and (12)]. In this limit, \"qbecomesqinde-\npendent as \"q= 2S(J+D), except for the \u000e-functional\nchanges at q= 0,\u0006Q, and\u00062Q, namely,\n\"q=8\n>>><\n>>>:0; (q= 0)\n2Sp\nD(J+D); (q=\u0006Q)\nSp\n(J+D)(J+ 3D);(q=\u00062Q)\n2S(J+D); (otherq): (74)\nThe results are plotted as functions of D=J in Fig. 8.\nWe note that the \rat dispersion with excitation energy\n2S(J+D) originates from the term 2 S(J+D)P\n`ay\n`a`,\nindicating that the corresponding excitations are the lo-\ncal ones with reduction of the ~Sz\n`component at every\nsite [see Eq. (53)]. This is a pathological feature of the\nin\fnite-range model.\nNext, let us consider the \fnite-range model while\nchanging the interaction range \r[Eqs. (42) and (43)].\nFigures 9(a) and 9(b) show \rdependences of Jq=J\u000b\u000b\nq\nandDq=jDqj, respectively, for \u0003 = 16. While in-\ncreasing\r, the distributions of JqandDqinqspace get\nwider and qualitatively approach those of the model with\nthe nearest-neighbor interactions only: Jq=Jcosqand\nDq=Dsinq. Figure 9(c) shows the excitation spectrum\n\"qatD=J = 0:2. We \fnd that the spikes at q= 0,\u0006Q,\nand\u00062Qfor\r= 0 are broadened by increasing \r;\"qis\nalways zero at q= 0, accompanied by a linear dispersion\naround the gapless point for nonzero \r.\nMeanwhile, as indicated in Fig. 8, the spikes at q=\u0006Q\nfor\r= 0 also come down to zero energy when D!0.\nFigure 9(d) shows \"qin this limit. In this case, the broad-\nening by nonzero \rgives rise to gapless linear excitations\nat not only q= 0 but also q=\u0006Q. These three gapless\nmodes are commonly seen in the HS states appearing in\nspin models without the antisymmetric interactions, such\nas aJ1-J2model in one dimension [66, 67].Last, we derive the analytic form of the dynamical spin\nstructure factor S\u0016\u0016(q;!) de\fned by Eq. (60). Within\nthe linear spin-wave theory, by using Eqs. (53) and (64),\nwe replace the spin operators S\u0016\nqby linear combinations\nof the bosonic operators as\n2\n64Sx\nq\nSy\nq\nSz\nq3\n75!r\nS\n22\n641\ni(a\u0000q\u0000ay\nq)\n\u00001\n2(a\u0000(q\u0000Q)+ay\nq\u0000Q+a\u0000(q+Q)+ay\nq+Q)\n1\n2i(a\u0000(q\u0000Q)+ay\nq\u0000Q\u0000a\u0000(q+Q)\u0000ay\nq+Q)3\n75:\n(75)\nThen, using the Bogoliubov transformation [Eqs. (68){\n(72)], we obtain the diagonal components of the dynam-\nical spin structure factor as\nSxx(q;!) =S\u000f\n2\u0019\u0014e\u00002\u0018q\n(!\u0000\"q)2+\u000f2\u0015\n; (76)\nSyy(q;!) =Szz(q;!)\n=S\u000f\n8\u0019\u0014e2\u0018q+Q\n(!\u0000\"q+Q)2+\u000f2+e2\u0018q\u0000Q\n(!\u0000\"q\u0000Q)2+\u000f2\u0015\n;\n(77)\nwhere\u0018qis given by Eq. (70). Noting\ne\u00002\u0018q=s\n2(J+D)\u0000Jq+Q\u0000Dq+Q\u0000Jq\u0000Q+Dq\u0000Q\n2(J+D\u0000Jq)\nq!0\u0000\u0000\u0000!jqj; (78)\nwe can show the asymptotic behaviors:\nSxx(q;!)q!0\u0000\u0000\u0000!jqj; (79)\nSyy(q;!) =Szz(q;!)q!\u0006Q\u0000\u0000\u0000\u0000!jq\u0007Qj\u00001: (80)\nFigures 10(a), 10(b), and 10(c) show Sxx(q;!),\nSyy(q;!), andSzz(q;!), respectively, for the model with\n\u0003 = 16,D=J = 0:2, and\r= 0:3; we take \u000f= 0:1\nin Eqs. (76) and (77). Note that Sxx(q;!) vanishes as\nq!0, whileSyy(q;!) =Szz(q;!) diverge as q!\u0006Q,\nas shown in Eqs. (79) and (80). In the inelastic neu-\ntron scattering experiments, only the transverse compo-\nnents to the wave number q=q^x, namely,Syy(q;!) and\nSzz(q;!), can be observed, as mentioned below Eq. (60).\nThis means that for the proper-screw HS state the di-\nvergent behaviors at q!\u0006QinSyy(q;!) andSzz(q;!)\nare observable, but the q-linear mode around q= 0 with\nincreasing intensity for larger qinSxx(q;!) cannot be\nobserved. Note that when we consider a cycloidal HS\nstate, in which the spins are rotated by \u0019=2 about the\nzaxis from the proper-screw one (Fig. 1), Sxx(q;!) and\nSyy(q;!) are interchanged, and hence, the q-linear mode\nwith increasing intensity for larger qis observed in the\nSyy(q;!) component. Thus, the neutron scattering spec-\ntra are sensitive to the direction of the helical plane.\nSimilar behaviors were discussed for short-range mod-\nels [68, 69].12\n2. E\u000bect of magnetic anisotropy\nFIG. 11. E\u000bect of the anisotropy \u0001 in the symmetric inter-\nactions [Eq. (15)] for the 1D \fnite-range model with \u0003 = 16,\nD=J = 0:2, and\r= 0:3. (a{c) Stable spin con\fgurations\nfor (a) \u0001 = 0 :1, (b) \u0001 = 0 :2, and (c) \u0001 = 0 :4. The color\nof arrows indicates the zcomponent of spin according to the\ncolor bar in (c). (d) Inner product of nearest-neighbor spins,\nS`\u0001S`+1, representing the spatial modulation of the spin twist.\n(e) Spin excitation spectra for di\u000berent \u0001. (f) \u0001 dependences\nof the spin excitation gap \u0001 gapand the ratio of the Fourier\ncomponents of spins, RS=jSz\nQj=jSy\nQj.\nWhen we introduce the anisotropy \u0001 in the symmet-\nric interactions as Eq. (15), the proper-screw HS state\nis modulated from Eq. (62). Figures 11(a){11(c) show\nour numerical results for the stable spin con\fgurations\nobtained by the variational calculation in Sec. III A. We\n\fnd that the twist of the helix is modulated and becomes\nspatially nonuniform in the presence of the anisotropy \u0001,\nas more clearly shown in Fig. 11(d). This is because the\nspins tend to align to the \u0006^ydirection to gain energy\nfor \u0001>0. Indeed, we \fnd that the ratio of the Fourier\ncomponents of spins, RS=jSz\nQj=jSy\nQj, monotonically de-\n\u0001=x=y=z\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0001\u0005\u0001\u0007\u0001\b\u0001\t\u0001\u0004\u0001\u0001\u0004\u0005\u0001<latexit sha1_base64=\"d+r+5N6dCU6uNixACh19btwXKGo=\">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0hE1ItQ1IPHCvYD21A22027dLMJuxOhlP4LLx4U8eq/8ea/cdvmoNUHA4/3ZpiZF6ZSGPS8L6ewtLyyulZcL21sbm3vlHf3GibJNON1lshEt0JquBSK11Gg5K1UcxqHkjfD4fXUbz5ybUSi7nGU8iCmfSUiwSha6aFzwyXSS8/1u+WK53ozkL/Ez0kFctS65c9OL2FZzBUySY1p+16KwZhqFEzySamTGZ5SNqR93rZU0ZibYDy7eEKOrNIjUaJtKSQz9efEmMbGjOLQdsYUB2bRm4r/ee0Mo4tgLFSaIVdsvijKJMGETN8nPaE5QzmyhDIt7K2EDaimDG1IJRuCv/jyX9I4cf0z17s7rVSv8jiKcACHcAw+nEMVbqEGdWCg4Ale4NUxzrPz5rzPWwtOPrMPv+B8fAM1oY/1</latexit>\u0000=0.1\n<latexit 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The dynamical spin structure factor S\u0016\u0016(q;!)=S\natq=Qfor (a) \u0001 = 0 :1, (b) \u0001 = 0 :2, and (c) \u0001 = 0 :4. The\nmodel and parameters are the same as Fig. 11.\ncreases while increasing \u0001, as shown in Fig. 11(f). We\nnote that similar HS states with inhomogeneous twist\nwere studied for a short-range model [70] and observed\nin CuB 2O4[71] and TbMnO 3[72].\nFigure 11(e) shows how the spin-wave dispersion is\nchanged by \u0001. When \u0001 >0, the gap opens at K= 0 (K\nrepresents the wave number in the folded Brillouin zone\nas de\fned in Sec. III B) and monotonically increases with\nincreasing \u0001. The \u0001 dependence of \u0001 gapis plotted in\nFig. 11(f).\nFigure 10 displays the dynamical spin structure fac-\ntor for the modulated proper-screw HS states with sev-\neral values of \u0001. The spectra for \u0001 >0 are gapped\nre\recting the spin-wave excitation, although the gap is\nsmall and hardly seen in the spectra for \u0001 = 0 :1 and\n0:2. In addition, while increasing \u0001, the intensities at\nq=\u0006Qbecome weaker and the overall spectra become\ndi\u000busive. Looking more closely, we \fnd that a nonzero\n\u0001 makesSyy(q;!) di\u000berent from Szz(q;!);Syy(q;!) be-\ncomes larger than Szz(q;!) in the low-energy part around\nq=\u0006Q. This is more clearly seen in the !dependence\natq=Qshown in Fig. 12. The results are consis-\ntent withRSbeing smaller than 1 while increasing \u000113\n[Fig. 11(f)]. On the other hand, unlike Syy(Q;!) and\nSzz(Q;!),Sxx(Q;!) takes the largest value at a higher\nenergy around !=(JS)\u00190:7 as shown in Figs. 10 and\n12, whose energy scale roughly corresponds to \"\u0006Qin\nthe in\fnite-range limit [Eq. (74)]. The transverse com-\nponent of the dynamical spin structure factor, S?(q;!)\nin Eq. (61), for the proper-screw HS state is obtained by\nsetting\u00161=yand\u00162=zas discussed in the end of\nSec. IV A 1, and then the results for the other HS states\nare obtained by using the corresponding rotations in spin\nspace:\nS= (Sx;Sy;Sz);proper-screw HS state\n!(\nS= (\u0000Sy;Sx;Sz);cycloid(I) HS state\nS= (Sz;Sx;Sy);cycloid(II) HS state:(81)\nSinceSxx(q;!) is (not) observed in the proper-screw (cy-\ncloidal) HS state, the presence or absence of the higher-\nenergy intensity around !\u0018\"\u0006Qatq=Q^xin the in-\nelastic neutron scattering experiments can be an indica-\ntor for distinguishing the proper-screw and cycloidal HS\nstates. In addition, it is worth noting that the spectra\nfor \u0001>0 exhibit higher harmonics at q=\u00063Q, orig-\ninating from the modulation of the spin con\fgurations\nby the anisotropy. Such satellite peaks were observed\nin neutron scattering experiments of CuB 2O4[71] and\nTbMnO 3[72].\n3. Mode analysis\nLet us discuss the nature of the lowest-energy excita-\ntion mode. For this purpose, we consider a wave function\nj'i=jvaci+\u0015e\u0000i\"extjexi; (82)\nwhere\u0015denotes a mixing between the ground state jvaci\n(vacuum of magnons) and the lowest-energy excited state\njexiatK= 0:jexi=jK= 0;p= 1iwith excitation\nenergy of\"ex\u0011\"K=0;p=1[see Eq. (59)]. For simplicity,\nin this section, we assume the large Slimit, in which\nhvacj~Sz\n`jvaci=S. Then, in the linear spin-wave theory,\nthe expectation values of spins are computed as\nh'j~S\u0016\n`j'i=hvacj~S\u0016\n`jvaci\n+\u0015h\ne\u0000i\"exthvacj~S\u0016\n`jexi+ h:c:i\n+O(\u00152)\n\u0019S\u000e\u0016z+\u0015\n2Reh\ne\u0000i\"exthvacj~S\u0016\n`jexii\n=:hS\u0016\n`i:\n(83)\nFigure 13 shows the results of numerical calculations\nfor \u0001 = 0:1, 0:2, and 0:4 with\u0015= 0:1. Att= 0, all\nthe spins for the excited state 'are in the helical plane,\nnamelyhSx\n`i= 0. While increasing t, each spin shows an\nelliptically distorted precession, as schematically shown\nin the inset of Fig. 13(a). After the precession by \"ext=\n\u0019=2, theyzcomponents of spins are indistinguishable\n\u0001\u0002\u0003\u0004\u0005\u0006\u0005\u0001\u0005\u0002-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\u0006\u0006\u0007\b\u0005\u0007\u0006\n\u0001\u0002\u0003\u0004\u0005\u0006\u0005\u0001\u0005\u0002-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\u0006\u0006\u0007\b\u0005\u0007\u0006\n\u0001\u0002\u0003\u0004\u0005\u0006\u0005\u0001\u0005\u0002-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\u0006\u0006\u0007\b\u0005\u0007\u0006<latexit 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sha1_base64=\"Gc0S2iQjIDkdeyxxBqlim0RLF0M=\">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4qkkV7bHgxWMF0xbaUDbbTbt0swm7E6GU/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTKUw6Lrfztr6xubWdmGnuLu3f3BYOjpumiTTjPsskYluh9RwKRT3UaDk7VRzGoeSt8LR3cxvPXFtRKIecZzyIKYDJSLBKFrJ76bistorld2KOwdZJV5OypCj0St9dfsJy2KukElqTMdzUwwmVKNgkk+L3czwlLIRHfCOpYrG3AST+bFTcm6VPokSbUshmau/JyY0NmYch7Yzpjg0y95M/M/rZBjVgolQaYZcscWiKJMEEzL7nPSF5gzl2BLKtLC3EjakmjK0+RRtCN7yy6ukWa14N5Wrh+tyvZbHUYBTOIML8OAW6nAPDfCBgYBneIU3RzkvzrvzsWhdc/KZE/gD5/MHLtWOQA==</latexit>⇡/20\n0\n0FIG. 13. Time evolution of the lowest-energy excitation\nmode atK= 0 for (a) \u0001 = 0 :1, (b) \u0001 = 0 :2, and (c) \u0001 = 0 :4.\nThe data connected by the solid, dotted, and dashed lines\nindicate the spin con\fgurations obtained by Eq. (83) with\n\"ext= 0,\u0019=4, and\u0019=2, respectively, for \u0015= 0:1. The model\nand parameters are the same as Fig. 11. Inset of (a) shows\na schematic view of a spin precession motion (black arrow)\naround the ground state (gray arrow).\nfrom those of the ground state, and only the xcomponent\nis di\u000berent from the ground state. These features are\ncommonly seen regardless of \u0001.\nThe amplitude of the precession motion, however,\nstrongly depends on \u0001. When \u0001 is small, the spin\ncomponents in the helical plane, hSy\n`iandhSz\n`i, show\nlarge motions, while the perpendicular component hSx\n`i\nchanges much smaller, as exempli\fed in Fig. 13(a); the\nexcitation mode can be regarded as a phase shift of\nthe helix. With an increase of \u0001, the changes of hSy\n`i\nandhSz\n`i(hSx\n`i) are suppressed (enhanced), as shown in\nFig. 13(b). For larger \u0001, the spins are almost pinned in\nthe\u0006ydirections and the amplitude of the precession be-\ncomes small, as shown in Fig. 13(c); hSy\n`iandhSz\n`ishow\nalmost no change, while hSx\n`ioscillates near the regions\nwherehSy\n`ichanges its sign.14\nB. Two-dimensional vortex crystals\nNext, we present the results for the 2D VCs. In\nSec. IV B 1, we show the ground-state phase diagram for\nthe 2D model in the limit of in\fnite-range interactions\ncomputed by the variational calculations. In the phase\ndiagram, we \fnd that the anisotropy \u0001 favors the 2D\nVCs. In Sec. IV B 2, we examine the details of the 2D\nVCs and their stabilization mechanism. In Sec. IV B 3,\nwe discuss the dependence of the spin-wave dispersion\non the interaction range \r. Finally, in Sec. IV B 4, we\ndiscuss the dynamical spin structure factor computed by\nthe linear spin-wave theory.\n1. Phase diagram\n\u0001\u0002\u0001\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0001\u0002\u0007\u0001\u0002\u0001\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0001\u0002\u0007\n<latexit sha1_base64=\"LF+HG+B+z9UyHi1x9Et9Cc7DePE=\">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSSi6LGoB48V7Ae0oWy2m3btZjfsToQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MBHcoOd9O4WV1bX1jeJmaWt7Z3evvH/QNCrVlDWoEkq3Q2KY4JI1kKNg7UQzEoeCtcLRzdRvPTFtuJIPOE5YEJOB5BGnBK3U7N4ygaRXrnhVbwZ3mfg5qUCOeq/81e0rmsZMIhXEmI7vJRhkRCOngk1K3dSwhNARGbCOpZLEzATZ7NqJe2KVvhspbUuiO1N/T2QkNmYch7YzJjg0i95U/M/rpBhdBRmXSYpM0vmiKBUuKnf6utvnmlEUY0sI1dze6tIh0YSiDahkQ/AXX14mzbOqf1H17s8rtes8jiIcwTGcgg+XUIM7qEMDKDzCM7zCm6OcF+fd+Zi3Fpx85hD+wPn8AWHejwA=</latexit>\u0000<latexit 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sha1_base64=\"msMVS6j4aIEjadNOZHvkG3Zncw8=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKe6KqMegHsRTRPOAZAmzk04yZHZ2mZkVwpJP8OJBEa9+kTf/xkmyB00saCiquunuCmLBtXHdbye3tLyyupZfL2xsbm3vFHf36jpKFMMai0SkmgHVKLjEmuFGYDNWSMNAYCMYXk/8xhMqzSP5aEYx+iHtS97jjBorPdyc3HWKJbfsTkEWiZeREmSodopf7W7EkhClYYJq3fLc2PgpVYYzgeNCO9EYUzakfWxZKmmI2k+np47JkVW6pBcpW9KQqfp7IqWh1qMwsJ0hNQM9703E/7xWYnqXfsplnBiUbLaolwhiIjL5m3S5QmbEyBLKFLe3EjagijJj0ynYELz5lxdJ/bTsnZfd+7NS5SqLIw8HcAjH4MEFVOAWqlADBn14hld4c4Tz4rw7H7PWnJPN7MMfOJ8/mwONWg==</latexit>D/J\u0001 \u0001\u0002\u0003 \u0001\u0002\u0004 \u0001\u0002\u0005 \u0001\u0002\u0006 \u0007\u0002\u0001\nFIG. 14. Variational results for the in\fnite-range model in\ntwo dimensions. We take \u0003 = 16. (a) Phase diagram, includ-\ning the 1Qstate, the anisotropic 2 Qstate withjSQ1j6=jSQ2j,\nand the isotropic 2 Qstate withjSQ1j=jSQ2j. (b) Con-\ntour plot of the ratio of the Fourier components of spins,\nRS\u0011jS\u0016?;1\nQmaxj=jS\u0016?;2\nQmaxj, representing the ellipticity of the con-\nstituent spin helix; see the text for the de\fnition.\nFigure 14(a) shows the ground-state phase diagram for\nthe in\fnite-range model in two dimensions (Sec. II C 2)\nobtained by the variational calculations (Sec. III A) while\nchanging \u0001 and D=J. We take \u0003 = 16. The result is\nFIG. 15. Isotropic 2 Qspin states stabilized in the in\fnite-\nrange model in two dimensions with D=J = 0:2, \u0001 = 0:3,\nand \u0003 = 16: (a) proper-screw(I), (b) cycloid(I), (c) proper-\nscrew(II), and (d) cycloid(II) VCs. The forms of J\u000b\u000b\nQ\u0011and\nDQ\u0011in each case are summarized in Table I. The con\fgura-\ntions of DQ\u0011as well as Q\u0011are shown in each \fgure. The\ncolor of arrows indicates the zcomponent of spin according\nto the color bar in (a).\ncommon to all the settings of the proper-screw and cy-\ncloidal VCs listed in Table I. We \fnd three phases: the\n1Qstate, the anisotropic 2 Qstate wherejSQ1j6=jSQ2j,\nand the isotropic 2 Qstate wherejSQ1j=jSQ2j. All the\nphase transitions among the three states look continuous.\nIn the isotropic case (\u0001 = 0), the system always stabilizes\nthe 1Qstate forD> 0; it remains stable against nonzero\n\u0001, and the range becomes wider for larger D, as shown in\nFig. 14(a). In the larger \u0001 region, the system stabilizes\nthe 2Qstates, whose spin con\fgurations are noncoplanar\nexcept forD= 0. In the 2 Qstates, the constituent two\nspin helices are deformed from circular. To evaluate the\nellipticity, we extend the ratio RSde\fned in Sec. IV A 2\nto the present situation as RS=jS\u0016?;1\nQmaxj=jS\u0016?;2\nQmaxj, where\nQmaxisQ\u0011for largerjSQ\u0011j, and\u0016?;1and\u0016?;2denote the\nx,y, orzdirections perpendicular to DQmaxsatisfying\njS\u0016?;2\nQmaxj>jS\u0016?;1\nQmaxj(for instance, \u0016?;1=z, and\u0016?;2=y\nwhen DQmaxk^xandjSy\nQmaxj>jSz\nQmaxj). Note that\nRS= 0 in the isotropic 2 Qstate atD= 0 and \u0001 &1:6\nbecause the spin state becomes coplanar ( Sz\nQ\u0011= 0) com-\nposed of a superposition of two sinusoidal spin density\nwaves with equal weight. The calculated result of RSis\nplotted in Fig. 14(b). We \fnd that RSincreases while\nincreasingDand decreasing \u0001.15\n2. Vortex crystals\nWhile the phase diagram is common to all the settings\nof the proper-screw and cycloidal VCs listed in Table I,\nthe actual spin con\fguration in the 2 Qstate depends\non the type of interactions. We present the variational\nresults in Fig. 15, focusing on the isotropic 2 Qstate at\nD=J = 0:2 and \u0001 = 0 :3 (RS'0:31). Figure 15(a) shows\nthe stable spin con\fguration when taking J\u000b\u000b\nQ\u0011andDQ\u0011\nas Eqs. (22) and (21), respectively. This is the proper-\nscrew(I) VC. On the other hand, Fig. 15(b) shows the\nspin con\fguration obtained for Eqs. (23) and (24), which\nis the cycloid(I) VC. Likewise, Figs. 15(c) and 15(d) dis-\nplay the spin con\fgurations for Eqs. (25) and (26) and\nEqs. (27) and (28), which are the proper-screw(II) and\ncycloid(II) VCs, respectively. Note that the VCs of type\n(I) [(II)] can be regarded as square lattices with a stag-\ngered arrangement of merons and antimerons with vor-\nticity +1 (\u00001) [73]. Similarly, the spin con\fgurations of\nthe anisotropic 2 Qstate also form VCs where the vortices\nare deformed (not shown).\nThese VCs are stabilized by the anisotropy \u0001 in the\nsymmetric interactions, as suggested in the phase di-\nagram in Fig. 14(a). This can be directly con\frmed\nby calculating the spin components of SQ\u0011. We \fnd\nthat all the VCs have large values of jSy\nQ1j=jSx\nQ2j\nandjSx\nQ1j=jSy\nQ2jfor the proper-screw and cycloidal\nVCs, respectively: For example, at D=J = 0:2 and\n\u0001 = 0:3, we \fnd (jSx\nQ1j;jSy\nQ1j;jSz\nQ1j)'(0;7:4;2:3)\nand (jSx\nQ2j;jSy\nQ2j;jSz\nQ2j)'(7:4;0;2:3) for the proper-\nscrew VCs, while ( jSx\nQ1j;jSy\nQ1j;jSz\nQ1j)'(7:4;0;2:3)\nand (jSx\nQ2j;jSy\nQ2j;jSz\nQ2j)'(0;7:4;2:3) for the cycloidal\nVCs. This leads to the energy gain in the interaction\nterms,\u0000J(1 + 2\u0001)Sy\nQ1Sy\n\u0000Q1and\u0000J(1 + 2\u0001)Sx\nQ2Sx\n\u0000Q2\nin Eqs. (22) and (25), for the proper-screw VCs,\nwhile\u0000J(1 + 2\u0001)Sx\nQ1Sx\n\u0000Q1and\u0000J(1 + 2\u0001)Sy\nQ2Sy\n\u0000Q2in\nEqs. (23) and (27) for the cycloidal VCs.\n3. Interaction range dependence\nLet us consider the \fnite-range model in two dimen-\nsions while changing the parameter for the interaction\nrange,\r. We perform the variational calculations for\n\r >0 starting from the stable spin con\fguration for the\nin\fnite-range limit of \r= 0, which results in the solu-\ntions retaining the type of each VC with modulated spin\ncon\fgurations; while this procedure does not ensure that\nthe resultant solution is the ground state, but it is, at\nleast, a metastable state, for which we can compute the\nspin excitations by the linear spin-wave theory. Figure 16\nshows the spin-wave dispersion \"Kin the folded Brillouin\nzone for three di\u000berent values of \r. The results are com-\nmon to all the VCs in Fig. 15. When \ris small enough,\nthe excitation spectra are almost \rat except around the \u0000\npoint. The multiple values of excitation energy arise from\na nonuniform twist in VCs. As in the 1D case (Fig. 9),\nFIG. 16. Spin excitation spectra in the isotropic 2 Qspin\nstates in two dimensions while changing the parameter for\nthe interaction range: (a) \r= 0:01, (b)\r= 0:1, and (c)\n\r= 0:2. We take D=J = 0:2, \u0001 = 0:3, and \u0003 = 16, as in\nFig. 15. (d) Symmetric lines in the folded Brillouin zone, used\nfor the plots in (a{c).\nwhile increasing \r, the dispersion becomes more disper-\nsive, whereas the bandwidth is barely changed.\n4. Dynamical spin structure factor\nWe discuss the transverse component of the dynamical\nspin structure factor, S?(q;!) in Eq. (61), for each VC,\nwhich is related to the observable in the inelastic neu-\ntron scattering experiment. Note that the direction of q\nis \fxed along the xdirection in the 1 Qcase in Sec. IV A,\nwhere we discussed Syy(q;!) andSzz(q;!), but in the\n2D case the qdirection is rotated and the relevant spin\ncomponents change with the direction. In the calcula-\ntion, we \frst compute hKpjS\u0016\nqjvaciin Eq. (60) using the\nmodel for the proper-screw(I) VC, and then obtain the\nresults for the other VCs by using the corresponding ro-\ntations in spin space:\nS= (Sx;Sy;Sz);proper-screw(I) VC\n!8\n><\n>:S= (Sy;\u0000Sx;Sz);cycloid(I) VC\nS= (Sx;\u0000Sy;\u0000Sz);proper-screw(II) VC\nS= (Sy;Sx;\u0000Sz);cycloid(II) VC:\n(84)\nFigure 17 shows S?(q;!) for the four types of VCs\nobtained at D=J = 0:2, \u0001 = 0:3, \u0003 = 16, and \r= 0:2.\nHere, we take \u000f= 0:05 in Eq. (60). The overall spectra16\nFIG. 17. Transverse component of the dynamical spin struc-\nture factor, S?(q;!) in Eq. (61), for (a) proper-screw(I), (b)\ncycloid(I), (c) proper-screw(II), and (d) cycloid(II) VCs, plot-\nted along the symmetric lines in the \frst Brillouin zone shown\nin (h). (e) and (f) show the enlarged views of the dotted areas\nin (a,c) and (b,d), respectively. (g) !dependence at q=Q1.\nThe interaction range and the relaxation rate are taken as\n\r= 0:2 and\u000f= 0:05, respectively. The other parameters are\n\u0003 = 16,D=J = 0:2, and \u0001 = 0 :3 as in Fig. 15.\nlook similar among the di\u000berent VCs. In particular, as\nexpected from the above calculation scheme, the spectra\nalong the \u0000{X line are common to the two types of the\nproper-screw VCs [Figs. 17(a) and 17(c)]; this holds also\nfor the cycloid(I) and (II) VCs [Figs. 17(b) and 17(d)]. Inthe same way, the spectra along the M{\u0000 line are common\nto the proper-screw(I) and cycloid(II) VCs [Figs. 17(a)\nand 17(d)], and to the cycloid(I) and proper-screw(II)\nVCs [Figs. 17(b) and 17(c)]. In addition, we note that\nthe spectra along the X{M line are common to the two\ntypes of VCs for both proper-screw and cycloidal cases\n(see Appendix A).\nOn the other hand, a stark di\u000berence between the\nproper-screw and cycloidal VCs is found along the \u0000{X\nline, especially in the vicinity of q=Q1: The inten-\nsity of the lowest-energy excitation mode is much larger\nfor the proper-screw VCs [Fig. 17(e)] than the cycloidal\nVCs [Fig. 17(f)]. This is more clearly shown in the !de-\npendence at q=Q1in Fig. 17(g). The large di\u000berence\nis consistent with the small RSplotted in Fig. 14(b),\nsince the ratio between the intensities at q=Q1and\n!=\"Q1p=1is well approximated by R2\nS. The reason is\nas follows. The intensity is computed from the dynam-\nical spin structure factor as S?(Q1;!) =Syy(Q1;!) +\nSzz(Q1;!) for the two types of the proper-screw VCs,\nwhileS?(Q1;!) =Sxx(Q1;!) +Szz(Q1;!) for those\nof the cycloid. At !=\"Q1;p=1,S?(Q1;!) is dominated\nbySyy(Q1;!) [Szz(Q1;!)] for the proper-screw (cycloid)\nVCs. Since the frequency integral of the dynamical spin\nstructure factor corresponds to the static spin structure\nfactor, i.e.,R\nS\u0016\u0016(q;!)d!/ jS\u0016\nqj2, the intensity ratio\nis approximately given by jSz\nQ1j2=jSy\nQ1j2=R2\nS. For the\npresent parameter set, RS'0:31 as shown in Sec. IV B 2,\nleading toR2\nS\u00190:10. The value well explains the peak\ndi\u000berence in Fig. 17(g). Thus, such a di\u000berence around\nq=Q1could be useful to distinguish the proper-screw\nand cycloid types of VCs in experiments.\nWe note that it is rather di\u000ecult to distinguish the\ntype (I) and (II) from the spectra, in both proper-screw\nand cycloidal cases. There is, however, a noticeable dif-\nference near ( q;!)\u0019(Q1+Q2;1:8JS) along the M{\u0000\nline, as shown in Figs. 17(a){17(d).\nC. Three-dimensional hedgehog lattices\nFinally, we present the results for the 3D HLs. The\nstructure of the following sections is similar to that in\nSec. IV B for the 2D case: the variational phase diagram\nin Sec. IV C 1, the details of the 3D HLs and their stabi-\nlization mechanism in Sec. IV C 2, the spin-wave disper-\nsion while changing the interaction range in Sec. IV C 3,\nand the dynamical spin structure factor in Sec. IV C 4.\n1. Phase diagram\nFigure 18(a) shows the ground-state phase diagram for\nthe in\fnite-range model in three dimensions (Sec. II C 3)\nobtained by the variational calculations (Sec. III A) while\nchanging \u0001 and D=J. We here take \u0003 = 12. The phase\ndiagram is common to all the settings of the proper-screw\nand cycloid HLs listed in Table I. We \fnd three phases:17\n\u0001\u0002\u0001\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0001\u0002\u0007\u0001\u0002\u0001\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0001\u0002\u0007\n\u0001 \u0001\u0002\u0003 \u0001\u0002\u0004 \u0001\u0002\u0005 \u0001\u0002\u0006 \u0007\u0002\u0001\n<latexit sha1_base64=\"LF+HG+B+z9UyHi1x9Et9Cc7DePE=\">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSSi6LGoB48V7Ae0oWy2m3btZjfsToQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MBHcoOd9O4WV1bX1jeJmaWt7Z3evvH/QNCrVlDWoEkq3Q2KY4JI1kKNg7UQzEoeCtcLRzdRvPTFtuJIPOE5YEJOB5BGnBK3U7N4ygaRXrnhVbwZ3mfg5qUCOeq/81e0rmsZMIhXEmI7vJRhkRCOngk1K3dSwhNARGbCOpZLEzATZ7NqJe2KVvhspbUuiO1N/T2QkNmYch7YzJjg0i95U/M/rpBhdBRmXSYpM0vmiKBUuKnf6utvnmlEUY0sI1dze6tIh0YSiDahkQ/AXX14mzbOqf1H17s8rtes8jiIcwTGcgg+XUIM7qEMDKDzCM7zCm6OcF+fd+Zi3Fpx85hD+wPn8AWHejwA=</latexit>\u0000<latexit 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Variational results for the in\fnite-range model\nin three dimensions. We take \u0003 = 12. (a) Phase diagram,\nwhich includes the 1 Qstate, the anisotropic 2 Qstate with\njSQ1j>jSQ2j6= 0 and SQ3= 0 (the cyclic permutations of\nQ\u0011are energetically degenerate) and the isotropic 3 Qstate\nwithjSQ1j=jSQ2j=jSQ3j. (b) Contour plot of the ratio of\nthe Fourier components of spins, RS, de\fned in Sec. IV B 1\nand Fig. 14.\nthe 1Qstate, the anisotropic 2 Qstate where one of three\njSQ\u0011jis zero and the other two have nonzero di\u000berent\nvalues, and the isotropic 3 Qstate wherejSQ1j=jSQ2j=\njSQ3j. The phase transition between the 3 Qand 2Q\nstates is discontinuous, while that between 2 Qand 1Q\nlooks continuous. Similar to the 2D case, the system sta-\nbilizes the 1 Qstate forD > 0 and small \u0001, as shown\nin Fig. 18(a). In the larger \u0001 region, the system sta-\nbilizes the 2 Qand 3Qstates, whose spin con\fgurations\nare noncoplanar. In the 3 Qstate, the constituent three\nspin helices are elliptical, as in the isotropic 2 Qcase in\nSec. IV B 1. The calculated RS, whose de\fnition is the\nsame as that in Fig. 14(b) ( QmaxisQ\u0011for the largest\njSQ\u0011j), is plotted in Fig. 18(b). Note that RS= 0 in the\nisotropic 3Qstate atD= 0 where three sinusoidal spin\ndensity waves are superposed with equal weight. 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sha1_base64=\"cOrA3AQ3rqZ6JSQiIZsCELEcMF4=\">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBA8hV0VzTHgxWNE88BkDbOT3mTI7OwyMyvEJX/hxYMiXv0bb/6Nk8dBEwsaiqpuuruCRHBtXPfbWVpeWV1bz23kN7e2d3YLe/t1HaeKYY3FIlbNgGoUXGLNcCOwmSikUSCwEQyuxn7jEZXmsbwzwwT9iPYkDzmjxkr3tw9PnayNQow6haJbcicgi8SbkSLMUO0UvtrdmKURSsME1brluYnxM6oMZwJH+XaqMaFsQHvYslTSCLWfTS4ekWOrdEkYK1vSkIn6eyKjkdbDKLCdETV9Pe+Nxf+8VmrCsp9xmaQGJZsuClNBTEzG75MuV8iMGFpCmeL2VsL6VFFmbEh5G4I3//IiqZ+WvIvS2c15sVKexZGDQziCE/DgEipwDVWoAQMJz/AKb452Xpx352PauuTMZg7gD5zPH8dJkPc=</latexit>Sz`FIG. 19. Isotropic 3 Qspin states in the in\fnite-range model\nin three dimensions with D=J = 0:3, \u0001 = 0:3, and \u0003 = 12:\n(a) proper-screw, (b) cycloid(I), and (c) cycloid(II) HLs. The\nforms ofJ\u000b\u000b\nQ\u0011andDQ\u0011in each case are summarized in Table I.\nInsets show DQ\u0011as well as Q\u0011for each case. The color of ar-\nrows indicates the zcomponent of spin according to the color\nbar in (a). (d) Positions of the hedgehogs (magenta spheres)\nand the antihedgehogs (cyan spheres), which are common to\nall the HLs shown in (a){(c). The dashed lines are the guides\nfor eyes.\n2. Hedgehog lattices\nWhile the phase diagram is common to all the set-\ntings in Table I, the actual spin con\fguration in the 3 Q\nstate depends on the type of interactions. We present the\nvariational results in Fig. 19, focusing on the isotropic\n3Qstate atD=J = 0:3 and \u0001 = 0 :3 (RS'0:22). Fig-\nure 19(a) shows the stable spin con\fguration when taking\nJ\u000b\u000b\nQ\u0011andDQ\u0011as Eqs. (30) and (29), respectively. This\nis the proper-screw HL. On the other hand, Fig. 19(b)\nshows the spin con\fguration obtained for Eqs. (31) and\n(32), which is the cycloid(I) HL. Likewise, Fig. 19(c) dis-\nplays the spin con\fguration for Eqs. (33) and (34), which\nis the cycloid(II) HL. Figure 19(d) shows the positions\nof the topological defects, i.e., hedgehogs and antihedge-\nhogs, which are identi\fed as sources and sinks, respec-\ntively, of the emergent magnetic \feld de\fned by a solid\nangle formed by neighboring three spins [58]. The po-18\nsitions of the hedgehogs and antihedgehogs are common\nto the three HLs, since the spin con\fgurations are mutu-\nally transformed by 2 \u0019=3 rotations about the [111] axis\nin spin space as described in Sec. II C 3.\nThese HLs are stabilized by the anisotropy \u0001 in the\nsymmetric interactions, as suggested in the phase dia-\ngram in Fig. 18(a). Similar to the 2D case in Sec. IV B 2,\nthis can be con\frmed by calculating the spin components\nofSQ\u0011. We \fnd that all the HLs have the large ampli-\ntudes forjSy\nQ1j=jSz\nQ2j=jSx\nQ3j,jSx\nQ1j=jSy\nQ2j=jSz\nQ3j,\nandjSz\nQ1j=jSx\nQ2j=jSy\nQ3jfor the proper-screw, cy-\ncloid(I), and cycloid(II), respectively, which gain the in-\nteraction energy in the presence of \u0001 in each case.\n3. Interaction range dependence\nFIG. 20. Spin excitation spectra in the isotropic 3 Qspin\nstates in three dimensions for (a) \r= 0:01, (b)\r= 0:1, and\n(c)\r= 0:2. We take D=J = 0:3, \u0001 = 0:3, and \u0003 = 12, as\nin Fig. 19. (d) Symmetric lines in the folded Brillouin zone,\nused for the plots in (a{c).\nLet us discuss the \fnite-range model in three dimen-\nsions. As in the 2D case, we perform the variational\ncalculations for \r > 0 starting from the solutions for\n\r= 0, and \fnd stable but modi\fed spin con\fgura-\ntions. Figure 20 shows the spin-wave dispersion \"Kin\nthe folded Brillouin zone for three di\u000berent values of \r\natD=J = 0:3, and \u0001 = 0 :3 with \u0003 = 12; the results are\ncommon to all the HLs in Fig. 19. Similar to the 1D and\n2D cases, the excitation spectra are almost \rat in most\nregions in momentum space for small \r, but they become\nmore dispersive while increasing \r.4. Dynamical spin structure factor\nFIG. 21. Transverse component of the dynamical spin struc-\nture factor, S?(q;!) in Eq. (61), for (a) proper-screw, (b)\ncycloid(I), and (c) cycloid(II) HLs, plotted along the sym-\nmetric lines in the \frst Brillouin zone shown in (h). (d), (e),\nand (f) show the enlarged views of the dotted areas in (a),\n(b), and (c), respectively. (g) !dependence at q=Q1. The\ninteraction range and the relaxation rate are taken as \r= 0:2\nand\u000f= 0:05, respectively. The other parameters are \u0003 = 12,\nD=J = 0:3, and \u0001 = 0 :3 as in Fig. 19.\nFigure 21 shows the transverse component of the dy-\nnamical spin structure factor, S?(q;!) in Eq. (61), for\nthe three types of HLs obtained at D=J = 0:3, \u0001 = 0:3,\n\u0003 = 12, and \r= 0:2. We take \u000f= 0:05 in Eq. (60). The\ncalculations are done in a similar manner to the 2D case\nin Sec. IV B 4, by using the spin rotation as\nS= (Sx;Sy;Sz);proper-screw HL19\n!(\nS= (Sy;Sz;Sx);cycloid(I) HL\nS= (Sz;Sx;Sy);cycloid(II) HL: (85)\nThe overall spectra look similar among the di\u000berent\nHLs. In particular, as expected from the above calcula-\ntion scheme, the spectra along the R{\u0000 line are common\nto all the three types of HLs.\nOn the other hand, similar to the 2D case, a stark dif-\nference among the three HLs is found in the vicinity of\nq=Q1: The large intensities of the lowest-energy exci-\ntation mode are seen in the proper-screw and cycloid(II)\nHLs [Figs. 21(d) and 21(f)], whereas not in the cycloid(I)\nHL [Fig. 21(e)]. This is more clearly shown in the !de-\npendence at q=Q1in Fig. 21(g). As in the 2D case,\nthis large di\u000berence is consistently understood from R2\nS:\nFor the present parameter set, RS'0:22 as shown in\nSec. IV C 2, leading to R2\nS\u00190:048. Since the intensity\nfor the cycloid(I) is particularly smaller than the other\ntwo, this di\u000berence could be useful to distinguish the cy-\ncloid(I) HL from the proper-screw and cycloid(II) HLs in\nexperiments.\nAlthough it is rather di\u000ecult to distinguish the proper-\nscrew and cycloid(II) HLs solely form the spectra in\nFig. 21, it would be useful to separately measure the spin-\n\rip and non-spin-\rip cross sections in inelastic neutron\nscattering experiments. We expect a larger (non-)spin-\n\rip component for the proper-screw [cycloid(II)] HL since\nit has a larger intensity in Syy(Q1;!) [Szz(Q1;!)].\nV. SUMMARY AND DISCUSSION\nWe have investigated the spin excitation spectra for\nvarious types of helimagnetic states in the spin mod-\nels with long-range exchange interactions. Starting\nfrom the model with in\fnite-range interactions, we have\nstudied the models with long- but \fnite-range interac-\ntions including the symmetric diagonal ones with spin\nanisotropy and the antisymmetric o\u000b-diagonal ones of the\nDzyaloshinskii-Moriya type. While changing the range\nof the interactions, we clari\fed the ground state and the\nspin excitation by the variational calculation and the lin-\near spin-wave theory, respectively. For the spin excita-\ntion, in addition to the spin-wave dispersion, we com-\nputed the transverse component of the dynamical spin\nstructure factor, S?(q;!), which is relevant to the in-\nelastic neutron scattering experiments.\nIn the 1D case, we obtained the analytical solution\nfor the spin excitation in the isotropic HS states with a\nspatially uniform twist angle, for both proper-screw and\ncycloid types. We showed that the spin-wave dispersion\nis completely \rat in the in\fnite-range model except for\nq= 0,\u0006Q, and\u00062Q, whereQis the helical wave num-\nber, but it becomes dispersive for the \fnite-range case.\nIrrespective of the spatial range of interactions, there is\na gapless excitation mode, which results in strong inten-\nsities atq=\u0006Qin the dynamical spin structure factor.\nMeanwhile, we also obtained the numerical results forthe e\u000bect of the spin anisotropy in the symmetric diag-\nonal interactions. We found that the anisotropy makes\nthe twist angle of the stable spin texture inhomogeneous,\nand accordingly opens a gap in the spin-wave disper-\nsion. We also showed that, as long as the anisotropy is\nweak, the lowest-energy excitation mode can be regarded\nas a phase shift of the helix. In addition, we found a\ndiscernible di\u000berence between the proper-screw and cy-\ncloid HS states in the high-energy spectra of S?(q;!) at\nq=Q^x. We also found additional intensities in S?(q;!)\nfor the higher harmonics at q=\u00063Qin the presence of\nthe spin anisotropy.\nExtending the analyses to the 2D case, we have dis-\ncussed the stability and excitations of 2 QVCs. By using\nthe variational calculation, we found that the 2 QVCs are\nstabilized by the spin anisotropy. More speci\fcally, while\nincreasing the spin anisotropy, the 1 QHS state stabi-\nlized by the antisymmetric interactions turns into the 2 Q\nVCs through the apparently continuous phase transition.\nThere are two di\u000berent 2 QVC phases: the anisotropic\none with a superposition of two spin helices with di\u000ber-\nent amplitudes and the isotropic one with equal ampli-\ntudes. The latter appears for larger anisotropy than the\nformer. While the phase diagram is common, the stable\nspin con\fgurations of the VCs depend on the form of the\ninteractions in the model, that is, the easy axes of the\nspin anisotropy and the directions of the Dzyaloshinskii-\nMoriya vectors. We obtained four types of VCs: two of\nthem are superpositions of proper screws and the other\ntwo are of cycloids. By using the linear spin-wave the-\nory, we showed that the spin-wave dispersion, which is\ncommon to the four VCs, becomes dispersive upon in-\ntroducing the spatial decay of the interactions, similar to\nthe 1D HS case. Meanwhile, we found discernible di\u000ber-\nences inS?(q;!) among the four types of VCs at q'Q1\nandq'Q1+Q2. The \fnding could be useful to deter-\nmine the type of VCs as well as the relevant e\u000bective spin\nmodel in inelastic neutron scattering experiments.\nIn the 3D case, we have examined the stable spin con-\n\fgurations and the spin excitation spectra for three types\nof 3QHLs: one proper-screw type and two cycloid types.\nIn the variational phase diagram, which is common to\nthe three HLs, we found that the HLs are stabilized in\nthe presence of the spin anisotropy, similar to the 2 Q\nVCs in the 2D case. In this 3D case, however, while in-\ncreasing the spin anisotropy, the 1 QHS state \frst turns\ninto the anisotropic 2 QVC, and then into the isotropic\n3QHL; the phase transition between the 2 QVC and the\n1QHS state looks continuous, while that between the\n3QHL and the 2 QVC is discontinuous. With regard to\nthe spin excitation spectra, we found qualitatively simi-\nlar behaviors to the 2 QVC cases. Thus, in this case also,\nthe di\u000berences in S?(q;!) would be useful to distinguish\nthe type of HLs and to identify the relevant interactions\nin experiments.\nFinally, we discuss candidate materials to which our re-\nsults are potentially relevant. As discussed in Sec. IV A 2,\nthe 1D HS states in CuB 2O4and TbMnO 3could be ac-20\ncounted for by our 1D model with the spin anisotropy,\nwhich predicts higher harmonics at q=\u00063Qin the dy-\nnamical spin structure factor as observed in the exper-\niments [71, 72]. For the 2D (3D) models, the magnetic\nmetals with the crystallographic point groups D4,C4v,\nandD2d(TandC3) can be candidate materials, as shown\nin Table I: for example, a Mn-Pt-Sn inverse Heusler com-\npound (D2d) where an antiskyrmion crystal has been\nfound [74], and MnSi 1\u0000xGexofB20 structure ( T) where\nthe magnetic HLs have been found [20{25]. To the best of\nour knowledge, inelastic neutron scattering experiments\nhave not been performed systematically for the multiple-\nQspin states of VC and HL thus far. We hope that our\nresults stimulate such experiments and the detailed com-\nparison between theory and experiment provides a hint\nfor understanding of the microscopic mechanism of the\nmultiple-Qstates.\nIn addition to the above substances, recently, multiple-\nQspin states in centrosymmetric systems, for which\nthe antisymmetric interactions of the Dzyaloshinskii-\nMoriya type are inactive, have been attracting consid-\nerable attentions, for example, GdRu 2Si2(D4h) [19, 75],\nGd2PdSi 3(D6h) [76{81], Gd 3Ru4Al12(D6h) [82, 83], and\nSrFeO 3(Oh) [84{87]. In our model, however, when the\nantisymmetric interactions are absent, the ground states\nare almost always given by superpositions of sinusoidal\nspin density waves, inconsistent with the experimental\nobservations. Thus, for these multiple- Qspin states, fur-\nther extensions of the model are necessary, e.g., addi-\ntional biquadratic interactions [58, 60, 61]. This interest-\ning issue is left for future research.\nACKNOWLEDGMENTS\nThe authors thank Y. Fujishiro, N. Kanazawa, and T.\nNakajima for fruitful discussions. This work was par-\ntially supported by Japan Society for the Promotion\nof Science (JSPS) KAKENHI Grant No. JP18K03447,\nJP19H01834, and JP19H05825, JST CREST Grant No.\nJPMJCR18T2, and JST PRESTO (JPMJPR20L8).\nAppendix A: Symmetry argument for S?(q;!)along\ntheX{Mline in 2D VCs\nIn this Appendix, we explain why S?(q;!) along the\nX{M line are common to the two types of VCs for both\nproper-screw and cycloidal cases as shown in Fig. 15,\nfrom a symmetry argument. Let q0be any wavenumber\non the X{M line. S?(q;!) atq=q0for the type (I)\nVCs are computed by using Eq. (60) as\nSI\n?(q0;!) =Szz(q0;!) +S\u00162\u00162(q0;!); (A1)\nwhere the spin component in the second term is taken\nalong ^\u0016= (\u0000^qy\n0;^qx\n0;0); here, ^q0= (^qx\n0;^qy\n0) =q0=jq0j. On\nthe other hand, those for the type (II) VCs, SII\n?(q0;!),\nare computed by replacing ^\u0016by^\u00160= (^qy\n0;^qx\n0;0) becausethe type (I) and (II) VCs are connected each other by \u0019\nrotation about [100] axis in spin space. 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In this Paper, we report on a sys-\ntematic investigation of the dynamics of spin polarization and spin-polarized currents produced by\nfemtosecond laser pulses, modeled within our theoretical framework evolve. The spin polarization\ndepends strongly on the properties of the laser pulse and on the sample composition, as is shown by\ncomparing results for Cu(100), Co(100), and a Co/Cu heterostructure. We ﬁnd a transition from co-\nherence before the laser pulse’s maximum to incoherence thereafter. Moreover, the time dependence\nof the spin-polarization components induced by spin-orbit coupling diﬀer signiﬁcantly in Cu and Co:\nin Cu, we ﬁnd long-period oscillations with tiny rapid modulations, whereas in Co prominent rapid\noscillations with long period ones are superimposed. The pronounced spatial dependencies of the\nsignals underline the importance of inhomogeneities, in particular magnetic/non-magnetic interfaces\n‘act as source’ for ultrafast spin-polarization eﬀects. Our investigation provides detailed insight into\nelectron dynamics during and shortly after a femtosecond laser excitation.\nKeywords: Condensed matter physics, ultrafast magnetization dynamics, spin dynamics simulations\nI. INTRODUCTION\nSpin-polarized photocurrents are ubiquitous in\nspin- and angle-resolved photoelectron spectroscopy\n(SARPES) [1, 2]. In non-magnetic samples, the spin po-\nlarization of the detected photocurrents—brought about\nby spin-orbit coupling (SOC)—depends on details of\nthe setup, in particular on those of the incident electro-\nmagnetic radiation (e.g. on photon energy, polarization,\nand incidence direction; see for example Ref. 3) and on\nthe symmetry of the surface [4–7]. In magnetic samples,\nthe same eﬀect results in magnetic dichroism [8, 9],\nand, as theoretical and experimental studies show, the\nspin-orbit-induced spin polarization of photoelectrons\nmust not be aligned with the magnetization direction\n(see for example Ref. 8 and references therein).\nIn ultrafast spin dynamics, electrons are excited by\nelectromagnetic radiation as well, for example by a fem-\ntosecond laser pulse. Focusing on the demagnetization\nof a magnetic sample [10–12], one investigates mainly\nthe reduction of the magnetization but disregards its\nchange in direction. The latter could be brought about\nby photo-induced spin-polarization components that are\nnot aligned with the ground state’s magnetization. In\nSARPES these ‘oblique’ components are those of elec-\ntrons measured at the detector , whereas in ultrafast\nspin dynamics they are those of electrons within a sam-\nple; thus one is concerned with diﬀerent boundary con-\nditions [13]. This idea immediately calls for a sys-\ntematic investigation of photo-induced spin polarization\nand spin-polarized currents caused by femtosecond laser\npulses [14].\n\u0003Correspondence email address: oliver.busch@physik.uni-halle.deIn the theoretical study reported in this Paper, we con-\ncentrate on the SO-induced spin-polarization eﬀects dur-\ning a laser excitation. In order to excise the main fea-\ntures we begin with a non-magnetic sample—Cu(100)—\nand then turn to a magnetic sample—face-centered cu-\nbic Co(100). Since real samples often contain inter-\nfaces, we investigate the role of the latter by means of a\nCo/Cu(100) heterostructure. The simulations were per-\nformed using our computational framework evolve[15].\nQuestions worth considering are, among others, which\ncomponents of the spin polarization are forbidden by\nsymmetry? How large are the allowed components, and\nare their magnitudes comparable to those observed in\nSARPES? What are their temporal and spatial distribu-\ntions? Does magnetism reduce the ‘oblique’ SO-induced\ncomponents (here: in samples containing Co)? What are\nthe detailed properties of the photo-induced currents?\nWe respond to these questions in this work.\nThis Paper is organized as follows. In Section II we\nsketchourapproachtoultrafastelectrondynamics(IIA),\ndiscuss spin polarization as well as currents (IIB), and\nperformasymmetryanalysis(IIC).Resultsarediscussed\nin Section III: beginning with Cu(100) (IIIA) we turn\nto magnetic systems, namely fcc Co(100) (IIIB) and a\nCo/Cu(100) heterostructure (IIIC). We conclude with\nSection IV.\nII. THEORETICAL ASPECTS\nA. Ultrafast electron dynamics\nThe samples are free-standing fcc(100) ﬁlms of 40lay-\ners thickness. We consider Cu(100), Co(100), and\nCo/Cu(100) (with 20layers each) ﬁlms. The CartesianarXiv:2303.09291v1  [cond-mat.mtrl-sci]  16 Mar 20232\nϑph\nz\nxy\nFigure 1. Geometry of a Co/Cu heterostructure. The fcc ﬁlm\nconsists of 40layers stacked in xdirection, with 20layers of\nCo atoms (cyan spheres) and 20layers of Cu atoms (magenta\nspheres). The Co magnetic moments point along the zdirec-\ntion (black arrows). The ﬁlm is inﬁnite in yandzdirection\nbut ﬁnite in xdirection. A laser pulse impinges with a polar\nangle#phof45\u000ewithin thexzplane onto the sample.\nxaxis is perpendicular to the ﬁlm, and we apply periodic\nboundary conditions within the ﬁlm, i.e. in yandzdi-\nrection. In the case of Co(100) and Co/Cu(100), the\nmagnetic moments are collinear and point along the zdi-\nrection (Fig. 1) [16].\nThe electronic structure of the samples is described\nby a tight-binding Hamiltonian ^H0of Slater-Koster\ntype [17], with parameters for the s-, p-, and d-orbitals\ntaken from Ref. 18. Collinear magnetism and spin-orbit\ncoupling are taken into account as described in Ref. 19.\nThe electron system is excited by a femtosecond laser\npulse with photon energy Eph=!(in atomic units, ~=\n1). The laser’s electric ﬁeld\nE(t) =l(t)X\nl=s;pElcos(!t+'l): (1)\nis a coherent superposition of s- and p-polarized partial\nwavesmodulatedwithaLorentzianenvelope l(t).Eland\n'lare the amplitudes resp. the phase shifts of the partial\nwaves.\nThe electromagnetic radiation impinges within the\nxzplane onto the ﬁlms, with a polar angle #ph= 45\u000e\nof incidence. For s-polarized light ( Ep= 0),E(t)points\nalong theyaxis which is perpendicular to the plane of\nincidence, the latter spanned by the incidence direction\nof the light and the surface normal. For p-polarized light\n(Es= 0),E(t)lies within the xzincidence plane. Cir-\ncular polarized radiation with helicity \u001b\u0006is obtained by\n's\u0000'p=\u000690\u000eand equal amplitudes ( Es=Ep).\nThe electron dynamics is described by the von Neu-\nmann equation\n\u0000id^\u001a(t)\ndt= [^\u001a(t);^H(t)] (2)\nfor the one-particle density matrix\n^\u001a(t) =X\nn;mjnipnm(t)hmj: (3)fjnigis the set of eigenstates of ^H0, with ^H0jni=\u000fnjni.\nThe time-dependent Hamiltonian ^H(t)comprises the\nelectric ﬁeld of the laser via minimal coupling [20]. The\nequation of motion (2) for ^\u001a(t)is solved within our the-\noretical framework evolve; for details see Ref. 15.\nB. Spin polarization and spin-polarized currents\nSite-, orbital-, and spin-resolved properties of an ob-\nservableOare obtained by taking partial traces in the\nexpectation values hOi(t) = tr[^\u001a(t)^O], with the density\nmatrix in an appropriate basis.\nIn matrix form an expectation value reads hOi(t) =\ntr[P(t)O]. We deﬁne matrices p\u001b\u001b0\nkland h\u001b\u001b0\nklfor the den-\nsity matrix and the Hamiltonian, respectively, with ele-\nments\n\u0010\np\u001b\u001b0\nkl\u0011\n\u000b\f=pk\u000b\u001b;l\f\u001b0; (4a)\n\u0010\nh\u001b\u001b0\nkl\u0011\n\u000b\f=hk\u000b\u001b;l\f\u001b0: (4b)\nkandlare site indices, \u001band\u001b0specify the spin orien-\ntation (\"and#with respect to the zdirection), and \u000b\nand\fare orbital indices. These matrices are combined\ninto site-resolved block matrices\nPkl=\u0012\np\"\"\nklp\"#\nkl\np#\"\nklp##\nkl\u0013\n; (5a)\nHkl=\u0012\nh\"\"\nklh\"#\nkl\nh#\"\nklh##\nkl\u0013\n: (5b)\nThe spin polarization at site lis given by\ns\u0016\nl= tr ( Pll\u0006\u0016); \u0016 =x;y;z;\nin which \u0006\u0016is a block Pauli matrix. Explicitly,\nsx\nl= 2 Re tr\u0010\np\"#\nll\u0011\n; (6a)\nsy\nl=\u00002 Im tr\u0010\np\"#\nll\u0011\n; (6b)\nsz\nl= tr\u0010\np\"\"\nll\u0000p##\nll\u0011\n; (6c)\nwith normalization tr(Pll) = 1. The site-averaged spin\npolarization\nS\u0016=1\nNsiteX\nls\u0016\nl; \u0016 =x;y;z; (7)\nis obtained by summation over all Nsitesites in a ﬁlm’s\nunit cell.\nThe current\njkl=\u0000i\n2tr (PlkHkl)\u0000hl$ki (8)\nfromsiteltositekaswellastherespectivespin-polarized\ncurrents\nj\u0016\nkl=\u0000i\n4tr\u0000\nPlk[\u0006\u0016;Hkl]+\u0001\n\u0000hl$ki\u0016=x;y;z;(9)3\nTable I. Eﬀect of symmetry operations (left column) on\nthe laser’s electric ﬁeld Edecomposed into its s- and p-\npolarization components EsandEp, the magnetization M\ninzdirection, and the electron spin polarization S=\n(Sx;Sy;Sz).^1is the identity operation, ^myis the reﬂection\nat thexzplane.\n^1EsEpMSxSySz\n^my\u0000EsEp\u0000M\u0000SxSy\u0000Sz\nare derived from Mahan’s equation for the current oper-\nator in spin-symmetrized form [21] (see also Refs. 22 and\n23;[\u0001;\u0001]+is the anticommutator). For collinear magnetic\ntextures, as discussed in this Paper, inter-site hopping\nwith spin ﬂip does not occur in ^H0, i.e. h\"#\nkl= 0resp.\nh#\"\nkl= 0. With this the above equations become\njkl=\u0000i\n2tr\u0010\np\"\"\nlkh\"\"\nkl+p##\nlkh##\nkl\u0011\n\u0000hl$ki; (10a)\njx\nkl=\u0000i\n4tr\u0010\np\"#\nlk+p#\"\nlk\u0011\u0010\nh\"\"\nkl+h##\nkl\u0011\n\u0000hl$ki;(10b)\njy\nkl=1\n4tr\u0010\np\"#\nlk\u0000p#\"\nlk\u0011\u0010\nh\"\"\nkl+h##\nkl\u0011\n\u0000hl$ki;(10c)\njz\nkl=\u0000i\n2tr\u0010\np\"\"\nlkh\"\"\nkl\u0000p##\nlkh##\nkl\u0011\n\u0000hl$ki: (10d)\nInterchanging the site and the spin indices yields jkl=\n\u0000jlkandj\u0016\nkl=\u0000j\u0016\nlk=j\u0000\u0016\nlk.\nC. Symmetry analysis\nInstead of a full group-theoretical analysis [8], we per-\nform a symmetry analysis which tells what components\nof the spin polarization are forbidden for a given setup.\nThe important symmetry is the reﬂection ^myat thexz\nplane: (x;y;z )!(x;\u0000y;z)since thexzplane is a sym-\nmetry plane of the lattice and is also the laser’s plane of\nincidence (spanned by the light incidence direction and\nthe surface normal).\nFor p-polarized light, ^myis a symmetry operation for a\nnon-magnetic sample ( M= 0; here: Cu) which tells that\nonlySyis allowed nonzero (Table I). A z-magnetization\nbreaks this symmetry ( M6= 0; here: Co(100) and\nCo/Cu(100)), and all three components of Sare allowed\nnonzero.\nFor s-polarized light, the electric ﬁeld of the laser\nis along the ydirection. Since for homogeneous non-\nmagnetic samples (Cu) the z-rotation by 180\u000eleaves the\nsetup invariant, Sy= 0andSz= 0. ForSythis symme-\ntry holds for the spin polarization at each site ( sy\nl= 0).\nForSz, however, it holds only for the site-averaged spin\npolarization, that is, sz\nlat equivalent sites lmay be\nnonzero but compensate each other (equivalent sites have\nthe same distance from the two surfaces of a ﬁlm).\nConsidering circular polarized light, ^myreverses the\nhelicity\u001b\u0006!\u001b\u0007[(Es;Ep)!(\u0000Es;Ep)], which tellsTable II. Components of the site-averaged electron spin po-\nlarization S= (Sx;Sy;Sz)allowed ( +) or forbidden (\u0000) by\nsymmetry, for the magnetic case in rectangular brackets. For\ndetails see text.\npolarization SxSySz\np\u0000[+] +[+]\u0000[+]\ns\u0000[\u0000]\u0000[+]\u0000[+]\ncircular +[+] +[+] +[+]\nthatSxandSzchange sign under helicity reversal for\na non-magnetic sample but Sydoes not. For magnetic\nsamples this strict relation is broken, which may be re-\ngarded as a magnetic spin dichroism (magnetic dichroism\nis an intensity change upon magnetization reversal [24];\nhere we are concerned with a change of the spin polar-\nization). The symmetry-allowed or -forbidden spin po-\nlarization components are summarized in Table II.\nIII. RESULTS AND DISCUSSION\nFor discussing our results, we proceed by increasing\nstep by step the order of complexity. We begin with a\nnon-magnetic Cu(100) ﬁlm, since it exhibits the phenom-\nena most clearly. The eﬀect of magnetism is addressed\nby fcc Co(100), and eventually the combination of both\nsystems into a Co/Cu(100) heterostructure allows exam-\nining the eﬀect of a magnetic/non-magnetic interface.\nIn all simulations discussed below, the laser has a pho-\nton energy of 1:55 eV, a ﬂuence of about 3:3 mJ cm\u00002,\nand is modulated with a Lorentzian l(t)with 10 fswidth.\nAll samples comprise 40layers, with sites 0and39deﬁn-\ning the bottom and top surfaces, respectively.\nA. Cu(100)\nIn accordance with the symmetry analysis (Table II),\nthe calculations for p-polarized light yield only a nonzero\nSythat is slightly modulated with the doubled laser fre-\nquency (Fig. 2a). The sizable magnitude is explained\nby the local contributions sy\nl(t)which oscillate in phase\nwith almost identical amplitude [constructive interfer-\nence; panel (b)]. After the laser pulse; deviations among\nthe site-resolved spectra increase marginally (cf. t >\n12 fs).\nThe above ‘unison’ oscillations found for sy\nl(t)show\nup as well in the currents jkl(t)before the laser pulse’s\nmaximum [Fig. 2(c) and (d)], but with a much smaller\nperiod. The laser’s photon energy of 1:55 eVcorresponds\nto a period of 2:7 fsor about 3:7oscillations within 10 fs,\nwhich is also seen in panels (c) and (d). This suggests\nthat the electron system follows the electric ﬁeld of the\nlaser, that is a collective motion across the ﬁlm (in x\ndirection). At about t=\u00003 fsincreasing interference,4\n(a)\n(b)\n20\n 10\n 0 10 20\nt (fs)0.004\n0.002\n0.0000.0020.004jkl (arb.u.)\nlink l  k:\n8  9\n18  19\n21  22\n31  32\n(c)\n(d)\n(e)\nFigure 2. Photo-induced spin polarization and currents for\na Cu(100) sample excited by p-polarized light. (a) Site-\naveraged spin polarization Sy(t). (b) Local spin polarization\nsy\nl(t)for selected sites, as indicated. (c) Currents jkl(t)be-\ntween neighboring sites l!k=l+ 1for selected site pairs as\nindicated. (d) Currents jkl(t)and (e) spin-resolved currents\njy\nkl(t); their magnitude is indicated by color bars with the\nsame range (red positive, blue negative). Data in panels (c),\n(d) and (e) in arbitrary units. Vertical dashed lines at t= 0 fs\nmark the maximum of the laser pulse.\n20\n 10\n 0 10 20\nt (fs)0.02\n0.01\n0.000.010.02sz\nl(t)\nsite 8\nsite 18\nsite 21\nsite 31(a)\n(b)\nFigure 3. Photo-induced spin polarization and current of a\nCu(100)ﬁlmexcitedbys-polarizedlight. (a) sz\nl(t)forselected\nsites as indicated. Sites 8(18) and 31(21) are equivalent. (b)\nCurrentsjkl(t)displayed as color scale (red positive, blue neg-\native; in arbitrary units). Dashed arrows serve as guides to\nthe eye. Vertical dashed lines at t= 0 fsindicate the maxi-\nmum of the laser pulse.\nstarting at the surfaces, reduces the coherence in the os-\ncillations, thereby obliterating the pattern at later times.\nThe oscillations of the currents are accompanied by\nthose of the spin-resolved currents jy\nkl(t)in opposite di-\nrection [panel (e); the xandzcomponents are zero]. A\ncurrent in positive xdirection (red in panel d) appears\nsimultaneously with a spin-polarized current in opposite\ndirection [blue in panel (e)], which implies a ﬂow of \u0000y-\npolarized electrons in xdirection. Again, the current\npattern becomes complicated after the laser pulse due to\nthe interferences mentioned before.\nFors-polarizedlight, thesymmetryanalysisyields S=\n0but allows for sz\nl6= 0. The photo-induced local spin\npolarizations at equivalent sites thus have to compensate\neach other. This is fully conﬁrmed by the simulations:\nthe spin polarization is spatially antisymmetric within\nthe Cu ﬁlm [Fig. 3(a)].\nThe antisymmetry of the spin polarization may be at-\ntributed to the surface normals of the freestanding Cu\nﬁlm being opposite to each other. This reasoning com-\nplies with spin polarization eﬀects in spin- and angle-\nresolvedphotoemission[4–8], sincetheserelyonthepres-\nence of a surface (they do not occur in bulk samples).\nHence, one may regard the present result as a ﬁrst indi-\ncation for the importance of surfaces and interfaces for\nultrafast spin dynamics; see for example Ref. 15 (for\nreviews on polarized electrons at surfaces we refer to\nRefs. 25 and 26).\nTheaboveargumentissupportedbythecurrents jkl(t)5\n(a) (c) (d) (b)\ncircular\npolarized σ+\n(e)\np-polarized(g) (f)\n (h)\nFigure 4. Photo-induced spin polarization S\u0016(t)and spin-resolved currents j\u0016\nkl(t)for a Cu(100) ﬁlm excited by circular polarized\nlightwithhelicity \u001b+(toprow)andforfccCo(100)excitedbyp-polarizedlight(bottomrow). (a)Site-averagedspinpolarization\nS\u0016(t)for Cu(100) ( \u0016=x;y;z). (b) – (d) Spin-resolved currents j\u0016\nkl(t)displayed as color scale, as in Fig. 2. (e) Sx(t)andSy(t)\nfor Co(100). (f) – (h) as (b) – (d) using the same color scale. Dashed vertical lines indicate the maximum of the laser pulse at\nt= 0 fs.\n[Fig. 3(b)] which are initiated at the surfaces: compare\nfor example the darker color scale at the surface sites 0\nand 39 in panel (b) with respect to the lighter colors in\nthe interior of the ﬁlm at t=\u00005 fs. The currents enter\nthe ﬁlm’s interior slightly after the laser’s maximum (at\nt\u00194 fs), as schematically indicated by the dashed arrows\n(due to the antisymmetry, the current at the ﬁlm’s center\nvanishes, giving rise to the white horizontal stripe) and\nare reﬂected at the surfaces at t\u001912 fs, leading to a\n‘crisscross’ pattern [cf. the dashed arrows in panel (b)].\nThe spin-resolved currents jz\nkl(t)exhibit a pattern (not\nshown here) reminiscent of that of jy\nkl(t)for p-polarized\nlight displayed in Fig. 2(e).\nFor circular polarized light it is suﬃcient to discuss\none helicity (here: \u001b+as deﬁned in Section IIA), since\nthexandzcomponents of both spin polarization and\nspin-resolved currents change sign upon helicity reversal,\nwhereas the ycomponent does not, as is conﬁrmed by\nour simulations.\nAll components of the site-averaged spin polarization\nS\u0016(t)and the spin-resolved currents j\u0016\nkl(t)are nonzero\n(Fig. 4; top row). In an admittedly simple picture Sx(t)\nandSz(t)may be viewed as due to optical orientation in\nphotoemission [27]. Recall that the laser impinges within\nthexzplane onto the ﬁlm; for a single atom optical ori-\nentation by circular polarized light would then cause spin\npolarization within the xzplane. Likewise, Sy(t)may be\nattributedtotheeﬀectpredictedbyTamura, Piepke, and\nFeder [4] for SARPES. Of course, this ‘decomposition of\neﬀects’ ignores that the superposition of the laser’s s-\nand p-polarized partial waves are coherent and shifted in\nphase. Moreover, the electron dynamics mixes the com-\nponents of the local spin polarization because of spin-orbit coupling; nevertheless Sy(t)is reminiscent of that\nfor p-polarized light [Fig. 2(a)].\nB. fcc Co(100)\nFor fcc Co(100) we focus on excitation by p-polarized\nlight(bottomrowinFig.4). Asexpectedandoftenfound\nin both experiment and theory, the site-averaged spin-\npolarization component Sz(t)associated with magnetism\nis reduced by the laser pulse, that is the sample becomes\ndemagnetized (cf. Ref. 28 and references therein). This\ndemagnetization is site-dependent (not shown) similar to\nthe induced spin polarization in Cu(100) discussed be-\nfore.\nIn contrast to non-magnetic Cu(100), the magnetiza-\ntion of Co(100) breaks the mirror symmetry at the xz\nplane and allows for nonzero Sx(t)andSy(t); confer Ta-\nble II. Both components are modulated by the doubled\nlaser frequency but shifted in phase [panel (e)]. Their\nmagnitudes are roughly 10 %of theSycomponent in\nCu(100) [Fig. 2(a)]. Moreover, both Sx(t)andSy(t)of\nCo(100)exhibitabeatingpattern(withmaximaatabout\nt\u00190 fs,10 fsand20 fs), whileSy(t)of Cu(100) displays\na clear sinusoidal shape.\nThe spin-polarization components Sx(t)andSy(t)ex-\nhibit a regular pattern before the maximum of the laser\npulse [Fig. 4(e)], which hints toward laser-driven preces-\nsion of the spin polarization S(t). Indeed, Sx(t)and\nSy(t)display a left-handed helix, starting at the origin\nand with increasing amplitude [Fig. 5(a)]. Moreover,\nthe noticeable shift of the spiral center to positive val-\nues is explained by spin-orbit coupling: a minimal tight-6\n(a) (b) (c)\nFigure 5. Laser-driven precession of the spin polarization in Co(100) excited by p-polarized light. The color scale visualizes\nthe time evolution from t=\u000020 fs(dark blue) to t= 0 fs(orange). (a) Correlation of Sy(t)andSx(t)using data presented in\nFig. 4e. Panels (b) and (c) show Sx(t)resp.Sy(t)versus the electric ﬁeld Eph(t)of the laser pulse.\nbinding model for the motion of S(t), including SOC,\nyields two features: a deformation of the precession cone\nand a shift of the cone axis oﬀ the magnetization direc-\ntion (zaxis). Without spin-orbit coupling, one ﬁnds the\nusual circular cone with its axis along the magnetization\ndirection.\nThe time sequences of Sx(t)andSy(t)versus the laser\namplitudeEph(t)prove that the precession is driven by\nthe laser [panels (b) and (c) in Fig. 5]. The diﬀerences\nin the patterns are attributed to the phase shift between\nSx(t)andSy(t).\nThe striking diﬀerences in the spin polarization of Cu\nand Co could be attributed to the electronic structure,\nto spin-orbit coupling or to exchange splitting. Concern-\ning the electronic structure, Cu has completely occupied\nd-orbitals, while the respective spin-resolved orbitals in\nCo are partially occupied. On the other hand, the elec-\ntronic structure of Co might be viewed as that of Cu but\nexchange-split (in a rigid-band model). In this scenario,\nthe diﬀerences could be ascribed to the magnetism in Co.\nIn order to shed light upon the origin we performed\nsimulations for Cu and Co with varied strengths of the\nspin-orbit coupling and of the exchange splitting (not\nshown here). Varying the SOC strength has minute ef-\nfect on the spin polarization in both Cu and Co. How-\never, reducing the exchange splitting in Co(100) re-\nveals that, ﬁrst, the long-period oscillations in Sx(t)and\nSy(t)are modiﬁed without clear trend, and second, the\nrapid oscillations associated with the laser pulse become\nsuppressed. Without exchange splitting Sx(t)vanishes\nandSy(t)displays features that are similar to those in\nCu(100) [Fig. 2(a)]. These ﬁndings prompt magnetism\nas origin.\nAs for the spin polarization, all three components of\nthe spin-resolved currents are nonzero [panels (f) – (h)\nin Fig. 4], with the z-component jz\nkl(t)being the largest\n[as exhibited by darker colors in panel (h)]. All compo-\nnents oscillate ‘unison’ before the laser pulse maximum;\ncomplicated current patterns arise after the pulse.\nSummarizing brieﬂy for Cu and Co, we ﬁnd that the\nsimulations conﬁrm the symmetry considerations. Gen-\neral trends are ‘unison’ oscillations before the laser max-imum and complicated patterns thereafter; the optically\ninduced spin-polarization components are smaller in a\nmagnetic sample but exhibit precession before the laser\npulse maximum.\nC. Co/Cu heterostructure\nWe now address a Co/Cu(100) heterostructure illumi-\nnated by p-polarized light. Decomposing Sx(t)andSy(t)\nof the entire sample [black in Figs. 6(a) and (b)] into the\nrespective parts in the Co (cyan) and in the Cu region\n(magenta) tells that Sx(t)[panel (a)] is ﬁrst induced by\nthe laser pulse in the Co region and enters subsequently\nthe Cu region (recall that Sx(t)is symmetry-forbidden\nin Cu(100); Section IIIA). This ﬁnding underlines the\nimportance of an interface for ultrafast spin dynamics.\nIn contrast, Sy(t)is by far the largest in the non-\nmagnetic Cu region [panel (b)], whereas it is strongly\nreduced in the Co region. This ﬁnding corroborates\nthe above argument that magnetism may reduce photo-\ninduced spin-polarization components. Both the magni-\ntude and frequency of the site-averaged components in\nthe two regions are reminiscent of those in the respective\nhomogeneous samples.\nThe currents jkl(t)exhibit an oscillating collective\nmotion across the sample before the pulse, similar to\nCu(100) [Fig. 2(c)]. However, beginning slightly be-\nfore the pulse maximum at t= 0 fs, the spatial homo-\ngeneity is lost; instead there are sizable currents initi-\nated at the interface (visualized by the horizontal dashed\nline at site 19), and propagating towards the Co re-\ngion [dark blue features in panel (c)]. This ﬁnding cor-\nroborates that the interface acts as a ‘source’ of ultra-\nfast spin currents. At the magnetic/non-magnetic in-\nterface, the imbalance of occupation facilitates the pro-\nduction of currents. Moreover, since the imbalance is\nspin-dependent, also the spin-resolved currents jz\nkl(t),\nthat is those with spin along the magnetization direc-\ntion, should be triggered at the interface. This is in-\ndeed veriﬁed by jz\nkl(t)[panel (d)]. More precisely, these7\n(a)\n(b)\n(c)\n(d)\nFigure 6. Photo-induced spin polarization and currents of\na Co/Cu(100) heterostructure excited by p-polarized light.\n(a) Component Sx(t)of the site-averaged spin polarization\n(black) decomposed into that in the Co region (cyan) and\nthat in the Cu region (magenta). The latter are normalized\nwith respect to Nsite; cf. Eq. (7). The maximum of the laser\npulse att= 0 fsis marked by the vertical dashed line. (b)\nAs (a), but for Sy(t). (c) Currents jkl(t)depicted as color\nscale (red positive, blue negative). The Co/Cu interface is\nidentiﬁed by the horizontal dashed line. (d) As (c), but for\nspin-resolved currents jz\nkl(t). Arrows serve as guide to the\neye.\ncurrents are homogeneous in the Co region before the\npulse; they become enhanced at the interface at about\nt=\u00005 fs(dark red patches; also illustrated by the black\narrows). The x- andy-spin-resolved currents (not shown\nhere) are not as much aﬀected by the interface as the\nz-component, which suggests that the imbalance of mag-netization (spin-dependent occupation) at the interface\nis the most relevant origin.\nThe above argument concerning the importance of in-\nterfaces is further supported by the diﬀerent velocities of\njz\nkl(t)in the Cu and in the Co region (cf. the slopes of the\narrows). In the latter, we ﬁnd the homogeneous oscillat-\ning current pattern before the pulse maximum. In the\nCu region, which is non-magnetic, the same pattern ap-\npears oblique, as indicated by the black dashed arrow in\npanel (d). This means that these currents ‘spill out’ from\nthe Co region into the Cu region and propagate toward\nthe Cu surface (site 39).\nIV. CONCLUSION AND OUTLOOK\nOur theoretical ﬁndings suggest that femtosecond laser\npulsesimpingingonthinﬁlmsmaybeusedforgenerating\nultrafast oscillating spin-polarized currents. Moreover,\ninterfaces amplify the production of these currents, as is\nevidenced in our study. And the spin polarization can be\ntuned by details of the laser’s electric ﬁeld, in particular\nby the polarization of the radiation.\nInhomogeneities in the sample (surfaces, interfaces)\nyield intrinsic imbalances of occupation which facilitate\nthe production of spin-polarized currents. This ﬁnding\nsupports reasoning given in Ref. 29, in which it is argued\nthat a spin voltage, that is a spin-dependent imbalance\nof occupation, results in both demagnetization and spin\ncurrents. Moreover, thetransferofspinpolarizationfrom\nthe magnetic Co region into the non-magnetic Cu region\nof a Co/Cu heterostructure may also be regarded as spin\npumping: ﬂuctuating magnetic moments in a ferromag-\nnet produce spin currents in an attached normal metal\n[30, 31]. Hence, our study gives further details on the\nmechanisms for the transfer of spin polarization across a\nmagnetic/non-magnetic interface due to laser excitation,\nas reported for example in Refs. 32–35.\nAs is shown in this Paper, already the combination\nof 3d materials (here: Co and Cu) produces sizable spin-\npolarizationeﬀects. Thelattercouldbeenhancedfurther\nby increasing the imbalance of spin-dependent occupa-\ntion at interfaces. Material combinations worth investi-\ngating could comprise heavier elements with larger SOC\n(e.g. Pt) and heavy magnetic materials (e.g. Gd).\nA direct observation of the photo-induced spin polar-\nization and the spin-polarized currents studied in this\nPaper challenges experiments because of their limited\ntemporal resolution. However, it is conceivable to probe\nthe currents via their emitted electromagnetic radiation.\nAs a source of THz radiation usually an electric dipole\nalongy-direction is discussed, which is generated by a\nspin current jz. In contrast a spin current jy, carrying a\nspin polarization orthogonal to the interface normal and\northogonal to the magnetization direction generates an\nelectric dipole along z-direction and could thus be distin-\nguishedfromtheformeroneexperimentally. Theemitted\nradiation can be computed from the density matrix and8\nthe currents using the Jeﬁmenko equations [36, 37] and\nthus be compared with measured signals (e.g. Ref. 38).\nACKNOWLEDGMENTS\nThis work is funded by the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation) –Project-ID 328545488 – TRR 227, project B04.\n[1] S. Hüfner, Photoelectron Spectroscopy: Principles and\nApplications , 2nd ed. (Springer, Berlin, 1996).\n[2] W. Schattke and M. A. 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Münzenberg, Nature Nanotechnology 8,\n256 (2013)."    },    {        "title": "1606.07280v1.Spin_Hall_nano_oscillator_with_oblique_magnetization_and_Dzyaloshinskii_Moriya_interaction_as_generator_of_skyrmions_and_nonreciprocal_spin_waves.pdf",        "content": "Spin -Hall nano -oscillator with oblique magnetization and \nDzyaloshinskii -Moriya interaction as generator of skyrmions and \nnonreciprocal spin-waves \n \nA. Giordano1, R. Verba2, R. Zivieri3, A. Laudani4, V. Puliafito5, G. Gubbiotti6, R. Tomasello7, G. \nSiracusano1, B. Azzerboni5, M. Carpentieri8, A. Slavin9, and G. Finocchio1* \n \n1 Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of \nMessina, Messina, Italy  \n2 Institute of Magnetism, National Academy of Sciences of Ukr aine, Kyiv, Ukraine  \n3 Department of Physics and Earth Sciences and CNISM Unit of Ferrara, University of Ferrara, Ferrara, Italy  \n4 Department of Engineering, University of Roma Tre, Roma, Italy  \n5 Department of Engineering, University of Messina, Messina, It aly \n6 Istituto Officina dei Materiali del CNR (CNR -IOM), Sede Secondaria di Perugia , c/o Dipartimento di Fisica \ne Geologia,  University of Perugia,  Perugia, Italy  \n7 Department of Engineering, Polo Scientifico Didattico di Terni, University of Perugia, Terni , Italy  \n8 Department  of Electrical and Information Engineering, Politecnico di Bari, I -70125 Bari, Italy  \n9 Department of Physics, Oakland University, Rochester, MI 48309, USA  \n \nAbstract  \nSpin-Hall oscillators (SHO) are promising sources of spin -wave signals for magnonics application s, \nand can serve as building blocks for magnonic logic in ultralow power computation devices. Thin \nmagnetic layers used as “free” layers in SHO are in contact with heavy metals having large spin -\norbital interaction, and, therefore,  could be subject to the spin -Hall effect  (SHE)  and the interfacial \nDzyaloshinskii -Moriya interaction ( i-DMI), which may lead to the nonreciprocity of the excited \nspin waves and other unusual effects. Here, we  analytically and micromagnetically  study \nmagne tization dynamics excited in an SHO with oblique magnetization when the SHE  and i-DMI \nact simultaneously . Our key results are: (i) excitation of nonreciprocal spin -wave s propagating \nperpendicular ly to the in -plane projection of the static magnetization; (i i) skyrmions generation by \npure spin -current; (iii) excitation of a new spin -wave mode with a spiral spatial profile originating \nfrom a gyrotropic rotation of a dynamical skyrmion. These results demonstrate that SHO s can be \nused as generators of magnetic s kyrmions and different types of propagating spin -waves for \nmagnetic data storage and signal processing  applications.  Introduction  \n Spin-orbitronics combined with other sub -fields of spintronics, such as magnonics and spin -\ncaloritronics, has create d a novel  paradigm in information processing which could become a viable \nalternative to CMOS electroni cs1.  \n Recent experimental and theoretical developments in spin -orbitronics have clearly shown a \ngreat potential  in generation of  spin-currents able to compensate damping in magnetic \nmaterials2,3,4,5,6. The spin -Hall effect (SHE) plays a dominant role in the above -mentioned  \nexperiments, as it converts the input charge current , flowing in a heavy metal , into a spin-current , \ndiffusing  perpendicularly into the adjacent  ferromagnet , and creating  a spin -transfer torque (STT)  \nthat acts on the ferromagnet magnetization7. Another interesting and highly non -trivial spin -orbital \neffect is the interfacial Dzyaloshinskii -Moriya interaction ( i-DMI)8. Both SHE and i-DMI have \nbeen used to improve the performance of “racetrack” device prototypes in magnetic storage9,10, to \nadd a new degree of freedom in the design of magnetoresistive memories3,11, to create \nnonreciprocity in the spin -wave p ropagation for signal processing applications12,13,14, to excite \ncoherent magnetization self-oscillations4,5, and for the manipulation of skyrmions in ultrathin \nferromagnetic material s15,16. However, to the best of our knowledge, the influence of i-DMI on the \nperformance of a spin-Hall oscillator  (SHO ) has not been studied so fa r4,5.  \n Here , we pr esent the magnetization dynamics induced by the SHE in a realistic SHO \nstructure , taking  into account the influence of the i-DMI8. We have chosen a state-of-the-art SHO \ngeometry  (Fig.1a) where the charge  curren t I flows in the Pt layer along the x-axis between the \ngolden electrodes and, due to the SHE in Pt, a spin-current is locally inject ed into the ultrathin \nextended CoFe  ferromagnet (SHO “free” layer). The CoFe  layer has an in -plane easy axis at zero \nbias fi eld, so when a sufficiently large out -of-plane bias field is applied at an oblique angle in the \n“yz” plane (Fig.1 b), the static magnetization M of the “free” layer al so goes out -of-plane, making \nthe angle M with the vertical axis “ z”. In such a geometry , the Slonczewski propagating  spin \nwaves17 can be excited in any in -plane direction18,19,20 and, due to the influence of  the i-DMI, they \nhave the maximum nonreciprocity when propagating along the x-axis, perpendicular to the in -plane \nprojection of the bias magn etic field.  Our numerical simulations have shown that the wave numbers \nof spin waves excited at a particular frequency  and propagating along the positive and negative \ndirections of the x-axis are different. The difference is proportional to the magnitude of the i-DMI \nparameter D. This result, well reproduced by a simple one -dimensional analytical model, can be \nused to e stablish a novel procedure for the experimental measurements of D. Micromagnetic \nsimulations have also demonstrated that ( i) a novel propagating spin -wave mode, characterized by a \nspiral spatial profile, can be excited at sufficiently large magnitudes of D and I, and ( ii) skyrmions  can be efficiently nucleated by the SHE in the SHO geometry (Fig.1c). Similarly to optics 21,22, the \nexcitation of spiral spin -waves in magnetism could be attractive for designing new information \ncoding  protocols. Recent experiment al observations have demonstrated that skyrmions15,16,23,24,25 \ncan be nucleated via conversion of domain walls in Ta/CoFeB/MgO26, or by applying an  out-of-\nplane field in Ir/Co/Pt27 and Pt/Co/MgO28 multi layers . Although  a single skyrmion can be nucleated \nby a spin-polarized scanning tunneling micro scope29, the control of its room temperature nucleation \nis still an experimental challenge . Earlier achievements  have shown the possibility to solve this \nproblem15,23,30. Our results show an alternative method to control the nucleation of single \nskyrmions, based on the use of the SHE .  \n \n \nFigure 1 | Sketch of the SHO device under investigation and dynamica l phase diagram of this device.  a, The \nsketch of a bilayer composed of the CoFe ferromagnetic layer and a layer of a heavy metal (Pt) and having the \nrectangular cross -section. The thick Au electrodes carry the charge current  everywhere, except the inter -electrode gap of \nthe width d, where the charge current flows inside the bilayer and excites perpend icular (vertical) spin current going \ninto the CoFe ferromagnetic layer. A rectangular coordinate system for the above described  SHO geometry is shown . b, \nThe angle \nM  characterizing the  equilibrium  direction of the static magnetization in the CoFe ferromagnetic  layer as a function of the magnitude of the external bias magnetic field B. The vertical line at  B=200mT separates the regions \nwhere it is possible to excite localized and propagating spin -wave modes, respectively. Inset: Cartesian coordinate \nreference system  where the angles \nM  and \nB  are shown expli citly. c, The phase diagram of the SHO excitations on \nthe D vs. I plane. Seven different regions can be distinguished in this phase diagram: uniform states (US), Slonczewski \nlinear modes (SLM), spiral modes (SpM), skyrmions (SKY), uniform states/skyrmions (US/SKY), Slonczewski linear \nmodes/spiral modes (SLM/SpM) and Slonczewski linear modes/skyrmions (SLM/SKY). The amplitude of the external \nfield is B=400mT. The Oersted field is included in the model. DC is the critical value of the i-DMI parameter (see \nexplanations below), Ith is the threshold current, Isky is the current needed to nucleate skyrmions (line between the points \n'A' and 'B'); d, Comparison between the threshold current  of the SLM excitation obtained by means of micromagnetic \nsimulations (sym bols) and using the analytical formula  (3) (solid line).  \n \nResults  \nStatic characterization of the SHO structure and phase diagram of the SHO excitations . We \nhave micromagnetically studied a Pt(5nm)/CoFe(1nm) SHO with a rectangular cross section of \n1500x 3000nm2 (see  Fig.1a for the  sketch of the device, including a Cartesian coordinate system \nwhere x and y are the in -plane axes, w hile z is the out -of-plane axis, Methods  and Supplementary \nNote 1 for the detailed description of the micromagnetic frame work and simu lation parameters ). \nFig.1b shows the angle \nM , characterizing the equilibrium orientation of the static magnetization in \nthe SHO, as a function of the external bias magnetic field B. This field is applied at the tilting angle \nB\n=15° with respect to the perpendicular of the SHO ferromagnetic layer in the y-z plane (see inset \nin Fig.1b). As the bias field increases, the magnetization vector tends to align along the field \ndirection.   \n Similarly  to what is obser ved in STT oscillators based on the point -contact geometry, the \ntype of the spin -wave mode excited by the SHE can be controlled by the direction of the bias \nmagnetic field and the effective anisotropy. In particular, the materials with in -plane easy axis \ndemonstrate excitation of self -localized spin -wave “bullets” for sufficiently large values of θM, and \nexcitation of Slonczewski  propagating spin -wave modes for sufficiently small values of θM18,31. In \nthis study, numerical simulations showed that, for the bias field larger than 200mT and θM<37o, the \nSlonczewski  propagating spin -wave modes were excited.   \n As it will be discussed below, the additional degree of freedom of the i-DMI can introduce \nqualitative differences in the spatial profile of the Slonczewski -type cylindrical mode, compared to \nthe case whe n i-DMI is ignored. Hereafter, we focus on the results obtained at the bias field of \n400mT and active region (distance between the Au electrodes in Fig.1a) of d=100nm , however \nsimilar findings have been obtained at d=200nm  and at larger bias fields (up to 800mT).    Fig. 1c shows a  phase diagram  of dynamical excitations in the SHO on the plane D-vs-I. \nSeven different regions can be identified: ( i) uniform states (US), ( ii) Slonczewski linear modes \n(SLM), ( iii) spiral modes (SpM), ( iv) skyrmions (SKY), ( v) uniform states/skyrmions (US/SKY), \n(vi) Slonczewski linear modes/skyrmions (SLM/SKY) and ( vii) Slonczewski linear/spiral modes \n(SLM/SpM). At small values of the driving current, the SHO is in the US, i.e. in a region \ncharacterized by a uniform magnetic confi guration. SLMs are excited at a critical current  Ith that \nslightly decreases as a function of D (see Fig.1 d). The excited modes in the SLM region exhibit a \ntwo-dimensional radiation pattern that changes from the isotropic (see Supplementary Movies 1 and \n2 for the SLM dynamics at I=4.22mA and I=5.28mA respectively,  B=400mT and D=0.0mJ/m2) to \nthe anisotropic cylindrical profile with the increase of the i-DMI parameter D (see Supplementary \nMovies 3 and 4 for the SLM at I=4.22mA and I=5.28mA, respectively,  B=400mT and \nD=1.5mJ/m2). The cylindrical profile of the spin -wave radiation evolves into a spiral -like profile for \n1.5mJ/m2<D<3.0mJ/m2, and a sufficiently large I (SpM region)  (see as an example Supplementary \nMovie 5 ). The identification of the scenario leadin g to the radiation of these spiral spin-wave modes \nis one of the most important results of this study.  \n The SKY region is observed starting from D values near the critical value of DC=3.16mJ/m2 \n(\n  0 4 ( 0.5 ) /S u C Z SDA K B M M    )32.The skyrmion nucleation process , driven by the SHE, \noccurs together with the excitation of propagating spin -waves (see for an example Supplementary \nMovie 6 ), and at the current Isky (solid line between the point 'A' and 'B'). The Isky curve coincides \nwith Ith for D values larger than 4m J/m2 (point 'A' in Fig.1c). This fact constitutes the second key \nresult of this study, i.e. the prediction that a pure spin -current with in -plane polarization can be used \nfor the nucleation of skyrmions. The regions US/SKY, SLM/SKY and SLM/SpM are the bist ability \nregions. In the first of these regions, we have a uniform ground state or skyrmions, depending on \nthe excitation history. The second region is characterized by the excitation of SLMs that can coexist \nwith the presence of skyrmions after they have b een nucleated at the current Isky. In the last region, \nan SLM (SpM) is observed, if the current is increased (decreased) from SLM (SpM) region. The \norigin of this hysteretic behavior will be discussed in detail below. We have also investigated the \nrole of the Oersted field, finding that it does not influence qualitatively the results of Fig.1c (s ee \nSupplementary Note 2  for more details).  \n \nExcitation of Slonczewski linear spin -wave mode. As it was pointed out earlier, the i-DMI leads \nto the excitation of non reciprocal spin -waves. It can be observed qualitatively in the Supplementary \nMovies 3 and 4, and by comparing the mode profile in Figs.2 a and 2b. The largest nonreciprocal \neffect induced by the i-DMI occurs in the direction perpendicular to the in -plane pr ojection of the static magnetization M0 (x-axis), while the propagation along the in -plane projection of M0 (y-axis) \nis reciprocal, and is characterized by the wave number that is the same as in the case of zero D (0.03 \nand 0.035nm-1 at I=4.22mA and I=5.28 mA, respectively, see Supplementary Note 2). Those results \nare consistent with the previous experimental measurements33 and the results of the analytical \ntheory12. The i-DMI -induced appearance of the nonreciproc al spin -waves leads to the decrease of \nthe threshold current (Fig.1d), and to a “red” shift of the generation frequency at a constant current \n(Fig.2 c). Fig. 2d summarizes the dependence of the wave numbers (| k-x| and | k+x|) on D computed \nfrom the spatial di stribution of the magnetization for I=4.22mA and I=5.28mA. The difference \nbetween the | k-x| and | k+x| is shown in Fig.2e, and, as it can be noticed, is independent of I and D. All \nthese numerically obtained features can be understood using a simple one -dimensional analytical \nmodel. In the framework of this model, we consider only the spin -waves propagating along the x-\ndirection, where the spin -waves exhibit the largest nonreciprocity . \n The frequency and wave vectors of the excited spin-waves  are de fined by the spatial \nquantization rule , which is determined by the spatial distribution of the spin -current\nsJ . In the case \nof a nonreciprocal spectrum, the general quantization rule can be written as  \n( , ( )) 0s f k k J x34, \nor, equiva lently \n1( ( ))s k k const f J x   , where \nk  and \nk  are the wave vectors of spin-waves  \npropagating in opposite directions along the direction of maximum nonreciprocity and having  the \nsame frequency  (for a reciprocal wave spectrum this rule is reduced to the condition \n||k const ). \nThe approximate spin -wave  spectrum in the x-direction can be written as \n22\n0 k M x M x k Dk       \n (see Methods) , where  \n0 is the angular  frequency of the ferromagnetic \nresonance  in the SHO , \n2\n0 2 / ( )sinSM D D M\n , \n2 2 2\n0 (2 (1 / )sin ) / 2H M an S M HM        \n , \n \nis the exchange length in the material of the SHO ferromagnetic layer, and \n0 2u an S K HM   is \nthe anisotr opy field. From this equation , the wave vectors of counter -propagating nonreciprocal \nspin-waves, having the same frequency \n , can be computed as : \n 2 2 2\n0 214 ( )2x M M M\nMk D D        \n                                        (1) \nwhere \nxk  (\nxk) is associated with the plus (minus) sign in the second term in the circular brackets \nin the equation (1). Substituting the wavenumber of equation  (1) in the quantization rule , we get  the \ncondition \n2 2 2\n01 4 ( )MMD const      \n  that gives the following dependence of the generation  \nfrequency on D: \n22/ (4 )MD   \n . Thus, the generation frequency has a “red” shift with the \nincreased D, as obtained from our micromagnetic simulations (see Fig. 2c). This ef fect could be \neasily understood by noting that the minimum spin-wave  frequency in the spectrum becomes lower with the increase of D. From equation (1), it is easy to calculate  the difference between the wave \nnumber s of the excited waves:  \n                       \n2/xxk k D \n                                                                (2) \nand to verify that this difference is independent  of the quantization constant and, t herefore , of the \nspatial distribution of the spin-current . Hence, the con dition (2) can be used for the experimental \ndetermination of the magnitude and sign of the i-DMI parameter . Equation (2) g ives a reasonable \ndescription of the simulation data , considering the same physical parameters of the SHO  (Fig.2e). \nThe fact that the dependence \nkD  is almost the same for different I is linked to a weak nonlinear \nvariation of the spin -wave  spectrum  with driving current,  due to a small difference in amplitudes of \nthe excited spin-waves. Therefore, the difference o f the spin -wave numbers is mainly determined by \nthe linear spin-wave  spectrum . From an experimental point of view, a direct determination of D can \nbe achieved by measuring the wavelength of the emitted spin -waves along the + x and –x direction, \nusing the phase-resolved micro -focused Brillouin light scattering35 or time-resolved Kerr \nmicroscopy36. However, this method of determination of D may have practical limitations due to the \nfact that the wavelength of the excited spin -waves (see Fig.2d) are in the range 0.13-0.63μm, i.e. \nbeing comparable with the lateral resolution of the above mentioned optical techniques.  \n Within the above described one -dimensional model, we can also calculate the dependence of \nthe threshold current for spin -wave excitation on D. Assumi ng a rectangular profile of the charge \ncurrent density in the active region (\nJ x J  within \n[0, ]xd , and \n0 Jx  otherwise ), one can \nget the following implicit expression (similar to equation (6c) in37), \n\n\n\n\n\n \n\n vikid\nvikvikG J G J G\n2tan\n                         (3) \nwhere \n/2xx k k k  is the average wave number of excited nonreciprocal spin -waves  (note, \nthat in our notation \n0xk ), d is the distance between the SHO golden e lectrodes characterizing the \nspatial localization of the spin -current, \n  21 2\n02 2 ~4~  M MD v  is the spin -wave group \nvelocity, \nG G  is the spin -wave damping, \nJJ  is the negative damping created by the \nspin-current, and \nsin / (2 )B H M S CoFeg eM t     determines the spin-Hall efficiency (\ng  is the \nLandè factor, \nB  the Bohr magneton,  \ne the electronic charge and \nCoFet  the Co Fe layer thickness ). \nThe threshold current calcu lated from  equation (3) is compared with numerical results in Fig.1d by \nusing the value Ith=3.7mA at D=0mJ/m2 as a fitting parameter. One can see a good coincidence \nbetween the analytical and numerical results. N ote that the decrease of the threshold current with D has the same nature, as a frequency “red” shift -lowering the bottom of the spin -wave spectrum with \nthe increase of D and, consequently, the decrease of spin -wave damping  \nG G . \n The SL M in SHOs have not been observed experimentally, since the threshold current for \ntheir excitation is expected to be very large (>109A/cm2)20, around three times larger than the \ncurrent necessary to excite a “bu llet” spin -wave mode in an SHO with in -plane magnetization. In \nthe SHO of this study, we were able to reduce the critical current density of one order of magnitude \n(<4×108A/cm2) thanks to the additional perpendicular interface anisotropy in the CoFe ferrom agnet. \nThis additional anisotropy allows one to achieve the positive nonlinear frequency shift, required for \nthe SLM excitation31, at a higher magnetization angle \nM , which results in t he  higher spin -Hall \nefficiency, since it is proportional to \nsinM . A further reduction of the current density can be \nachieved by including an additional Ta layer above the CoFe ferromagnet24. \n \nFigure 2. | Non-reciprocal Slonczewski linear modes.  a, Oscillation frequency of the excited SLM as a function of D \nfor two values of the driving current . b and c, Example of the spatial profile of the reciprocal and non reciprocal \nSlonczweski spin  waves, respectively, calculated for  I=5.28 A  and  D=0 and 1.5mJ/m2 , respectively.  d, Wave numbers \nalong the -x and + x directions as functions of D for two values of current; e, Difference between the wave numbers \nalong the positive and negative x-direction s as a function D for the same two values of current (solid lines) and the same \ndifference calculated analytically from equation  (2) (dashed line).    \nExcitation of spin -wave modes with a spiral spatial profile.  Fig.3a summarizes the spin -wave \nfrequency as a function of I computed for D=0.0mJ/m2 and D=1.5mJ/m2 (d=100nm). In the absence \nof the i-DMI, the oscillation frequency shows a monotonic increase with current, or a “blue” \nfrequency shift, typical for the Slonczewski linear propagating spin -wave mode . A different \nfrequency behavior is seen for  D=1.5mJ/m2, where the frequency tunability with current becomes \nnon-monotonic. This behavior is robust under the variation of d, as seen from Fig.3b where \nd=200nm . At sufficiently large I and D, the spin -wave is con verted from the cylindrical to a spiral -\nlike (SpM region in Fig.1c).  Fig.3c shows a spiral -type profile (the color  is linked to the y-\ncomponent of the magnetization).  \n In order to understand the origin of the spiral mode, we have performed a detailed analy sis \nof the spatial distribution of the dynamic magnetization in the SHO ferromagnetic layer in this \nregime. Figs.3d -g illustrate four snapshots ( I=6.33mA) which clearly reveal the physics of the spiral \nmode formation. In the SpM region, the SHE is able to nucleate a dynamical soliton38,39,40. It is \ncharacterized by a central core with the magnetization pointing along the negative out -of-plane \ndirection (opposite to the equilibrium axis of the magnetization), and by the rotation of its boundary \nspins through 36 0° (see Figs.3d -g). The dynamical skyrmion exhibits a rotational motion (gyration) \nalong a circular trajectory within the region of the high current density, that is typical for solitons \nwith nonzero topological charge under the influence of spin -current41 (see Supplementary Movie 7 ). \nDynamical skyrmion plays a role of a “source” for magnetization oscillations in the outer region, \nand, since the source is gyrating, the radiation acquires the form of a spiral wave, as it happens in \nmany other fields with gyra ting source42,43. Note, also, that once it has been excited the SpM is still \nstable at lower current magnitudes in the SpM/SLM  region, because the excitation of the dynamical \nskyrmion is linked to a sub -critical Hopf bifurcation40. \n  \nFigure 3 | Excitation of a  spin-wave with spiral profile.  a, Oscillation frequency of the excited mode as a function of \nI without and with i-DMI, for d=100 nm . b, Same as a, but with d=200 nm.  c, Example of a spatial profile of the spiral-\ntype spin -wave for I=6.22mA and D=1.5mJ/m2. d-g, Spatial distributions of the  magnetization characterizing a \ntopologi cal-type magnetic soliton, the current -induced gyration of which causes the radiation of a spiral -type spin wave \nmode.  \n \nGeneration of  single skyrmions and “gas” of skyrmions.  The last regions of the phase diagram \nof Fig.1c are related to skyrmions.  For the critical DC, the skyrmions become energetically stable32 \nand, after the  nucleat ion dri ven by the  SHE  (SKY region)  (see Supplementary Movie 6 for the \nnucleation of a single skyrmion ), they remain stable even when the driving current is switched off \n(US/SKY region). Once the skyrmion is nucleated, it is shifted along the spin -current directio n, as \nexpected for Néel skyrmions16. For D below 4.0mJ/m2 (point 'A' in Fig.1c), Isky and Ith split into \ndifferent curves, and, hence, in the SKY/SLM region when the current increases from the uniform \nstate, on ly the SLMs are excited. The presence of this region in the phase diagram is interesting \nfrom a fundamental point of view, as it identifies a scenario where the interaction between the spin -\nwaves and skyrmions44 can be studied. Fig.4a shows the nucleation t ime of a single skyrmion as a function of the current magnitude for two values of D (3.5 and 4.0mJ/m2). It can be seen from \nFig.4a that a sub -nanosecond skyrmion nucleation time can be achieved (see Fig.4b for a single \nskyrmion snapshot). Our results predi ct a new scenario for a single skyrmion nucleation driven by a \npure spin -current. This method can be used as an alternative to the method based on the STT from a \nperpendicular spin -polarized current15, with the  possible advantage of the simpler fabrication \nprocess of the device. If the current is not switched off, the skyrmions are nucleated continuously. \nHowever, since the current is non -uniformly applied, the skyrmions tend to accumulate in one side \nof the fer romagnet. A skyrmion “gas” is, therefore, formed45 (see Fig.4c for an example of the \nspatial distribution of the skyrmions). When the “gas” saturates, i.e. when no more skyrmions can \nbe nucleated in the “gas” because of the skyrmion -skyrmion magnetostatic r epulsion, each \nskyrmion further nucleated is immediately annihilated (see Supplementary Movie 8 ). This result \npaves the way to study the magnetic properties of skyrmion “gas” described theoretically in45.  \n \nFigure 4 | Skyrmion nucleation . a, Nucleation time of a single skyrmion as a function of the current amplitude for D = \n3.5 and 4.0mJ/m2. b, Snapshot of a single skyrmion and zoom of  the skyrmion nucleation region.  c, Snapshot of a \nskyrmion gas and zoom of th e region indicated by a green square in the right frame.  \n \nDiscussion  \n In our study, we propose an SHO device geometry that, combining SHE and i-DMI, offers a \nunique opportunity to study nonreciprocal effects of spin -wave propagation in two dimensional \nsystems and to observe a new type of dynamical spin -wave modes having a spiral spatial profile. \nThis novel spin -wave mode originates from the gyrotropic rotation of a dynamical  skyrmion. From \nthe technological point of view, the proposed SHO geometry could be useful for the development of \nnovel generators of short propagating spin -waves in future magnonic signal processing devices. \nFrom the fundamental point of view, it is also very interesting, as it allows to study the interaction of spin -wave and skyrmions, as well as to control the number of the nucleated skyrmions by \napplying a properly designed current pulse.  \n \nMethods  \nMicromagnetic framework.  Micromagnetic simulations were  carried out by means of a state -of-\nthe-art parallel micromagnetic solver, which nume rically integrates the LLG equation including the \nSlonczewski -like torque due to SHE46,20: \n  \n2\n0ˆ\n2B\nGH\nS CoFeg ddzd d eM t         EFFmmm h m m m J\n                         (4) \n \nwhere m and \nEFFh  are the normalized magnetization and th e effective  field of the ferromagnet. The \neffective field includes the standard magnetic field contributions, as well as the i-DMI and Oersted \nfield (see  also Supplementary Note 1 ).  is the dimensionless time \n0 SMt , where \n0  is the \ngyromagnetic ratio, and  Ms is the saturation magnetization of the ferromagnet. \nG is the Gilbert \ndamping, g is the Landè factor, \nB  is the Bohr Magneton , e is the electron charge, tCoFe is the \nthickness of the ferromagnetic layer, H is the spin -Hall angle obtained from the ratio between the \nspin current and the electrical current. \nˆz  is the unit vector of the out -of-plane direction and J is the \nin-plane current density injecte d via the heavy metal. The i-DMI  energetic density expression, as \nderived considering the ultra -thin film hypothesis\n0zm , is \ni DMI z z D m m   mm  \nD being the parameter taking into account the intensity of the DMI, and mz is the z-component of \nthe normalized ma gnetization. By making the fun ctional derivative of equation, the normalized i-\nDMI effective field is given by:  \n22 z\n0 02ˆ·1i DMI\ni DMI\nSSMMDzm\n \n   hmm\n                                 (5) \nThe boundary conditions related to the interfacial  DMI are expressed by \n1ˆdzdn  mnm  where \nn is the unit vector normal to the edge and \n2A\nD  (being A the exchange constant) is a \ncharacteristic length in the presence of i-DMI.   \n We have studied a bilayer system Pt(5nm)/CoFe(1nm) with a rectangular cro ss section of \n1500x3000nm2. The electric current was locally injected into the ferromagnet via a thick Au electrode (thickness of 150nm) with two tips located at a distance d from each other.  The charge \ncurrent flowing in the Pt layer gives rise to the SHE  and then, to flow of perpendicular (along the \n“z” axis) pure spin current at the Pt/CoFe interface creating an anti -damping Slonczewski -like \ntorque in the ferromagnetic layer. At sufficiently large magnitudes of the charge current his torque \ncompensates t he Gilbert losses in the ferromagnetic layer and excites in it persistent magnetization \noscillations. For the results discussed in the main text, we have considered the following physical \nparameters of the SHO (Fig.1a): saturation magnetization MS=1x106A/m47, exchange stiffness \nconstant A=2.0x10-11J/m, interfacial perpendicular anisotropy induced at the boundary between \nCoFe and Pt characterized by the anisotropy constant Ku=5.5x105J/m3,48 damping constant \nG\n=0.0349, and the spin -Hall angle \nH =0.19. The ferromagnetic CoFe layer has an in -plane \nequilibrium magnetization at zero field which is directed along the y- in-plane direction due to the \nshape anisotropy of the ferromagnetic layer. The real spatial distributions of the density \neJ  of the \ncharge current, density \nsJ of the spin current and the Oersted field were  calculated numerically, as it \nis described in the  Supple mentary Note 1 . \n \nDerivation of analytical equations.  The spin wave dispersion relation along the x-direction in the \npresence of i-DMI can be calculated analogously to Ref. [12] and has the following form  \n     2 2 2 2 21 sink H M x H M x M an S M M x k k H M Dk               \n,           (6) \n where  \nH eff B , \neffH  is the static effective field, \n0 Ms M  . In the range \n22\n0 Mk  \n  it can \nbe approximated as \n22\n0 k M x M x k Dk       \n , where \n   2\n0 1 / sinH H M an s M HM         \nis the angular frequency of the ferromagnetic resonance  in the ferromagnetic layer, and \n2 2 2\n0 (2 (1 / )sin ) / 2H M an S M HM        \n. \nMaking a formal substitution \n/xk i d dx  in this dispersion equation, it is possible to obtain  the \nfollow ing dynamical equation describing the  spatial and temporal evolution of the spin wave \ncomplex amplitude a:  \n2\n2\n0 2()M M Gai a i i D a a J x at x x                   \n.             (7) \nThe spin wave damping is accounted for by the term \nG  (spin wave ellipticity, which could \nmodify the damping term50 in our case is small), while the influence of the spin current c ould be \neasily calculated from equation (4 ) within the framework of the perturbation theory50 and is given \nby the term \n()J x a , with \nsin / (2 )B H M S CoFeg eM t    . 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Collective spin -wave excitations in a two -\ndimensional array of coupled magnetic nanodots, Phys. Rev. B.  85, 014427 (2012).  \n \nAcknowledgments  \nThis work was supported by the project PRIN2010ECA8P3 from Italian MIUR , the bilateral \nagreement Italy -Turkey (TUBITAK -CNR) project (CNR Grant #: B52I14002910005, TUBITAK \nGrant #113F378) “Nanoscale m agnetic devices based on the coupling of S pintronics and \nSpinorbitronics”, and the executive programme of scientific and technological cooperation between Italy and China for the years 2016 -2018 (code CN16GR09 ) title \"Nanoscale broadband spin -\ntransfer -torque microwave detector\" funded by Ministero degli Affari Esteri e della Cooperazione \nInternazionale.  A.S and R.V. acknowledge support from the Grant ECCS -1305586 from the \nNational Science Foundation of the USA, from the contract with the US Army TARDEC, \nRDE COM, from DARPA MTO/MESO grant N66001 -11-1-54114, and from the Center for \nNanoFerroic Devices (CNFD) and the Nanoelectronics Research Initiative (NRI).  The authors thank \nDomenico Romolo for the graphical support . R. T. acknowledges Fondazione Carit - Proje cts – \n“Sistemi Phased -Array Ultrasonori ”, and  “Sensori Spintronici ”. \n \nAuthor  contributions  \nA.G., R.Z., M.C., G. G. and G.F. initiated the work and designed the numerical experiments. R.V. \nand A.S. developed the analytical theory and performed the analytica l calculations. A.L. performed \nthe computation of the spatial distribution of the current density and the Oersted field and wrote the \nsupplementary note 1. A.G. performed micromagnetic simu lations supported by V.P., G.S., B.A \nand M.C. V.P. wrote the supple mentary note 2  and prepared the last version of the figures . A.G., \nand G.F. analyzed the data. G. F. wrote  the paper with input from R.V., A.S. and R.T. All authors \ncontributed to th e general discussion of the results and commented  on the manuscript.  \n \nAddit ional information  \nSupplementary Information  accompanies this paper at ...  \n \nCompeting financial interests : The authors declare no competing financial interests.  \n \nReprints and permission information  is available online at ...  \n \nHow to cite this article : Xxxx  \n \n "    },    {        "title": "1409.1809v1.Room_temperature_spin_thermoelectrics_in_metallic_films.pdf",        "content": "Room temperature spin thermoelectrics in metallic \flms\nSebastian T olle,1Cosimo Gorini,1, 2and Ulrich Eckern1\n1Universit at Augsburg, Institut f ur Physik, 86135 Augsburg, Germany\n2Service de Physique de l' \u0013Etat Condens\u0013 e, CNRS URA 2464, CEA Saclay, 91191 Gif-sur-Yvette, France\nConsidering metallic \flms at room temperature, we present the \frst theoretical study of the spin\nNernst and thermal Edelstein e\u000bects which takes into account dynamical spin-orbit coupling, i.e.,\ndirect spin-orbit coupling with the vibrating lattice (phonons) and impurities. This gives rise to\ntwo novel processes, namely a dynamical Elliott-Yafet spin relaxation and a dynamical side-jump\nmechanism. Both are the high-temperature counterparts of the well-known T= 0 Elliott-Yafet\nand side-jump, central to the current understanding of the spin Hall, spin Nernst and Edelstein\ne\u000bects at low T. We consider the experimentally relevant regime T > T D, withTDthe Debye\ntemperature, as the latter is lower than room temperature in transition metals such as Pt, Au and\nTa typically employed in spin injection/extraction experiments. We show that the interplay between\nintrinsic (Bychkov-Rashba type) and extrinsic (dynamical) spin-orbit coupling yields a nonlinear T-\ndependence of the spin Nernst and spin Hall conductivities.\nI. INTRODUCTION\nE\u000ecient heat-to-spin conversion is the central goal of\nspin caloritronics.1When considering metallic systems,\ntwo interesting phenomena stand out in this \feld: the\nspin Nernst2,3and thermal Edelstein e\u000bects.4,5They con-\nsist in the generation of, respectively, a spin current or\na spin polarization transverse to an applied temperature\ngradient. That is, they are the thermal counterparts of\nthe well known spin Hall6and Edelstein e\u000bects.7,8These\nphenomena are due to spin-orbit coupling and do not re-\nquire the presence of magnetic textures or Zeeman \felds,\nand are typically classi\fed as intrinsic orextrinsic de-\npending on their origin { respectively band and device\nstructure or impurities.\nSpin Hall measurements are typically performed in\ntransition metals such as Au, Pt or Ta,9{13where such\ne\u000bects are orders of magnitude larger than in standard\nsemiconductors,14and, very importantly, at room tem-\nperature. In this temperature regime the dominant\nmomentum-degrading scattering mechanism in bulk is\nelectron-phonon scattering. Therefore the latter will,\nthrough spin-orbit coupling, heavily a\u000bect the spin Hall\nsignals. An identical reasoning applies to the Edelstein,\nthermal Edelstein and spin Nernst e\u000bects, though the\nlast two have yet to be experimentally observed. Indeed,\nthe spin-orbit interaction adds an interesting twist to the\ncoupling between electrons and phonons: electrons in a\ndisordered lattice at T= 0 move in a \\frozen\" electro-\nstatic potential U(r) =Vcrys(r)+Vimp(r) arising from the\ncrystal lattice and the impurities, yielding in the Hamil-\ntonian the terms\nU(r) +\u00152\n0\n4~\u001b\u0002rU(r)\u0001p; (1)\n\u001b;pand\u00150being, respectively, the vector of Pauli ma-\ntrices, the electron momentum and the Compton wave-\nlength. The potential becomes, however, time-dependent\nat \fniteT,U(r)!U(r;t):Thus, the lattice (impurity)\ndynamics will not only give rise to standard electron-\nphonon (dynamical impurity) scattering through thetermU(r;t), but will also couple directly to the car-\nrier spin through the dynamical spin-orbit interaction\n\u00152\n0\u001b\u0002rU(r;t)\u0001p=4~. Remarkably, such a direct \\spin-\nphonon\" (\\spin-dynamical impurities\") coupling has not\nyet been studied, and even standard electron-phonon\nscattering has received minimal attention in the present\ncontext. To the best of our knowledge, the only theoret-\nical work considering the impact of standard electron-\nphonon interaction on the spin Hall e\u000bect is that of\nGrimaldi et al.15, which is focused on a 2-Dimensional\nElectron Gas (2DEG) with Bychkov-Rashba16spin-orbit\ncoupling at T!0.\nOur purpose is to start \flling this gap, considering\nthe spin Hall, spin Nernst, Edelstein and thermal Edel-\nstein e\u000bects in a metallic thin \flm at room temperature.\nMoreover, we wish to identify the possible connections\nbetween the four phenomena. It is known, for example,\nthat in a 2DEG at low Tthe spin Hall and Edelstein ef-\nfects are closely related,17,18and that such a relation can\nbe extended to thin (quasi-2D) \flms as well.19Whether\nthis connection exists, possibly in a modi\fed form, also\nat highTor in 3D is an open question. Another impor-\ntant point concerns the T-dependence of the above cited\ne\u000bects. For example, whereas this is expected to be linear\nfor a purely extrinsic spin Nernst e\u000bect2, it is not known\nhow the interplay between extrinsic and intrinsic mech-\nanism will modify such behavior. Similarly, for the spin\nHall e\u000bect its T-dependence should allow to establish its\nspeci\fc intrinsic or extrinsic origin.20,21The latter is still\na somewhat controversial issue, in particular in Au and\nPt.9,21{23\nOur treatment relies on two central assumptions. The\n\frst one is based on the observation that the Debye\ntemperature TDof bulk Au (165 K), Pt (240 K) or\nTa (240 K) is lower than room temperature, and in\nthis regime electron-phonon scattering is predominantly\nelastic .24This leads to a remarkable simpli\fcation of the\nquantum kinetic equations we will employ, allowing to\nextend to the present case the analysis of the T= 0\nscenario.24The second one concerns the type of spin-\norbit interaction in a metallic \flm. There is yet noarXiv:1409.1809v1  [cond-mat.mes-hall]  5 Sep 20142\ngeneral theory capable of identifying its precise e\u000bective\nform, but experiments show that a strong Rashba-like\nspin-orbit interaction appears at the interface between\ntransition metals and insulators/vacuum,25{27where in-\nversion symmetry is broken; density functional theory\nhas been recently used to estimate its size in Ag, Au or Al\non W(110) structures.28In general, spin-orbit splittings\nof up to several hundreds of meV are reported { that is,\nconsiderably larger than in a standard GaAs 2DEG. We\nwill thus assume the intrinsic spin-orbit mechanism to\nbe described by a Rashba term in the Hamiltonian. The\nextrinsic one will be treated in analogy with the semi-\nconductor case, where the spin-orbit interaction with the\nimpurity potential is mediated by an e\u000bective Compton\nwavelength renormalized by the lattice.29,30\nExperimentally realized \flms explore the full 2D\nto 3D range, thicknesses ranging from one or few\nmonolayers25{27, up to few to tens of nanometers.11,13,31\nWe will start by considering a strictly 2D metallic layer,\nand later comment on its 3D counterpart. For the latter\ncase our approach follows the spirit of Ref. [32], which\ntakes the Rashba-like term to be homogeneous across the\n\flm thickness. Notice that this is complementary to what\nis done in Refs. [19, 33, and 34], where the Rashba inter-\naction is a \u000e-function di\u000berent from zero only exactly at\nthe \flm edges.\nFinally, we will rely on the SU(2)-covariant kinetic\nformulation introduced in Ref. [35]. This considerably\nsimpli\fes the collision integrals to be faced,18,35and en-\nsures the unambiguous de\fnition of spin-related physical\nquantities even when the spin itself is not conserved (due\nto spin-orbit interaction).35,36In particular, as shown in\nRef. [37], it provides the framework to properly de\fne\nOnsager reciprocal relations in the presence of spin-orbit\ncoupling, e.g., between the direct and inverse spin Hall37\nor Edelstein38e\u000bects. This ensures that our results will\nhave an immediate bearing on the inverse counterparts\nof the phenomena considered below.\nThe paper is organized as follows. We \frst introduce\nthe model and the linear response formulation in Sec. II,\nthen move on to the kinetic approach in Sec. III. Sec-\ntion IV discusses the core results, namely the spin Nernst\nand thermal Edelstein e\u000bects. The focus is on their T-\ndependence and their relation with each other, as well\nas with the spin Hall and Edelstein e\u000bects. We conclude\nwith a brief summary. Certain general but cumbersome\nformulas are given in Appendix A, whereas the estima-\ntion of di\u000berent spin lifetimes appear in Appendix B.\nII. THE MODEL AND THE ONSAGER\nFORMULATION\nLet us start from the following e\u000bective (static) model\nHamiltonian for conduction electrons in a parabolic\nband:39\nH0=p2\n2m\u0000\u000b\n~\u001b\u0002^z\u0001p+Vimp(r)\u0000\u00152\n4~\u001b\u0002rVimp(r)\u0001p:(2)As customary, the static lattice potential Vcrys(r) does\nnot appear explicitly anymore, its e\u000bects having been\nincorporated in the e\u000bective mass ( m0!m) and ef-\nfective Compton wavelength ( \u00150!\u0015).29,30Above, ^ z\nis the unit vector pointing towards the metal-substrate\ninterface, whereas p;rcan be either vectors in the x-y\nplane for strictly 2D \flms, or also have a z-component\nfor thicker, 3D systems. The second term on the r.h.s. is\nthe Bychkov-Rashba intrinsic spin-orbit coupling due to\nstructure symmetry breaking (metal-substrate interface),\ncharacterized by a coupling constant \u000b, whose strength\ncan be measured by angle-resolved photoemission,25{27\nand estimated by ab-inito methods.28Vimp(r) is the ran-\ndom impurity potential, see Sec. III. Impurities give also\nrise to the fourth term, which represents extrinsic spin-\norbit interaction. In the strictly 2D limit the Hamilto-\nnian (2) was used to study the spin Hall18,37,40,41and\nEdelstein e\u000bect18in the presence of both intrinsic and\nextrinsic mechanisms at T= 0. Such mechanisms were\nshown notto be simply additive, and their interplay leads\nto a nontrivial behavior.18,41\nFor \fnite temperatures ( T6= 0) the now time-\ndependent potential U(r;t) is expanded around its static\ncon\fguration:\nU(r;t) =Vimp(r) +\u000eVcrys(r;t) +\u000eVimp(r;t) +:::;(3)\nwhere\u000eVcrys(r;t); \u000eV imp(r;t) are linear in the small\nion/impurity displacements. Note that the static lattice\npotentialVcrys(r) has already been e\u000bectively taken into\naccount, and so it does not appear in Eq. (3) above. Nei-\nther does the phononic term, since we are not interested\nin the phonon dynamics; the phonons are assumed to be\nin equilibrium. The Hamiltonian thus becomes\nH=H0+\u000eVcrys(r;t) +\u000eVimp(r;t)\n\u0000\u00152\n4~\u001b\u0002r[\u000eVcrys(r;t) +\u000eVimp(r;t)]\u0001p:(4)\nThe second term on the r.h.s. gives rise to the electron-\nphonon interaction, the third to electron scattering with\ndynamical impurities, and the fourth describes dynam-\nical spin-orbit coupling (see Fig. 1). This last one is\nnovel and crucial for our purposes, as it yields the dy-\nnamical Elliott-Yafet spin relaxation and the dynamical\nside-jump mechanism. Neither of these two processes\nhave been considered previously, even though their static\ncounterparts are central in T= 0 treatments of the spin\nHall and related e\u000bects.18,42,43A third potentially rele-\nvant process is phonon skew scattering.23This will be dis-\ncussed elsewhere,44since its treatment requires going be-\nyond the Born approximation, which is beyond the scope\nof the present work.\nIn order to employ the SU(2)-covariant kinetic\nformulation35mentioned in the Introduction, the intrin-\nsic Bychkov-Rashba term is rewritten as a non-Abelian\nvector potential:35,36,45,46\n\u0000\u000b\n~pi\"iaz\u001ba=piAa\ni\u001ba\n2m; (5)3\nwithAx\ny=\u0000Ay\nx= 2m\u000b=~, all other components of Aa\nbeing zero, whereas \"iazis thez-component of the anti-\nsymmetric tensor. Here and throughout the paper upper\n(lower) indices will indicate spin (real space) components,\nwhile repeated indices are summed over unless otherwise\nspeci\fed.\nThe \fnal step is de\fning the relevant transport coef-\n\fcients within linear response. Assuming homogeneous\nconditions and taking as driving \felds an electric \feld\nExand a temperature gradient rxT, we are interested in\nthe generation of (i) a y-spin polarization sy(Edelstein7,8\nand thermal Edelstein4,5e\u000bects); (ii) a z-polarized spin\ncurrent \rowing along y,jz\ny(spin Hall6and spin Nernst2,3\ne\u000bects). In the presence of spin-orbit coupling, i.e., when\nspin is not conserved, the spin current has a di\u000busion\nterm even under homogeneous conditions:35\njz\ny= 2m\u000bDsy+jz\ny;drift; (6)\nwithDthe di\u000busion constant. Extending the standard\nOnsager formulation of thermoelectric transport to the\npresent spin-thermoelectric context, we then write\nsy=PsEEx+PsTrxT; (7)\njz\ny;drift=\u001bsE;driftEx+\u001bsT;driftrxT: (8)\nThe conductivities \u001bsE;drift;\u001bsT;driftcorrespond, in Kubo\ndiagrammatics, to \\bare\" response bubbles. For the full\nspin current jz\nyone has\njz\ny=\u001bsEEx+\u001bsTrxT: (9)\nwhere\u001bsE;\u001bsTare bubbles with \\dressed\" vertices, the\nsame holding for PsE;PsT. The spin Hall conductivity\n\u001bsH\u0011\u001bsE, whereas the spin Nernst one is de\fned un-\nder open circuit conditions, \u001bsN\u0011S\u001bsE+\u001bsT, withS\nthe Seebeck coe\u000ecient. Similarly, the Edelstein e\u000bect is\ndirectly given by the spin polarization response to the\nelectric \feld,P\u0011PsE, while for its thermal counterpart\nPt\u0011SPsE+PsT.\nOur goal is the computation of the transport coe\u000e-\ncientsPsE;PsT;\u001bsE;\u001bsTde\fned above. For the sake of\nclarity we have introduced them within a drift-di\u000busion\npicture, however Eqs. (7) and (9) are general, and our\ntreatment works in the ballistic limit as well. Finally,\nOnsager reciprocity is duly respected,37,38and is here be-\ntweenjz\ny$jx(spin Hall$inverse spin Hall e\u000bect) and\nsy$jx(Edelstein$inverse Edelstein or spin-galvanic\ne\u000bect47,48).\nIII. THE KINETIC EQUATIONS\nThe kinetic (Boltzmann-like) equation for the 2 \u00022\ndistribution function fp=f0+\u001b\u0001f, wheref0andfare\nthe charge and spin distribution functions, respectively,35\nreads\n@tfp+~r\u0001hp\nmfp+ \u0001jsji\n+1\n2fF\u0001rp;fpg=I0+Isj+IEY;\n(10)where we introduced the covariant spatial derivative and\ntheSU(2) Lorentz force due to the Rashba spin-orbit\ncoupling:\n~r=r+i\n~\u0014\nAa\u001ba\n2;\u0001\u0015\n; (11)\nF=\u0000p\nm\u0002Ba\u001ba\n2; (12)\nBa\ni=\u00001\n2~\"ijk\"abcAb\njAc\nk: (13)\nA summation over identical indices is implied unless\nstated otherwise. Note that an external magnetic \feld\nis not included in these equations (since it is not needed\nfor the present purpose). The term \u0001 jsjin Eq. (10) is a\ncorrection to the current due to side-jumps.\nNext we consider the collision operators on the r.h.s.\nof Eq. (10), where I0describes scattering with dynami-\ncal impurities and phonons, Isjthe contribution due to\nside-jumps, and IEYElliott-Yafet spin relaxation due to\nspin-\rip processes. At zero temperature the collision\noperators are obtained from the impurity averaged self-\nenergies within the self-consistent Born approximation\n(see Fig. 1). For isotropic scattering, the impurity corre-\nlations are given by\nVimp(r)Vimp(r0) =nimpv2\n0\u000e(r\u0000r0) =~\n2\u0019N0\u001cimp\u000e(r\u0000r0);\n(14)\nwithnimpthe impurity concentration, v0the scattering\namplitude, and 1 =\u001cimpthe momentum relaxation rate\ndue to impurities; N0is the density of states per area\n(volume) and spin in two (three) dimensions. More gen-\nerally,v2\n0!hjv(q)j2i, whereh:::idenotes the angular\naverage, and q2= (p\u0000p0)2=~2= 2p2\nF(1\u0000cos\u0012), since\njpj=jp0j=pF.\nIn order to include the impurities' thermal \ructua-\ntions, we consider small time-dependent displacements\n\u000eri(t) of thei-th impurity, which leads to\n\u000eVimp(r;t) =\u0000r\u0001X\ni\u000eri(t)v(r\u0000ri); (15)\nwherevis the single-impurity potential. We further as-\nsume that the displacement \ructuations of di\u000berent im-\npurities are independent, and can be approximated by\nthe classical harmonic oscillator expression, i.e,\n\u000er\u000b\ni(t)\u000er\f\nj(t0)'\u000eij\u000e\u000b\fkBT\nM!2\nD; (16)\nwhereMand!Dare the typical mass and frequency; we\nalso considered short times, !Djt\u0000t0j\u001c 1. Then we\nobtain\n\u000eVimp(r;t)\u000eVimp(r0;t0)'~\n2\u0019N0\u001cdyn\u000e(r\u0000r0) (17)\nwith\n1\n\u001cdyn=2\u0019nimpv2\n0N0\n~2kBTp2\nF\n~2M!2\nD: (18)4\nFIG. 1. Shown are the self-energies which determine the col-\nlision operators in the Boltzmann equation. The arrowed line\nrepresents the Green's function in Keldysh space, a cross (dot)\nthe potential due to an impurity (a crystal displacement). The\ndashed line depicts the impurity correlation either for static\n(straight line) or for dynamical impurities (wavy line). The\nwavy solid line illustrates the phonon propagator and a box\naround a vertex the spin-orbit coupling due to the boxed po-\ntential.\nMore precisely, as follows from the corresponding self-\nenergy expression (Fig. 1), v2\n0!h(1\u0000cos\u0012)jv(q)j2iin\n(18). In order of magnitude, \u001cimp=\u001cdyn'kBT=\u000fFsince\n(~!D)2'(m=M )\u000f2\nF. Note that the \u000e-function in Eq. (17)\nhas to be interpreted in connection with the correspond-\ning self-energy diagram. A detailed analysis shows that\nthe result given in Eq. (18) applies for high temperatures,\nkBT\u001d~!D, where scattering processes essentially are\nelastic.\nA similar reasoning can be employed for electron-\nphonon scattering at high T, which leads to\n\u000eVcrys(r;t)\u000eVcrys(r0;t0)'~\n2\u0019N0\u001cph\u000e(r\u0000r0); (19)\nwhere 1=\u001cph= 2\u0019N0g2kBT=~is the standard (high T)\nmomentum relaxation rate.49Based on the Keldysh tech-\nnique, the collision operators can be derived as usual.50\nThe result corresponds, in the classical limit, to\n\u000eVcrys(r;t)\u000eVcrys(r0;t0) =ig2\n2DK(r\u0000r0;t\u0000t0);(20)\nwheregis the electron-phonon coupling constant and DK\ndenotes the Keldysh component of the phonon Green's\nfunction in equilibrium.\nSince 1=\u001cphcan be several orders of magnitude larger\nthan 1=\u001cimp,51the total momentum relaxation rate 1 =\u001c=\n1=\u001cimp+1=\u001cdyn+1=\u001cphis typically dominated by electron-\nphonon scattering, 1 =\u001c'1=\u001cph, in the high-temperature\nregime.\nThe above discussion shows that one may use the re-\nsults for the collision operators and the side-jump correc-tion given in Refs. [18] and [35],\nI0=\u00001\n\u001c(fp\u0000hfpi); (21)\nIsj=\u00152\n8~\u001c\"abcn\n(~ra\u001bb);pcfp\u0000hpcfpio\n;(22)\nIEY=\u00001\n\u001c\u0012d\u00001\nd\u0013\u0012\u0015p\n2~\u00134\n\u0002X\na=x;y;(z)\u00121\n3d\u00002fa+hfai\u0013\n\u001ba(23)\n\u0001jsj=\u00152\n8~\u001chf(p0\u0000p)\u0002\u001b;fp0gi^p0; (24)\nwhere 1=\u001cis now the total scattering rate. The wavy\nbrackets represent the anti-commutator and d= 2;3 the\ndimensionality.52Formally, the diagrams in Fig. 1, to-\ngether with Eqs. (21)-(23), show that the phenomeno-\nlogical substitution 1 =\u001cimp!1=\u001cforT= 0!T6= 0\nis fully justi\fed for all spin-dependent processes at the\nBorn approximation level of accuracy.\nFinally, the yspin polarization syand thez-polarized\nspin current \rowing along y,jz\ny, are de\fned according to\nRef. [18],\nsy=Zdp\n(2\u0019~)dfy=Z\nd\u000fpN0hfyi; (25)\nand\njz\ny= Tr\u001bz\n2Zdp\n(2\u0019~)d\u0014py\nmfp+\u0015\n8~\u001cf(p\u0002\u001b)gy;fp\u0015\n:\n(26)\nIV. SPIN NERNST AND THERMAL\nEDELSTEIN EFFECTS\nIn this section we present and discuss our results, i.e.,\nthe spin transport coe\u000ecients PsE;PsT;\u001bsE, and\u001bsT. We\n\fnd that the competition between intrinsic and extrinsic\nspin-orbit mechanisms can lead the former to have a non-\nlinear temperature dependence. Notice that when only\nextrinsic mechanisms are considered, the spin Nernst con-\nductivity was instead predicted to be simply linear in\nT.2Though the spin Nernst nonlinearity will prove to be\nrather weak in a wide range of parameters, it is in princi-\nple a signature of the relative strength between intrinsic\nand extrinsic spin-orbit coupling.\nWe \frst consider a two-dimensional system and com-\nment on the three-dimensional case at the end of this\nsection. Furthermore, we focus on the di\u000busive (\\dirty\")\nregime, in which a very transparent drift-di\u000busion picture\nfor both charge and spin degrees of freedom is possible.35\nHowever, the ballistic (\\clean\") limit is also discussed in\nthe closing Subsection IV A, since estimates show it to be\nrelevant for certain experimentally realized systems. In-\ndeed, spin di\u000busion takes place as long as the spin-orbit\nsplitting is smaller than the lifetime broadening, which5\nin a Rashba-like system means 2 \u000bpF=~<~=\u001c,pFbe-\ning the Fermi surface momentum. At room temperature\n~=\u001c\u001910\u00002eV, whereas 2 \u000bpF=~can vary substantially in\nmetallic \flms, 10\u00003eV.2\u000bpF=~.10\u00001eV.25{27Thus,\nthe full di\u000busive-to-ballistic spectrum can in principle be\nexplored.\nIn the di\u000busive regime the Boltzmann equation (10)\nforhfyican be solved within the p-wave approximation\n(fp'hfpi+^p\u0001\u000efp), in terms of the x-spatial derivative\nof the local equilibrium charge distribution function,\nrxfeq=\u0012\u000fp\u0000\u000fF\nTrxT+rx\u0016\u0013\u0012\n\u0000@feq\n@\u000fp\u0013\n:(27)\nHere\u000fp(\u000fF) is the particle (Fermi) energy. The chemi-\ncal potential gradient is identi\fed with the electric \feld,\neEx\u0011rx\u0016withe=jej. The temperature gradient and\nthe electric \feld act as driving terms in the charge sec-\ntor of the Boltzmann equation, which is easily solved.\nVia Eqs. (22) and (24), the charge distribution enters\nthe spin sector, from which we determine hfyiand hence\nthe spin polarization linear in ExandrxTaccording to\nEq. (25). In the last step, integrating the yspin compo-\nnent of Eq. (10) we obtain\n@tsy+2m\u000b\n~2jz\ny=\u0000Z\nd\u000fpN0\n\u001cshfyi: (28)\nFrom this relation, we then calculate jz\ny. Note that no\nspatial gradients (beyond rxTandrx\u0016) are considered.\nIn Eq. (28), the (weakly energy-dependent) Elliott-Yafet\nrelaxation rate is proportional to the momentum relax-\nation rate, and given by\n1\n\u001cs=1\n\u001c\u0012\u0015p\n2~\u00134\n: (29)\nSpeci\fcally, in order to obtain hfyiwe perform a Fourier\ntransformation in time, t!!, multiply the zspin com-\nponent of the Boltzmann equation by py, the charge com-\nponent bypx, and perform the momentum angular aver-\nage of these two equations as well as of the yspin com-\nponent of the Boltzmann equation. The result is\nhfyi=\u0000F!\u0001rxfeq(30)\nwith\nF!=p2\u000b\n~3\u001cs\n1\u0000i!\u001c\u0014\n2\u0010\u000b\u001c\n~\u00112\n+\u00152\n2(1\u0000i!\u001c)\u0015\n\u0002\"\n2\u00124\u000b\u001c\n\u00152p\u00132\n+ (1\u0000i!\u001cs)(1\u0000i!\u001c)#\u00001\n:(31)\nFrom this expression, we are now able to determine\nthe transport coe\u000ecients, similar to Mott's formula inthermoelectrics.53We \fnd\nPsE(!) =\u0000eZ\nd\u000fpN0F!\u0012\n\u0000@feq\n@\u000fp\u0013\n; (32)\nPsT(!) =\u0000Z\nd\u000fpN0F!\u000fp\u0000\u000fF\nT\u0012\n\u0000@feq\n@\u000fp\u0013\n; (33)\n\u001bsE(!) =e~2\n2m\u000bZ\nd\u000fpN0\n\u001cs(1\u0000i!\u001cs)F!\u0012\n\u0000@feq\n@\u000fp\u0013\n;(34)\n\u001bsT(!) =~2\n2m\u000bZ\nd\u000fpN0\n\u001cs(1\u0000i!\u001cs)F!\n\u0002\u000fp\u0000\u000fF\nT\u0012\n\u0000@feq\n@\u000fp\u0013\n: (35)\nIn the following we consider the \frst non-vanishing order\nof the Sommerfeld expansion51of Eqs. (32){(35). Also,\nall energy-dependent quantities are given at the Fermi\nenergy unless mentioned otherwise. The following Mott-\nlike formulas are obtained:\nPsT=\u0000S0\u000fFP0\nsE; (36)\n\u001bsT=\u0000S0\u000fF\u001b0\nsE; (37)\nwithS0=\u0000\u00192k2\nBT=(3e\u000fF),P0\nsE\u0011@\u000fpPsEj\u000fF, and\u001b0\nsE\u0011\n@\u000fp\u001bsEj\u000fF.\nFirst we discuss the simple case of a 2DEG with an\nenergy-independent relaxation rate 1 =\u001cin the static case\n(!= 0). We refer to App. A for more general formulas.\nConcerning the spin polarization, we \fnd\nPsE=\u00002m\u000b\n~2\u001cs\n\u001cs=\u001cDP+ 1\u0000\n\u001bsH\nint+\u001bsH\nsj\u0001\n; (38)\nPsT=\u0000S02m\u000b\n~2\u001cs\n(\u001cs=\u001cDP+ 1)2\u0000\n\u001bsH\nint+\u001bsH\nsj\u0001\n:(39)\nHere, 1=\u001cDP= (2m\u000b=~2)2Dis the Dyakonov-Perel relax-\nation rate in the di\u000busive regime, with D=v2\nF\u001c=2 the\ndi\u000busion constant, whereas \u001bsH\nint= (N0e~=4m)(2\u001c=\u001cDP)\nand\u001bsH\nsj=en\u00152=(4~) are the intrinsic and side-jump spin\nHall conductivity, respectively. Note that for a 2DEG we\nhaveN0e~=4m=e=8\u0019~, giving the \\universal\" intrinsic\nspin Hall conductivity.54Clearly,PsTis in general nonlin-\near in temperature due to the T-dependence of the spin\nrelaxation rates,\n1\n\u001cDP\u0018\u001c\u00181\nT;1\n\u001cs\u00181\n\u001c\u0018T: (40)\nAn experimental relevant setup would be an open circuit\nalongx, i.e., along the direction where the thermal gradi-\nent is applied. Then, the electric \feld can be expressed by\nthe thermal gradient as Ex=SrxT, whereSis the See-\nbeck coe\u000ecient. For a 2DEG with an energy-independent\nrelaxation rate S=S0. With the open circuit condition\nthe thermal Edelstein polarization coe\u000ecient is given as\na sum of electrical and thermal contributions, and reads\nPt=S0PsE+PsT; (41)\nwhich is shown in Fig. 2 for \u001c=\u001cs= 0:01. The parameter\n\u001cs;r=\u001cDP;r, the subscript rindicating that the value of a6\na)\n−75−50−250\n0.5 1 1.5 2Pt/(S0,r2mατ r\n¯h2e\n8π¯h)\nT/Trtotal\nelectrical contribution\nthermal contributionτs,r/τDP,r=1\nb)\n−10−50\n0.5 1 1.5 2Pt/(S0,r2mατ r\n¯h2e\n8π¯h)\nT/Trtotal\nelectrical contribution\nthermal contributionτs,r/τDP,r=20\nFIG. 2.Pt, compare Eq. (41), versus temperature, in units of\nS0;r(2m\u000b\u001c r=~2)(e=8\u0019~), split into its thermal and electrical\ncontributions. The Elliottt-Yafet spin relaxation is chosen\nas\u001c=\u001cs= 0:01. For a) we have \u001cs;r=\u001cDP;r= 1 and for b)\n\u001cs;r=\u001cDP;r= 20.Trdenotes the temperature scale (room\ntemperature).\ngiven quantity is taken at room temperature, gives the\nratio of intrinsic to extrinsic spin-orbit coupling and is\nusually large. As discussed in Appendix B, one typically\nexpects 1 .\u001cs=\u001cDP.102. The thermal contribution\nPsTis in general less relevant when intrinsic spin-orbit\ncoupling dominates [Fig. 2b)], and only gives a signi\f-\ncant contribution above room temperature. According\nto Ref. [23] this should correspond to a metal like Pt,\nwhereas Fig. (2a) to one like Au. The temperature de-\npendence is clearly nonlinear around room temperature.\nAnalogously, we \fnd for the spin current\n\u001bsE=1\n\u001cs=\u001cDP+ 1\u0000\n\u001bsH\nint+\u001bsH\nsj\u0001\n; (42)\n\u001bsT=\u0000S02\u001cs=\u001cDP+ 1\n(\u001cs=\u001cDP+ 1)2\u0000\n\u001bsH\nint+\u001bsH\nsj\u0001\n; (43)\nand for an open circuit condition the spin Nernst con-\nductivity\n\u001bsN=S0\u001bsE+\u001bsT: (44)\nA plot of\u001bsNversus temperature is shown in Fig. 3, witha)\n−2−101\n0.5 1 1.5 2σsN/(S0,re\n8π¯h)\nT/Trtotal\nelectrical contribution\nthermal contributionτs,r/τDP,r=1\na)\n−0.4−0.200.2\n0.5 1 1.5 2σsN/(S0,re\n8π¯h)\nT/Trtotal\nelectrical contribution\nthermal contributionτs,r/τDP,r=20\nFIG. 3. Spin Nernst conductivity in units of the \\univer-\nsal\" value of the intrinsic spin Hall conductivity times the\nSeebeck coe\u000ecient at room temperature, S0;re=8\u0019~, against\nT=Tr, with\u001c=\u001cs= 0:01. We show the electrical and ther-\nmal contribution separately; the parameters are chosen as\n\u001cs;r=\u001cDP;r= 1 for a) and \u001cs;r=\u001cDP;r= 20 for b).\n\u001c=\u001cs= 0:01. The interplay of intrinsic and extrinsic spin-\norbit coupling leads to a nonlinear temperature depen-\ndence, provided the intrinsic spin-orbit coupling domi-\nnates [Fig. 3b)]. On the other hand, when intrinsic and\nextrinsic spin relaxation times are comparable [Fig. 3a)]\nthe spin Nernst conductivity is small since the thermal\nand the electrical contribution cancel each other. In-\ndeed, for vanishing intrinsic spin-orbit coupling, the spin\nNernst conductivity is zero for a 2DEG.\nFinally, we comment on the three-dimensional case.\nAs can be seen in App. A, the quantitative change is\nrather small since only \u001cschanges by a numerical pre-\nfactor of 8=9, while the other relevant quantities remain\nunchanged. We remark, however, that in 3D we en-\ncounter an energy-dependent density of states. In ad-\ndition, the momentum relaxation rate is in general also\nenergy-dependent. This manifests itself directly in the\nthermal part of the spin transport coe\u000ecients where we\nencounter the factors \u0011\u0011\u000fFN0\n0=N0and\f\u0011\u000fF\u001c0=\u001c,\nnamely the change in energy of the density of states\nand the momentum relaxation rate at the Fermi energy.\nTherefore, the relative weight between thermal and elec-7\ntrical contribution can be modi\fed. Note that in case\nof an open circuit along the thermal gradient, the elec-\ntrical contribution is also modi\fed by \u0011and\fsince the\nSeebeck coe\u000ecient is then given by S=S0(1 +\u0011+\f),\nas follows from the charge component of the Boltzmann\nequation.\nA. The \\clean\" limit\nAt room temperature, one enters the \\clean\" regime\nfor 2\u000bpF=~>10\u00002eV. Under homogeneous conditions,\nEq. (10) can be solved in this limit as well { note that\nEq. (28) is valid irrespective of the regime considered.\nThe procedure is straightforward, yet lengthy and not\nparticularly illuminating,55therefore we simply give the\nresults for the 2D case when 2 \u000bpF\u001c=~2\u001d1.\nThe transport coe\u000ecients read\nPsE=\u00002m\u000b\n~22\u001c \n\u001bsH\nint+\u001bsH\nsj\n2!\n; (45)\nPsT=S02m\u000b\n~22\u001c\u001bsH\nsj\n2; (46)\n\u001bsE=2\u001c\n\u001cs \n\u001bsH\nint+\u001bsH\nsj\n2!\n; (47)\n\u001bsT=\u0000S02\u001c\n\u001cs\u001bsH\nsj; (48)\nwhere now \u001bsH\nint=e=8\u0019~. From Eqs. (41) and (44) it\nis immediate to see that the thermal Edelstein e\u000bect is\nconstant in T, whereas the spin Nernst is linear. This\noverall simpler behavior is expected, as in the \\clean\"\nlimit 1=\u001cDP!1=2\u001c, i.e., both the Dyakonov-Perel and\nthe Elliott-Yafet relaxation rates are proportional to T.\nV. CONCLUSIONS\nWe have explicitly considered the dynamical spin-orbit\ninteraction of conduction electrons with phonons, which\ngives rise to dynamical Elliott-Yafet spin relaxation and\nside-jump mechanism. The focus has been on the high-\ntemperature regime T > TD. Symmetric, Mott-like for-\nmulas for the (thermal) Edelstein and spin Hall and\nNernst coe\u000ecients have been derived. The temperature-\ndependence of the spin transport coe\u000ecients was shown\nto be nontrivially a\u000bected by the competition between ex-\ntrinsic and intrinsic spin-orbit coupling mechanisms, the\norigin lying in the mixing of the spin relaxation times\n\u001cDPand\u001cs. In the di\u000busive regime the latter have dif-\nferent temperature dependences, which ultimately causes\nthe thermal Edelstein and spin Nernst e\u000bect to exhibit\na nonlinear T-behavior. The nonlinearity is in general\nstronger for the thermal Edelstein e\u000bect, and, especially\nin the spin Nernst case, it becomes weaker with decreas-\ning intrinsic spin-orbit coupling strength.ACKNOWLEDGMENTS\nWe acknowledge stimulating discussions with R. Rai-\nmondi, as well as \fnancial support from the German Re-\nsearch Foundation (DFG) through TRR 80, and from the\nCEA through the DSM-Energy Program (project E112-\n7-Meso-Therm-DSM).\nAppendix A: General Expressions for the spin\npolarization/current\nThis appendix shows more general expressions for\nPsE;PsT;\u001bsE, and\u001bsT, valid at \fnite frequency for both\n2D and 3D systems. The transport coe\u000ecients are ob-\ntained by the Sommerfeld expansion of Eqs. (32){(35).\nThis implies that all quantities appearing below are eval-\nuated at the Fermi energy unless otherwise speci\fed.\nThe dynamical Edelstein coe\u000ecient and the spin Hall\nconductivity are given as follows:\nPsE=\u00002m\u000b\n~2\u0014\u001cs\n2\u001cs=\u001cDP+d(1\u0000i!\u001cs)(1\u0000i!\u001c)\u0015\n\u0002\"\n2\u001bsH\nint+d\u001bsH\nsj(1\u0000i!\u001c)\n1\u0000i!\u001c#\n;(A1)\n\u001bsE=\u00141\u0000i!\u001cs\n2\u001cs=\u001cDP+d(1\u0000i!\u001cs)(1\u0000i!\u001c)\u0015\n\u0002\"\n2\u001bsH\nint+d\u001bsH\nsj(1\u0000i!\u001c)\n1\u0000i!\u001c#\n:(A2)\nHere, the form of \u001bsH\nintand\u001bsH\nsj[see Eq. (39)] remains\nunchanged in 3D and the Dyakonov-Perel relaxation rate\nremains exactly as it is in 2D, i.e., the di\u000busion constant\nwhich there appears is the 2D one. Only the Elliott-Yafet\nrelaxation rate exhibits a pre-factor of 8 =9 compared to\n\u001csin 2D. A plot of the spin Hall conductivity \u001bsE(!) is\nshown in Fig. 4.\n−0.500.511.5\n0 1 2 3 4σsE/(N0e¯h/4m)\nωττs/τDP=1\nτs/τDP=20\nτs/τDP=100\nFIG. 4. The spin Hall conductivity in 3D in units of N0e~=4m\nfor various ratios of \u001cs=\u001cDPvs.!\u001c, separated into its real\npart (solid lines) and its imaginary part (dashed lines). The\nextrinsic spin-orbit strength is chosen such that \u001c=\u001cs= 0:01.8\nThe thermal contribution is now obtained by Eqs. (36)\nand (37). Since the resulting equations are rather cum-\nbersome, we just show formulas for the static case, != 0.\nWe \fnd\nPsT=\u0000S02m\u000b\n~2\u001cs\n(2\u001cs=\u001cDP+d)2\n\u0002\u001a\n2\u001bsH\nint\u0014\nd\u0000\u0011\u00122\u001cs\n\u001cDP+d\u0013\n\u0000\f\u00122\u001cs\n\u001cDP+ 3d\u0013\u0015\n+d\u001bsH\nsj\u0014\nd\u0000\u0011\u00122\u001cs\n\u001cDP+d\u0013\n+\f\u00122\u001cs\n\u001cDP\u0000d\u0013\u0015\u001b\n;\n(A3)\n\u001bsT=\u0000S01\n(2\u001cs=\u001cDP+d)2\n\u0002\u001a\n2\u001bsH\nint\u00144\u001cs\n\u001cDP+d+\u0011\u00122\u001cs\n\u001cDP+d\u0013\n+ 2d\f\u0015\n+d\u001bsH\nsj\u00144\u001cs\n\u001cDP+d+\u0011\u00122\u001cs\n\u001cDP+d\u0013\u0015\u001b\n:(A4)\nNote that here the energy derivative of the density of\nstates (momentum relaxation rate) at the Fermi energy\ncomes into play by \u0011=\u000fFN0\n0=N0(\f=\u000fF\u001c0=\u001c) which\ndoes have an in\ruence on the thermal contribution to\nthe spin Nernst conductivity and the spin polarization\nin case of an open circuit. But we remark that also the\nelectrical contribution, S\u001bsE, is a\u000bected by \u0011and\fsincethe Seebeck coe\u000ecient then reads S=S0(1 +\u0011+\f).\nAppendix B: On the ratio \u001cs=\u001cDP\nWe estimate the size of the ratio \u001cs=\u001cDP, de\fning the\nrelative importance of extrinsic and intrinsic spin-orbit\ncoupling. The general form of the Dyakonov-Perel relax-\nation rate, valid from the \\clean\" to the \\dirty\" regime,\nreads\n1\n\u001cDP=1\n2\u001c(2\u000bpF\u001c=~2)2\n(2\u000bpF\u001c=~2)2+ 1; (B1)\nwhere 2\u000bpF=~is the spin-orbit splitting. Therefore\n\u001cs\n\u001cDP=1\n2(2\u000bpF\u001c=~2)2\n(2\u000bpF\u001c=~2)2+ 1\u00122~\n\u0015pF\u00134\n: (B2)\nIn doped semiconductors one typically \fnds 10\u00002.\n\u0015=\u0015F.1.20,29,30Though there is yet no theory capable\nof estimating \u0015in a metal,20one can argue that, since\nthe spin-orbit energy is small compared to the Fermi one,\nthe relation \u0015=\u0015F\u001c1 will hold in a metallic \flm. 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(10) in 4 \u00024 matrix form, sep-\narating angular averaged quantities from those which are\nnot, and to solve it in linear response to the driving electric\n\feld and temperature gradient. See Ref. [57] for an explicit\nexample.\n56A. V. Shytov, E. G. Mishchenko, H.-A. Engel, and B. I.\nHalperin, Phys. Rev. B 73, 075316 (2006).\n57R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa,\nPhys. Rev. B 74, 035340 (2006)."    },    {        "title": "1112.1596v2.Spin_Pumping_Driven_by_Bistable_Exchange_Spin_Waves.pdf",        "content": "arXiv:1112.1596v2  [cond-mat.mtrl-sci]  11 May 2012Spin Pumping Driven by Bistable Exchange Spin Waves\nK. Ando1,∗and E. Saitoh1,2,3\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2CREST, Japan Science and Technology Agency, Sanbancho, Tok yo 102-0075, Japan\n3The Advanced Science Research Center, Japan Atomic Energy A gency, Tokai 319-1195, Japan\n(Dated: August 26, 2018)\nSpin pumping driven by bistable exchange spin waves is demon strated in a Pt/Y 3Fe5O12ﬁlm\nunder parametric excitation. In the Pt/Y 3Fe5O12ﬁlm, the spin pumping driven by parametric\nexcitation selectively enhances the relaxation of short-w avelength exchange spin waves, indicating\nstrong coupling between the exchange spin waves and spin cur rents at the interface through eﬃcient\nspin transfer. The parametric spin pumping, furthermore, a llows direct access to nonlinear spin\nwave dynamics in combination with the inverse spin Hall eﬀec t, revealing unconventional bistability\nof the exchange spin waves.\nPACS numbers: 72.25.Ba, 72.25.Pn, 72.25.Mk, 75.76.+j\nNonlinear phenomena have been essential for explor-\ning physics and technology in condensed matter. Nonlin-\near dynamics is ubiquitous in nature, including mechan-\nical [1], optical [2], and magnetic [3] systems. This plays\na signiﬁcant role also in the ﬁeld of spintronics, where\nmagnetization dynamics is coupled with spin currents.\nThe coupling between dynamical magnetization and spin\ncurrents gives rise to generation of spin currents from\nprecessing magnetization in nonmagnetic/ferromagnetic\njunctions: the spin pumping [4–13]. In particular, the\nspin pumping in metal/magnetic insulator junctions of-\nfers a route for exploring nonlinear spin physics owing to\nthe exceptionally low magnetic damping.\nDue to the nonlinearlity of magnetization dynamics,\nparametric excitation of dynamic magnetization is possi-\nblewithoutfulﬁllmentofferromagneticresonance(FMR)\nconditions, providedthattheamplitudeofapumpingmi-\ncrowave magnetic ﬁeld his large enough to overcome the\nspin wave relaxation. Recent studies have revealed that\nthe parametrically-excited short-wavelength spin waves\ndrivethespinpumpinginmetal/magneticinsulatorjunc-\ntions [14–16]; spin waves parametrically excited by a mi-\ncrowave in the magnetic insulator induce a spin current\nin the adjacent metal, giving rise to an electric voltage\nthrough the inverse spin Hall eﬀect (ISHE) in the metal.\nThus the spin pumping allows not only to generate spin\ncurrents in a wide range of materials [10, 17] but also\nto explore interaction between spin currents and dipo-\nlar/exchange spin waves using the ISHE [14–16, 18].\nIn this Letter, we demonstrate spin pumping driven\nby bistable exchange spin waves. We generate two\ntypesofparametricspinwaves,small-wavevectordipolar-\nexchange spin waves and large-wavevectorexchange spin\nwaves, through three-magnon splitting by controlling the\nstrength of an external static magnetic ﬁeld. The ex-\nchange spin waves are found to couple strongly with spin\ncurrentsthrougheﬃcient spin transferdue to spin pump-\ning in a metal/magnetic insulator junction. Further-\nmore, using the spin pumping driven by the exchangespin waves in combination with the ISHE, we found that\nonly exchange spin waves show hysteresis with respect\nto an external magnetic ﬁeld and microwave excitation\npower, demonstrating bistable excitation of spin waves\ndue to a negative nonlinear damping.\nThe sample used in this study is a Pt/La-substituted\nY3Fe5O12(Pt/La:YIG) bilayer ﬁlm. The single-crystal\nLa:YIG (111) ﬁlm [2 ×2 mm2] with a thickness of 2 µm\nwas grown on a Gd 3Ga5O12(111) substrate by liquid\nphase epitaxy, where La was substituted to match the\nlattice constant between the ﬁlm and the substrate. The\n10-nm-thick Pt layer was then sputtered in an Ar atmo-\nsphere on the top of the ﬁlm and two electrodes are at-\ntachedtotheedgesofthePtlayertodetectaspincurrent\ngenerated by the spin pumping; a spin current generated\nby the spin pumping is converted into an electric volt-\nageVthrough the ISHE in the Pt layer [19–23]. Here,\nfor the measurement, the Pt/La:YIG ﬁlm was placed at\nthe center of a TE 011cavity, where a microwave with\nthe frequency f= 9.44 GHz exists. The background\nvoltageVbwas subtracted from the measured voltage ¯V:\nV=¯V−Vb. An external magnetic ﬁeld Hwas ap-\nplied along the ﬁlm plane perpendicular to the direction\nacross the electrodes. All measurements were performed\nat room temperature.\nIn Fig. 1(a), we show the Vsignals measured for the\nPt/La:YIG ﬁlm at P= 50 and 200 mW microwave ex-\ncitation powers. Figure 1(a) shows that a voltage signal\nappears around the ferromagnetic resonance (FMR) ﬁeld\nHFMR∼260 mT. This is the electric voltage induced by\nthe ISHE and spin pumping driven by the uniform mag-\nnetization precession [15]. Notable is that an additional\nVsignal appears far below the FMR ﬁeld [see H∼130\nmT] onlyfor P= 200mW. This additionalvoltagesignal\nisinducedbythe spinpumpingdrivenbyparametricspin\nwave excitation [15, 16]. Spin waves with the wavevector\nofkp∝negationslash= 0 can be excited through a three-magnon split-\nting process; when the microwavemagnetic ﬁeld with fre-\nquencyω0= 2πfis applied perpendicular to the static2\nmagnetic ﬁeld H, a pair of modes with the wavevectors\nofkpand−kpare excited parametrically via a k=0\nvirtual state when a pair of modes are fed suﬃcient en-\nergytoovercomethedissipation,i.e., whentheexcitation\nmicrowave power Pexceed the threshold power Pth[see\nFig. 1(b)] [24–27]. The energies of the parametrically ex-\ncited spin waves, ωkpandω−kp, areω0/2 because of the\nenergy conservation ω0=ωkp+ω−kpandωkp=ω−kp.\nThe spin pumping from parametrically excited spin\nwaves strongly aﬀects their damping. Figure 1(c) shows\ntheHdependence of P1/2\nthfor the Pt/La:YIG ﬁlm and\na La:YIG ﬁlm, where the Pt layer is missing. Here, the\nspin-wave damping is proportional to the square root of\nthe threshold power P1/2\nthof parametric excitation [27].\nP1/2\nthwas obtained not by the voltage measurements but\nby monitoring the microwave absorption signal to com-\nparePthfor the Pt/La:YIG ﬁlm and that for the La:YIG\nﬁlm; we measured the microwave absorption signal using\na ﬁeld lock-in technique [15] and deﬁned Pthas the min-\nimum microwave power at which the absorption signal\nbecomes nonzero [28]. In Fig. 1(c), no diﬀerence is found\nin the threshold values for the Pt/La:YIG and La:YIG\nﬁlms in the gray area. This shows that the applied mi-\ncrowave power is identical for these ﬁlms. In contrast,\nnotably, the threshold values are increased by attaching\nthe Pt layer in the blue area, indicating that the spin-\nwave damping is enhanced in this ﬁeld range. Notable is\nthat, in the gray and blue areas, diﬀerent types of spin\nwaves are excited. These spin waves have been assigned\nto small-wavevector kp∼5×104cm−1dipolar-exchange\nspin waves with θkp∼0−90◦[the gray area] and large-\nwavevector kp∼3×105cm−1exchange spin waves with\nθkp∼45◦[the blue area], where θkpis the angle between\nthe external magnetic ﬁeld Hand the wavevector kpof\nthe spin waves [29, 30].\nThe aboveexperimental resultssuggestthe strongcou-\npling between the short-wavelength exchange spin waves\nand spin currents through the spin pumping. The spin\npumping is known to enhance the relaxation of uniform\nmagnetization precession, or the k=0mode, because\nthe spin-current emission from a ferromagnetic ﬁlm de-\nprives the magnetization of the spin-angular momen-\ntum, giving rise to an additional damping [4, 6]. In\nthe Pt/La:YIG ﬁlm, however, we found that the change\nof the Gilbert damping αdue to the spin pumping is\nnegligibly small by comparing the FMR spectra for the\nLa:YIG and Pt/La:YIG ﬁlms. This is attributed to\nthe thick, 2 µm, La:YIG layer, since the relaxation en-\nhancement is inversely proportional to the thickness of\nthe ferromagnetic layer [6]. The experimental result\nshown in Fig. 1(c) indicates that this is also the case for\nthe dipolar-exchange spin waves; the relaxation of the\ndipolar-exchange spin waves is insensitive to the spin-\ncurrent emission. In contrast, the exchange spin waves\nwithθkp∼45◦excited through the three-magnon in-(a) (b) (GHz)\nk (10 5 cm -1 )0 10 5 10 -5 /2 0 9.4 ω\nω\n0ωMSSW \nBVMSW \n8101214Pth  (mW   ) 1/2 1/2 \n90 150 \nH (mT) 60 120  Pt/La:YIG La:YIGPt \nLa:Y 3Fe 5O12 hHLa:Y 3Fe 5O12 hH(c) 100 200 250 \nH (mT) 300 50 150 V ( µV) \n048\n50 mW \n200 mW \nFIG. 1: (a) Field Hdependence of an electric voltage V\nat microwave excitation powers P= 50 and 200 mW for\nthe Pt/La:YIG ﬁlm. (b) The spin wave dispersion for the\nPt/La:YIGﬁlmwhentheparametricexcitation condition. (c )\nThe square root of the threshold power Pthfor the La:YIG\nandPt/La:YIG ﬁlms obtained from the microwave absorption\nsignal.\nteraction may tend to localize near the interface in the\nlength scale of 1 /kp∼100 nm because of the large-\nwavevector kp. This induces eﬃcient transfer of spin-\nangularmomentum from the exchange spin wavesto spin\ncurrents at the Pt/La:YIG interface, giving rise to signif-\nicant enhancement of the relaxation.\nThe exchange spin wavesfurther show a nontrivial fea-\nture: bistability. Figure 2(a) shows the Vsignals mea-\nsured at P= 200 mW by sweepingthe external magnetic\nﬁeld from 60 to 160 mT [the red curve] or from 160 to\n60 mT [the black curve]. In Fig. 2(a), a “jump” in the\nVspectrum is observed around Hup= 85 mT when His\nincreased from 60 mT. By decreasing H, the ISHE volt-\nage shows hysteresis; the “jump” ﬁeld, HupandHdown,\ndepends on the history of the ﬁeld sweep as indicated\nby the black and red arrows, demonstrating bistability\nof the parametrically excite spin waves. The Vsignals\nat diﬀerent microwave excitation powers are shown in\nFig. 2(b). Figure 2(b) shows systematic variation of the\njump ﬁelds HupandHdown[see also Fig. 2(c)]. The dif-\nference between HupandHdown, ∆H=Hup−Hdown\nincreases with P[see Fig. 2(d)], a feature similar to the\nfoldover measured at FMR [31]. However, the origin of\nthe bistability of the parametrically excited spin waves\nis diﬀerent from that of FMR. The bistability of FMR is\ninduced by the reduction of a demagnetization ﬁeld due\nto high power excitation; the bistability appears in the\nﬁeld range where the number of excited magnons is large\nenough to change the resonance condition [32]. Thus\nbistability of parametrically excited spin waves is also\nexpected to appear for ﬁelds where Vis large, i.e., for3\n60 80 100 120 140 160 024V ( µV) \nH (mT) 200 mW \nHup(down)  (mT) \n758085\n190 195 200789\n75 80 85 90 00.5 1.0 V ( µV) \nH (mT) 200 mW 198 mW 196 mW 194 mW 192 mW 190 mW \n∆H (mT) \nP (mW) H up \nH down H up H down (a) \n(b) (c) \n(d) \nFIG. 2: (a) Hdependence of Vfor the Pt/La:YIG ﬁlm at\nP= 200 mW. The red curve was measured with increasing H\n(up sweep). The black curve was measured with decreasing\nH(down sweep). (b) The hysteresis of Vwith respect to\nHmeasured at diﬀerent microwave excitation powers for the\nPt/La:YIG ﬁlm. (c) Pdependence of HupandHdown, where\nHupandHdownrepresent the “jump” ﬁeld of Vfor the up\nsweep and down sweep, respectively. (d) Pdependence of\n∆H=Hup−Hdown.\ndipolar-exchange spin waves, which is in contrast to the\nobservation: bistabilityoftheexchangespinwaves. Here,\nthe Pt layer is not essential for the bistability; bistable\nstates appear also in the La:YIG ﬁlm, which was con-\nﬁrmed by monitoring the microwave absorption signal.\nWe further show the evidence that bistable states ap-\npear only for the exchange spin waves. Figure 3(a) shows\ntheVsignalsmeasuredwithincreasing P(theredcircles)\nor decreasing P(the black circles) for diﬀerent external\nmagnetic ﬁelds H. Clear bistable states are observed at\nH= 84, 94, and 96 mT. In contrast, at higher magnetic\nﬁelds, 120 mT and 150 mT, we found no hysteresis in the\nVsignal. As shown in Fig. 3(a), the appearance of the\nbistable state is irrelevant to the threshold powers; the\nexternal magnetic ﬁeld strength is essential for the bista-\nbility. The width of the hysteresis ∆ P=Pup−Pdown[see\nFig. 3(a) for 84 mT] for diﬀerent external magnetic ﬁelds\nHis plotted in Fig. 3(b). Figure 3(b) shows that the\nbistable state appears only in the magnetic ﬁeld range\nof 80−100 mT, where large-wavevector exchange spin\nwaves are excited [see Fig. 1(c)].\nThe observed bistability can be attributed to a neg-\native nonlinear damping of parametrically excited spin\nwaves; the damping of these spin waves decreases with\nincreasing the number nkp,∂ηkp/∂nkp<0, where ηkp\nis the damping of the parametrically excited spin waves.94 mT\n150 165 1800.00.51.01.5\nP (mW)  V ( µV) \n96 mT \n135 150 165 \nP (mW) 0.00.51.0 V ( µV) 120 mT \n75 90 105 \nP (mW) 0.00.51.0 V ( µV) 150 mT\n150 165 180\nP (mW) 0.00.30.6 V ( µV) 84 mT \n160 180 200 \nP (mW) 0.00.51.0 V ( µV) 80 mT\n180 190 200 \nP (mW) 0.00.51.0V ( µV) (a) \n(b) \n   \n80 100 120 140 160\nH (mT) 02040∆P (mW) \n0 20 400.51.01.5\n∆P (mW) (c) \n90 10001VP   (µV) \nup \nH (mT) \nVP   (µV) \nup Pup Pdown \nFIG. 3: (a) Vas a function of Pfor the Pt/La:YIG ﬁlm at\nH= 80−150 mT. The red(black) circles were measured with\nincreasing(decreasing) P. (b)Hdependence of the width of\nthe hesterysis ∆ P=Pup−Pdown, where Pup(down) denotes\nthe microwave power where Vjumps as shown in the Vdata\natH= 84 mT. The inset shows Hdependence of VPup, where\nVPuprepresents the magnitude of the electric voltage at Pup.\n(c) ∆Pdependence of VPup.\nThe damping of a spin wave with a negative nonlinear\ndamping can be expressed as ηkp=η0−η1nkp. The neg-\nativenonlineardampingisnegligiblebelow PupwhenPis\nincreased from 0, since nkpis negligibly small in this sit-\nuation. Thus by increasing P, parametric spin waves are\nexcited abovethe thresholdfora systemwithout the neg-\native nonlinear damping [32] Pup=ηkp/Gkp=η0/Gkp,\nwhereGkpis the coupling parameter. In a system with-\nout the negative nonlinear damping, nkp= 0 atPup[26].\nThe situation changes drastically by taking into account\nthe negative nonlinear damping; assuming the relation\nbetween nkpandPabove threshold power Pthasnkp=\nn0√P−Pth, we ﬁnd nkp= 0 and 2 n2\n0η0η1/(V2+n2\n0η2\n1)\natPup; a bistable state of the parametric spin waves ap-\npears at Pup. Since the existence of the parametric spin\nwaves reduces their damping when η1∝negationslash= 0, the threshold\npower when Pis decreased from Pup,Pdown=ηkp/V,\nis smaller than Pupand thus hysteresis appears in the V4\nsignal.\nThe negative nonlinear damping arises from a three\nmagnon conﬂuence process, where a parametrically ex-\ncited spin wave with the wavevector kpis annihilated\ntogether with a thermal spin wave with k1, creating an-\nother thermal spin wave with k2:kp+k1=k2and\nωkp+ωk1=ωk2. The damping of the parametrically\nexcited spin waves due to this three magnon conﬂuence\nprocess is expressed as [33]\nηc\nkp∝4π/integraldisplay\n(nk1−nk2)δ(kp+k1−k2)δ(ωkp+ωk1−ωk2)dk1dk2,\n(1)\nwherenk1andnk2are the number of thermal spin waves\nwith the wavevector of k1andk2. When nk1andnk2\nare close to their thermodynamic equilibrium values, ¯ nk1\nand ¯nk2, thedampingduetothethreemagnonconﬂuence\nprocess can be expressed as ηc\nkp=c(¯nk1−¯nk2)≡ηl\nkp,\nwherecis a proportionality constant; the diﬀerence in\nthe thermodynamic equilibrium number of the thermal\nspin waves leads to a positive linear damping. How-\never, the three magnon conﬂuence process increases nk2\nand reduces nk1. Thus this process reduces nk1−nk2,\nwhereas relaxation processes of the thermal spin waves\ntend to return this diﬀerence to the thermodynamic equi-\nlibrium value. This competition is expressed as rate\nequations, ˙ nk1=−ηk1(nk1−¯nk1)−ηc\nkpnkpand ˙nk2=\n−ηk2(nk2−¯nk2)+ηc\nkpnkp. In the equilibrium condition,\n˙nk1= ˙nk2= 0, when cnkp(η−1\nk1+η−1\nk2)≪1, the damp-\ning of the parametrically excited spin waves due to the\nconﬂuence process is given by ηc\nkp=ηl\nkp−ηn\nkpnkp, where\nηn\nkp= [ηl\nkpc(η−1\nk1+η−1\nk2)]. The second term represents the\nnegative nonlinear damping. For the conﬂuence process\nofparametricallyexcitedspinwaveswithsmall kp,k1and\nk2must be very large due to the energy and momentum\nconservation laws. However, the damping of spin waves,\nηk1andηk2, increaseswithincreasingthewavevector;the\nnegativenonlineardamping is ineﬃcient for small kpspin\nwaves, which is consistent with the experimental obser-\nvation; bistability appears only for the large-wavevector\nexchange spin waves.\nFigure 3(c) shows the relation between the magnitude\nof the electric voltage at Pup,VPup, and the width of the\nhysteresis ∆ P. This result indicates that the decrease of\n∆Pwith decreasing Hbelow90 mT shown in Fig. 3(b) is\ndue to the reduction of VPup, orthe number ofparametri-\ncally excited spin waves at Pup. The origin of this reduc-\ntionisthesuppressionofthenegativenonlineardamping.\nThe negative nonlinear damping can be suppressed for\nlargekpspinwavesduetothreemagnonsplittingprocess,\nwhere a parametrically excited spin wave with kpsplits\nintotwothermalspinwaveswith k1andk2:kp=k1+k2\nandωkp=ωk1+ωk2, since the splitting process leads to\na positive nonlinear damping [33]. This process is impor-\ntant only for suﬃciently large-wavevectorspin waves; the\nsplitting process is allowed for kp> kmbecause of theenergy and momentum conservation. Here kmis of the\norder of 105cm−1[33], indicating that the splitting pro-\ncess is essential for exchange spin waves. In the present\nsystem, theminimumwavevector kmisobtainedfromthe\nrelationωkm= 2ωkm/2for the approximated dispersion\nrelation[34] ωk=γH+Dk2+(1/2)γ(4πMs)sin2θkpwith\nθkp=π/4 askm=/radicalbig\nγ(4H+4πMs)/2D. This indicates\nthatthe minimum wavevector kmdecreaseswith decreas-\ning the external ﬁeld H, whereas the wavevector of para-\nmetrically excited spin waves kpincreases with decreas-\ningH[see Fig. 1(b)]; the wavevector of parametrically\nexcited spin waves kpexceedskmbelow 90 mT, which\nsuppresses the negative nonlinear damping through the\nthree magnon splitting process.\nInsummary,wedemonstratedstrongcouplingbetween\nspin currents and short-wavelength exchange spin waves\nat a Pt/Y 3Fe5O12interface using parametric excitation.\nThis strong coupling is responsible for the spin pumping\ndriven by parametrically excited spin waves, which, in\ncombination with the inverse spin Hall eﬀect, allows di-\nrect access to bistable states of the exchange spin waves.\nFurther studies, such as time evolution of the bistable\nstates for diﬀerent ﬁelds and microwave powers, will pro-\nvide crucial piece of information for understanding the\nbistability. Thus the combination of the inverse spin\nHall eﬀect and spin pumping driven by parametric spin\nwaves promises a signiﬁcant progress for understanding\nthe physics of nonlinear spin-wave dynamics and is of\ncrucial importance for further development of spin-wave-\nbased spintronic devices.\nThe authors thank to T. An and H. Kurebayashi for\nvaluable discussions. This work was supported by the\nCabinet Oﬃce, Government of Japan through its “Fund-\ning Program for Next Generation World-Leading Re-\nsearchers,” the Asahi Glass Foundation, the Sumitomo\nFoundation, Research Foundation for Materials Science,\nand JST-CREST “Creation of Nanosystems with Novel\nFunctions through Process Integration”.\n∗Electronic address: ando@imr.tohoku.ac.jp\n[1] L. D. Landau and E. M. Lifshitz, Mechanics , 3rd ed.\n(Butterworth-Heinemann, Oxford, 1976).\n[2] H. M. Gibbs, Optical bistability: controlling light with\nlight(Academic Press, Orlando, 1985).\n[3] I. D. Mayergoyz, Nonlinear Magnetization Dynamics in\nNanosystems (Elsevier, New York, 2009).\n[4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[5] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404(R) (2002).\n[6] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[7] B. Heinrich et al., Phys. Rev. Lett. 90, 187601 (2003).\n[8] M.V.Costache et al., Phys.Rev.Lett. 97, 216603(2006).\n[9] O. Mosendz et al., Phys. Rev. B 82, 214403 (2010).5\n[10] Y. Kajiwara et al., Nature 464, 262 (2010).\n[11] F. D. Czeschka et al., Phys. Rev. Lett. 107, 046601\n(2011).\n[12] A. Azevedo et al., Phys. Rev. B 83, 144402 (2011).\n[13] B. Heinrich et al., Phys. Rev. Lett. 107, 066604 (2011).\n[14] C. W. Sandweg et al., Phys. Rev. Lett. 106, 216601\n(2011).\n[15] K. Ando, T. An, and E. Saitoh, Appl. Phys. Lett. 99,\n092510 (2011).\n[16] H. Kurebayashi et al., Appl. Phys. Lett. 99, 162502\n(2011).\n[17] K. Ando et al., Nat. Mater. 10, 655 (2011).\n[18] H. Kurebayashi et al., Nat. Mater. 10, 660 (2011).\n[19] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n[20] T. Kimura et al., Phys. Rev. Lett. 98, 156601 (2007).\n[21] S. O. Valenzuela and M. Tinkham, Nature 442, 176\n(2006).\n[22] T. Seki et al., Nat. Mater. 7, 125 (2008).\n[23] K. Ando et al., Phys. Rev. B 78, 014413 (2008).\n[24] H. Suhl, J. Phys. Chem. Solids 1, 209 (1957).\n[25] P. Kabos, G. Wiese, and C. E. Patton, Phys. Rev. Lett.\n72, 2093 (1994).\n[26] V. S. L’vov, Wave Turbulence Under Parametric Excita-tion(Springer-Verlag, Berlin, 1994).\n[27]Nonlinear Phenomena and Chaos in Magnetic Materi-\nals, edited by P. E. Wigen (World Scientiﬁc, Singapore,\n1994).\n[28] The modulation ﬁeld for the lock-in detection was 0.01\nmT, which is small enough so that experimental artifacts\ndue to hysteresis of exchange spin waves for determining\nthe threshold are negligible. We plotted the minimum\nand maximum ﬁelds within which the absorption signal\nis nonzero as a function of the applied microwave power,\nwhich provides the relation between PthandH.\n[29] C. E. Patton, Phys. Rep. 103, 251 (1984).\n[30] G. Wiese, P. Kabos, and C. E. Patton, Phys. Rev. B 51,\n15085 (1995).\n[31] Y. S. Gui, A. Wirthmann, and C.-M. Hu, Phys. Rev. B\n80, 184422 (2009).\n[32] A. G. Gurevich and G. A. Melkov, Magnetization Oscil-\nlations and Waves (CRC Press, New York, 1996).\n[33] E. Zakharov, V. S. L’vov, and S. S. Starobinets, Sov.\nPhys. Usp. 17, 896 (1975).\n[34] M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-\nHill, New York, 1964)."    },    {        "title": "1502.04107v2.Spin_Dynamics_with_Inertia_in_Metallic_Ferromagnets.pdf",        "content": "arXiv:1502.04107v2  [cond-mat.mes-hall]  29 Jun 2015Spin Dynamics with Inertia in Metallic Ferromagnets\nToru Kikuchi∗and Gen Tatara\nRIKEN Center for Emergent Matter Science (CEMS),\n2-1 Hirosawa, Wako, Saitama, Japan\nJune 30, 2015\nAbstract\nNon-adiabatic contribution of environmental degrees of fr eedom yields eﬀective\ninertia of spin in eﬀective spin dynamics. In this paper, we st udy several aspects of\nthe inertia of spin in metallic ferromagnets. (i) a concrete expression of the spin inertia\nms:ms=/planckover2pi1Sc/(2gsd), whereScis the spin polarization of conduction electrons and\ngsdis thesdcoupling constant. (ii) dynamical behavior of spin with ine rtia, discussed\nfrom viewpoints of a spinning top and of a particle on a sphere . (iii) behavior of spin\nwaves and domain walls in the presence of inertia, and behavi or of spin with inertia\nin the case of a time-dependent magnetic ﬁeld.\n1 Introduction\nDiﬀerent from other fundamental quantities such as mass and cha rge, spin is a dynamical\nquantity, and its dynamics have been widely studied and applied in scien ce and technology.\nIn particular, recent rapid growth of spintronics provides a stage where deeper understand-\nings of spin dynamics directly lead to practical applications.\nThedynamics ofspinaregoverned bythespinBerryphase[1], andit sequationofmotion\nincludesonlytheﬁrst-ordertimederivativeofspin. Thisisnaturalb ecausespinisanangular\nmomentum and its equation of motion takes the familiar form: the time -derivative of the\nangular momentum (i.e. spin) is given by the torque acting on it. Withou t any torque, the\nsolution of the equation of motion of spin is only a static one. This is in co ntrast with, for\n∗toru.kikuchi@riken.jp\n1example, the case of a massive point particle, which has inertia and ca n move at non-zero\nspeed as its free motion. In this sense, spin does not have inertia.\nHowever, for systems where spin interacts with other environmen tal degrees of freedom,\nspindynamicsareaﬀectedbythoseenvironmental degreesoffre edom, andthedynamicallaw\nof spin is changed to be an eﬀective one. For example, in metallic ferro magnets, conduction\nelectrons aﬀectthedynamics oflocalizedspins (i.e. spins ofatomson latticecites). Atypical\neﬀect is spin-damping (e.g. [2]), where the energy and the angular mo mentum of spins are\ntransferred to the environmental degrees of freedom and, as t he result, spins relax to their\nground state within a certain time scale. Without this eﬀect, spins un dergo the Larmor\nprecession around the applied magnetic ﬁeld forever. In this respe ct, the existence of the\ndegrees of freedom other than spins changes the dynamical beha vior of spins signiﬁcantly.\nIn the equation of motion of spins, this damping is represented by th e Gilbert damping term\n[3]. This damping term includes only the ﬁrst-order time derivative of s pins. Therefore, spin\nstill does not have inertia even when we take into account the Gilbert damping eﬀect.\nThe eﬀects of the environments other than the Gilbert damping can be studied system-\natically by the derivative expansion, where the eﬀects are expande d in powers of the time\n(and spatial) derivative of spin. From that point of view, the Gilbert d amping term gives\nthe leading order term in that expansion. In the higher orders, the re appear terms which\ninclude the second-order time derivative, the third-order time der ivative and so on, in the\nequation of motion of spin. These terms with higher order time deriva tives are interesting\nin that, like the Gilbert damping term, they change the dynamical law o f spin itself, more\nthan give additional torque on spin. In particular, from a compariso n with the form of the\nNewton’s equation of motion of a massive point particle, the term with the second-order\ntime derivative of spin plays the role of the inertia of spin.\nSuch spin inertia has been discussed in the literature (e.g. ref.[4]). In particular, recent\nprogress in ultrafast magnetization [5, 6] motivated several work s. In refs.[7, 8], the inertia\nof spin was introduced phenomenologically and was shown to give addit ional nutation to the\nmotion of spin. The time scale where the eﬀect of the inertia is signiﬁca nt was discussed,\nbased on the work of Brown [9], to be sub-picosecond order. In re f.[10], the equivalence\nbetween the dynamics of spin with inertia and a spinning top was discus sed. Microscopic\nderivation of spin inertia was performed in refs.[11] and [12]. In ref.[11 ], an extended breath-\ning Fermi surface model was used, and the relation between the Gilb ert damping coeﬃcient\nand the spin inertia was given in terms of physical quantities (such as Fermi-Dirac occu-\npation numbers) of conduction electrons. The time scale for the nu tational motion to be\n2damped by the Gilbert damping was estimated to be sub-picosecond o rder. In ref.[12], a\ngeneral expression of the contribution of the conduction electro ns to spin dynamics was\ndiscussed. The spin eﬀective dynamics was shown to be non-local in g eneral, which can be\napproximated as local dynamics by the derivative expansion of spin. The inertial term of\nspin arises in that derivative expansion and its general expression w as given in term of the\nGreen’s function of the conduction electrons.\nSince the inertia of spin is conceptually interesting in its own and gives t he ﬁrst step\ntoward the understanding of non-adiabatic contribution of enviro nmental degrees of freedom\nto spin eﬀective dynamics, further investigations are worthwhile. A lthough the expressions\nof the inertia of spin were given as integral forms[11, 12], an explicit e xpression of the inertia\nof spin in terms of parameters of a model has not been obtained so f ar. Furthermore, the\neﬀect of the inertia of spin on the dynamics of spin has been discusse d only for spatially\nhomogeneous spin system under time-independent magnetic ﬁeld. I n this paper, we present\ndetailed theoretical study of the eﬀects induced by the spin inertia based on an sdmodel.\nIn section 2, we derive a concrete expression of spin inertia in terms of the parameters in the\nsdmodel. In section 3, the basic behavior of spin with ﬁnite inertia is stud ied with the help\nof its two equivalents: a symmetric spinning top and a massive charge d particle on a sphere\nsubject to a monopole ﬁeld. In section 4, we study spatially inhomoge neous system, and\ndiscuss that spin waves and magnetic domain walls acquire an additiona l oscillation mode\ndue to spin inertia. We also study the behavior of spin under large and time-dependent\nmagnetic ﬁeld, and ﬁnd an unusual behavior of spin where the velocit y of spin is parallel to\nthe direction of the time-derivative of magnetic ﬁeld.\n2 Spin eﬀective action and inertia\nThe inertia of spin arises naturally in its eﬀective dynamics, which take s into account the\neﬀects of theenvironmental degrees offreedom [12]. Consider a system where aclassical ﬁeld\nof localized spin, S(x,t) =Sln(x,t) (withSl=|S|ﬁxed), and a ﬁeld c(x,t) representing\nthe other environmental degrees of freedom are interacting with each other. For concrete-\nness, we consider the case of metallic ferromagnets in this paper. C onduction electrons are\nrepresented by annihilation and creation operators, cand ¯c. The total action is given by\nSs[n] +Se[c,¯c,n] whereSs[n] is the action of spin nandSe[c,¯c,n] is that of conduction\nelectrons with their interaction with spin n. When we are interested only in the dynamics\nof spinn, it is convenient to integrate out cand ¯c, and derive the eﬀective action of spin n.\n3The contribution from electrons is given by path integration as\nexp/parenleftbiggi\n/planckover2pi1∆Seﬀ[n]/parenrightbigg\n≡/integraldisplay\nD¯cDcexp/parenleftbiggi\n/planckover2pi1Se[c,¯c,n]/parenrightbigg\n. (2.1)\nThe sum of this ∆ Seﬀ[n] and the original spin action Ss[n] gives the total spin eﬀective\naction.\nIt is diﬃcult to calculate ∆ Seﬀ[n] exactly, so we should rely on a perturbative analysis.\nWe here perform derivative expansion, where ∆ Seﬀ[n] is expanded in powers of ∂µn(µ=\nt,x,y,z). When the system is isotropic, the general form is\n∆Seﬀ[n] =/integraldisplayd3x\na3dt/bracketleftbigg\nSc˙φ(cosθ−1)−JcS2\nc\n2(∂in)2+ms\n2˙n2/bracketrightbigg\n+O((∂µn)3),(2.2)\nwheren= (sinθcosφ,sinθsinφ,cosθ) andi=x,y,z. We have divided the Lagrangian\ndensity entirely by lattice volume a3so that each coeﬃcient represents a quantity per each\nlattice cite. The ﬁrst term is the spin Berry phase with Scthe spin polarization of the con-\nduction electrons and the second is the spin-spin exchange interac tion induced by electrons\nwithJcthe coupling constant. In the ﬁnal term, there arises the inertial term of spin with\nthe inertia ms. Thismshas the dimension of [kg ·m2], the same as that of the moment of\ninertia. In ref.[12], a general expression of spin inertia msis derived in term of the Green’s\nfunction of the conduction electrons. Let us here calculate a conc rete expression of spin\ninertia. As in ref.[12], a typical example of Se[c,¯c,n] is thesdmodel [13], where conduction\nelectrons interact with localized spins nas\nSe[c,¯c,n] =/integraldisplay\nd3xdt¯c/parenleftbigg\ni/planckover2pi1∂t+/planckover2pi12∂2\ni\n2m+ǫF+gsdn·σ/parenrightbigg\nc. (2.3)\nHere,mis the mass of the conduction electrons, ǫFis the Fermi energy, gsdis thesdcoupling\nconstant, and σis the Pauli matrix vector. To obtain the derivative expansion of ∆ Seﬀ[n],\nwe perform SU(2) gauge transformation c→U(x,t)cwith anSU(2) matrixUacting on the\nspinor indices, so that the sdinteraction becomes diagonal, ¯ c(n·σ)c→¯cσ3c. Due to this\nunitary transformation, there appears a so-called spin gauge ﬁeld Aµ≡−iU†∂µUin (2.3)\nthrough∂µc→U(∂µ+iAµ)c. ThisAµcontains the ﬁrst-order derivative ∂µn. Therefore,\nexpanding exp( iSe[c,¯c,n]//planckover2pi1) in powers of Aµ, we can calculate ∆ Seﬀ[n] perturbatively in\npowers of∂µn. The spin polarization Scand the inertia mscan be calculated as (see\nAppendix A for details)\nSc=a3/planckover2pi1\n2k3\nF+−k3\nF−\n6π2, ms=/planckover2pi1Sc\n2gsd(2.4)\n4with/planckover2pi12k2\nF±/(2m)≡ǫF±gsd. The inertia can be rewritten as\nms= (kFa)3/planckover2pi12\n8π21\nǫFf(gsd/ǫF), f(x)≡1\n3x/bracketleftig\n(1+x)3\n2−(1−x)3\n2/bracketrightig\n. (2.5)\nFor 0< x <1, the function f(x) is a only slightly decreasing function from f(0) = 1 to\nf(1)/equaldotleftright0.94. Therefore, spin inertia does not depend much on the sdcoupling constant\ngsdand is proportional to a3√ǫF, whenaandkF=√2mǫF//planckover2pi1are regarded as independent\nparameters.\nAdding this ∆Seﬀ[n] to the original spin action of the form\nSs[n] =/integraldisplayd3x\na3dt/bracketleftbigg\nSl˙φ(cosθ−1)−JlS2\nl\n2(∂in)2/bracketrightbigg\n, (2.6)\nwithJlthe exchange coupling between spins, we obtain the total spin eﬀec tive action as\nSeﬀ[n] =Ss[n]+∆Seﬀ[n]. Including the Zeeman coupling with external magnetic ﬁeld1we\nobtain\nSeﬀ[n] =/integraldisplayd3x\na3dt/bracketleftbigg\nSB·n+S˙φ(cosθ−1)−JS2\n2(∂in)2+ms\n2˙n2/bracketrightbigg\n+O((∂µn)3),(2.7)\nwithS≡Sc+Slthe total spin amplitude per lattice cite and J≡(JcS2\nc+JlS2\nl)/S2. We\nhave set the gyromagnetic ratio as unity.\nAswe will see below, spin with ﬁnite inertia hasa typical precession mod ewithfrequency\nω0∼S/ms. Usingms(2.4) or (2.5), and assuming S∼/planckover2pi1,kFa∼πandǫF∼1eV, the\nenergy scale of this frequency becomes /planckover2pi1ω0∼1eV, so that its period is 2 π/ω0∼0.1ps.\nTherefore, as far as this simple estimation suggests, the existenc e of the inertia is signiﬁcant\nfor the dynamics of sub-picosecond scale.\nThe equation of motion derived from this eﬀective action (2.7) is\nS˙n=−SB×n−JS2∂2\nin×n+ms¨n×n. (2.8)\nThus, the inertial term produces acceleration-dependent torqu e. We can rewrite this equa-\ntion of motion, by taking vector product with n, as\nms¨n=Sn×˙n+SB+JS2∂2\nin−(SB·n+ms˙n2−JS2(∂in)2)n.(2.9)\nThe Gilbert damping eﬀect adds a term −αS˙n, withαthe dimensionless constant, to the\nright hand side of the equation of motion (2.9). Therefore, the Gilbe rt damping plays the\n1To be precise, there emerge other coupling terms between spin and electromagnetic ﬁeld in addition to\nthe Zeeman coupling, such as in ref.[14], by integrating out conductio n electrons. In this paper, we simply\nassume that they are negligible.\n5same role as the familiar linear damping force for a point particle, and t he time scale for\nthis damping term to be signiﬁcant is tdamp∼ms/(αS). On the other hand, the time scale\nfor the inertial term to be eﬀective is, as we will see below, tinertia∼ms/S. Therefore, for\nα≪1, we canneglect the Gilbertdamping termaslong aswe areintereste d inthe dynamics\nwithin the time scale tinertia.\nThe equation of motion (2.8) can be rewritten (when B= 0) in the conservation form\nof the angular momentum current ( j0,ji),\n∂0j0+∂iji= 0,wherej0=Sn+msn×˙n,ji=JS2∂in×n (2.10)\n(∂0≡∂/∂t). Note that the angular momentum j0(per lattice cite), which is the Noether\ncharge corresponding to the invariance of the action (2.7) under SO(3) rotation in the in-\nternal spin space (see Appendix B for details), is no longer proport ional tonbut includes\nthe non-adiabatic contribution of the conduction electrons, msn×˙n. Originally, the total\nangular momentum consists of that of the localized spin and that of t he conduction elec-\ntrons:j0=Sln+(/planckover2pi1a3/2)/angbracketleft¯cσc/angbracketrightwithSlthe amplitude of the localized spin. In the lowest\norder of the derivative expansion, i.e. in the adiabatic limit, the spin of the conduction\nelectron aligns with that of the localized spin, so that ( /planckover2pi1a3/2)/angbracketleft¯cσc/angbracketright=ScnwithScthe spin\npolarization of the conduction electrons. Beyond the adiabatic limit, the direction of the\nspin of the conduction electron is generally diﬀerent from that of th e localized spin. The\nderivative expansion incorporates this diﬀerence systematically, a nd the next order term in\n(/planckover2pi1a3/2)/angbracketleft¯cσc/angbracketrightis given by msn×˙n.\nRemarkably, the relation between the inertia msand the spin polarization Scin (2.4),\nms=/planckover2pi1Sc/(2gsd), is easily obtained without any microscopic calculation, as follows. As we\nhave discussed in the last paragraph, the angular momentum ∆ j0derived from ∆Seﬀ[n]\nrepresents the spin polarization of the conduction electrons,\n∆j0≡Scn+msn×˙n=/planckover2pi1a3\n2/angbracketleft¯cσc/angbracketright. (2.11)\nSince (/planckover2pi1a3/2)/angbracketleft¯cσc/angbracketrightobeys the following equation of motion,\n∂0/parenleftbigg/planckover2pi1a3\n2/angbracketleft¯cσc/angbracketright/parenrightbigg\n=−2gsd\n/planckover2pi1n×/parenleftbigg/planckover2pi1a3\n2/angbracketleft¯cσc/angbracketright/parenrightbigg\n(2.12)\n(we consider here only spatially homogeneous case, for simplicity), ∆ j0also satisﬁes\n∂0∆j0=−2gsd\n/planckover2pi1n×∆j0. (2.13)\n6Substitution of eq.(2.11) into eq.(2.13) leads to\n/parenleftbigg\nSc−2gsd\n/planckover2pi1ms/parenrightbigg\n˙n+O((∂0)2) = 0. (2.14)\nSince this equation is true for an arbitrary n, we arrive at the relation ms=/planckover2pi1Sc/(2gsd).\nThus, we can obtain msfromScwithout any detailed calculation. The point is that the\nequation (2.13) is not the equation of motion of n, although it involves the time derivative\nofn. [The equation of motion of nis∂0j0= 0, or,∂0(Sln+∆j0) = 0.] Before integrating\nout the electrons, it was the equation of motion of /angbracketleft¯cσc/angbracketright(2.12) and, after integrating out\nthe electrons, the equation (2.13) determines the structure of ∆ Seﬀ[n]. Conversely, the\nrelationms=/planckover2pi1Sc/(2gsd) must hold in order that the eﬀective action ∆ Seﬀreproduces\nthe equation of motion of /angbracketleft¯cσc/angbracketright. We can repeat this procedure to arbitrary orders in the\nderivative expansion: ﬁrst, write down all possible terms in the eﬀec tive action ∆Seﬀ, with\ntheir coeﬃcients left undetermined; second, derive the angular mo mentum ∆ j0from that\n∆Seﬀand substitute it into eq.(2.13); then we can obtain the recursion re lations between\nthe coeﬃcients2. For example, ∆ Seﬀto the fourth-order can be obtained as follows (see\nAppendix C for details):\n∆Seﬀ[n] =Sc/integraldisplayd3x\na3dt/bracketleftig\n˙φ(cosθ−1)+1\n4/planckover2pi1\ngsd˙n2\n−1\n8/parenleftbigg/planckover2pi1\ngsd/parenrightbigg2\nn·(˙n×¨n)+1\n16/parenleftbigg/planckover2pi1\ngsd/parenrightbigg3\n¨n2−5\n64/parenleftbigg/planckover2pi1\ngsd/parenrightbigg3\n(˙n2)2/bracketrightig\n+O(∂5\n0).\n(2.15)\nThe same procedure can be applied also for the terms in the eﬀective action ∆Seﬀwhich\ninvolve the spatial derivative.\n3 Dynamical behavior of spin with inertia\nIn this section, we describe classical dynamics of spin with inertia. Fo r that purpose, it is\nhelpful to use two equivalent pictures, which are summarized in Fig.1 . One is a symmetric\n2Such recursion relations can be obtained also via the equation of mot ion of spin, as follows. The original\nequation of motion of localized spin is Sl˙n=−SlB×n−JlS2\nl∂2\nin×n−gsda3/angbracketleft¯cσc/angbracketright×n. Substitution of\nthe expression of /angbracketleft¯cσc/angbracketright[eq.(2.11)] into this original equation of motion gives the term propor tional to ˙n, i.e.\n−gsda3/angbracketleft¯cσc/angbracketright×n=−(2gsdms//planckover2pi1)˙n+O(∂2\n0). The coeﬃcient of this term is identical with the spin polarization\nScoftheconductionelectron, whichgivestherenormalizationofthes pinamplitude, Sl→Sl+Sc. Therefore,\nwe obtain the relation ms=/planckover2pi1Sc/(2gsd).\n7g\n/g373/g381/g374/g381/g393/g381/g367/g286\n/g17\n/g393/g258/g396/g410/g349/g272/g367/g286\n/g410/g381/g393 /g400/g393/g349/g374\nFigure 1: The dynamics of spin with inertia, a symmetric spinning top, and a mass ive charged particle\non a sphere subject to a monopole magnetic ﬁeld Bm, are classically equivalent. They undergo precession\nmotion accompanied by nutation under applied ﬁelds, which are a magn etic ﬁeldBfor spin, a gravitational\nﬁeldgfor the top, and an electric ﬁeld Efor the particle, respectively. See the main text for the detail.\nspinning top, and the other is a massive charged particle on a sphere subject to a monopole\nmagnetic ﬁeld. We consider here only spatially homogeneous spin, ∂in= 0, under time-\nindependent magnetic ﬁeld, for simplicity.\n3.1 Equivalence to a symmetric spinning top\nThe equivalence between the classical dynamics of spin and a spinning top has been recog-\nnized in the literature, e.g. refs.[15] and [10]. The content of this sub section is essentially\na recapitulation of these facts, which we describe here in order for this paper to be self-\ncontained.\nFirst, let us write down the Lagrangian of spin with ﬁnite inertia and th at of a spinning\ntop. The Lagrangian of spin (2.7) is\nLspin=ms\n2(˙θ2+˙φ2sin2θ)+S˙φ(cosθ−1)+BScosθ, (3.1)\nwhile the Lagrangian of a spinning top in terms of the Euler angles ( θ,φ,ψ) is\nLtop=I1\n2(˙θ2+˙φ2sin2θ)+I3\n2(˙ψ+˙φcosθ)2+µglcosθ, (3.2)\nwhereI1,I3are the principle moments of inertia, µis the mass of the top, gis the gravita-\ntional acceleration constant, and lis the distance between the center of mass and the ﬁxed\nextremity of the top. Here we have taken a symmetric spinning top a nd set two moments of\ninertia equal, I1=I2. We take the positive directions of the external magnetic ﬁeld Band\n8the gravity both in the positive zdirection. Note that Lspinis a function of ( θ,φ) whileLtop\nis that of (θ,φ,ψ). Let us see below that they have equivalent dynamics concerning ( θ,φ).\nThe equivalence can be directly seen at the level of their equations o f motion. The\nequation of motion of spin is\nms(¨θ−˙φ2sinθcosθ)+S˙φsinθ+BSsinθ= 0,\nd\ndt/bracketleftig\nms˙φsin2θ+S(cosθ−1)/bracketrightig\n= 0, (3.3)\nwhile the equation of motion of the spinning top is\nI1(¨θ−˙φ2sinθcosθ)+I3(˙ψ+˙φcosθ)˙φsinθ+µglsinθ= 0,\nd\ndt/bracketleftig\nI1˙φsin2θ+I3(˙ψ+˙φcosθ)cosθ/bracketrightig\n= 0,\nd\ndt/bracketleftig\nI3(˙ψ+˙φcosθ)/bracketrightig\n= 0. (3.4)\nFrom the last equation in eq.(3.4), the canonical momentum M3conjugate to ψ,\nM3≡I3(˙ψ+˙φcosθ), (3.5)\nis conserved. Substituting this M3for˙ψin the other two equations in (3.4), we obtain the\nsame equations as (3.3) with replacements\nms↔I1, S↔M3, BS↔µgl. (3.6)\nThus, the classical behaviors of θandφare the same for spin and a spinning top.\nWe can see the correspondence more explicitly through their Hamilto nians :\nHspin=p2\nθ\n2ms+1\n2ms(Mφ−Scosθ)2\nsin2θ−BScosθ (3.7)\nand\nHtop=p2\nθ\n2I1+1\n2I1(Mφ−M3cosθ)2\nsin2θ−µglcosθ+M2\n3\n2I3, (3.8)\nwherepθ≡∂L/∂˙θandMφ≡∂L/∂˙φare the canonical momenta of θandφ, respectively.\nThese two Hamiltonians are completely the same under the replaceme nts (3.6). [The last\nterm in (3.8) does not contribute to the dynamics of θandφ.]\nThe Lagrangians (3.1) and (3.2) are related by the Legendre trans formation about ψ:\nLspin(θ,˙θ,˙φ;S) =Ltop(θ,˙θ,˙φ,˙ψ)−S˙ψ|˙ψ=˙ψ(θ,˙φ,S) (3.9)\nwith replacements (3.6). In the right hand side, ˙ψis substituted by S(orM3) via (3.5).\nThe situation is quite similar to that of the familiar centrifugal force p roblem. There, the\n9/g400/g393/g349/g374/g894/g258/g895 /g894/g271/g895\n/g400/g393/g349/g374/g17\n/g410/g381/g393\nFigure 2: (a) Without any magnetic ﬁeld, general motion of spin with ﬁnite inert ia is a free precession\nmotion, just like a spinning top. The spin Snprecesses around the total angular momentum j0. (b) When\na constant magnetic ﬁeld Bis applied, the total angular momentum j0precesses around B. Therefore, the\n‘free precession cone’ in (a) precesses around Bas a whole, which corresponds to the Larmor precession in\nthe absence of the inertia. What was called the free precession in (a ) is now called the nutation.\noriginal Lagrangian is given as L(r,˙r,˙φ) = (m/2)(˙r2+r2˙φ2)−U(r), and we can obtain\nφ-reduced Lagrangian by the Legendre transformation about φ:Lred(r,˙r;M)≡L−M˙φ=\n(m/2)˙r2−M2/(2mr2)−U(r). Inexchange for reducing φ, there appearsaﬁctitious potential\nM2/(2mr2). Likewise, spin dynamics is the ψ-reduced dynamics of a spinning top, and the\nspin Berry phase (the second term in eq.(3.1)) appears as the ﬁctit ious potential arising\nfrom theψ-reduction. When we perform the Legendre transformation also a boutθandφ\non both sides in eq.(3.9), we are led to the same Hamiltonians (3.7) and ( 3.8).\nSince the classical dynamics of spin with inertia and of a spinning top ar e equivalent,\nspin with inertia behaves in the same manner as a spinning top does (Fig .2). When a\nmagnetic ﬁeld is not applied, spin undergoes free precession: the sp in precesses around the\ntotal angular momentum j0=Sn+msn×˙n(eq.(2.10)), which is a constant of motion.\nWhen a magnetic ﬁeld is applied, spin undergoes the Larmor precessio n accompanied by the\nnutation: the total angular momentum j0precesses around the magnetic ﬁeld (the Larmor\nprecession), and the spin precesses around j0at each time (the nutation) [7]. The free\nprecession solution of the equation of motion (3.3) (with B= 0) is\nθ=θ0(const.),˙φ=S\nmscosθ0. (3.10)\nTherefore, the free precession frequency, or the nutation fre quency,ω0, isω0∼S/ms,\nassuming that cos θ0is order of unity (that is, the radius of the spin free precession or\nnutation is not very large). This gives the typical time scale for the in ertial term, tinertia∼\nms/S.\n10Finally, we mention that usual spin with zero inertia, ms= 0, can be regarded as follows.\nIt corresponds to the case of I1(=I2) = 0 under replacements (3.6). This means that the\nother principle moment of inertia, I3, vanishes since, for an actual rigid body, any one of\nthe principle moments of inertia is equal to or less than the sum of the other two, e.g.,\nI3≤I1+I2[16]. Thus, setting I1=I2= 0 leads to I3= 0. Therefore, there does not exist\na spinning top corresponding to spin with zero inertia, in a non-relativ istic framework.\n3.2 Equivalence to a massive charged particle\nSince the spin Berry phase corresponds to a monopole gauge ﬁeld [1], the classical dynamics\nof spin is also equivalent to that of a charged particle on a sphere sub ject to a monopole\nbackground. Let us describe this equivalence and use it to underst and the behavior of spin\nwith inertia.\nA magnetic monopole yields a magnetic ﬁeld in the radial direction with th e strength\ninversely proportional to the square of the distance from the mon opole. Being put at the\noriginx= 0, the gauge ﬁeld Amand the magnetic ﬁeld Bmof the monopole are\nAm=q1−cosθ\nrsinθeφ,Bm=∇×Am=q\nr3x, (3.11)\nwherex=r(sinθcosφ,sinθsinφ,cosθ),eφis the unit vector in φ-direction, and qis the\nmagnetic charge of the monopole. The Lagrangian of a massive char ged particle coupled to\nthis monopole magnetic ﬁeld and a constant electric ﬁeld Eis\nL=m\n2˙x2−Am·˙x−Φ\n=m\n2˙x2+q˙φ(cosθ−1)+E·x, (3.12)\nwhere we take the electromagnetic scalar potential Φ = −E·xso that−∇Φ =E. This\nLagrangian is identical with that of spin with inertia (2.7) when we assu me that the particle\nis constrained on a unit sphere and identify the direction of the part iclexwith the spin\ndirection n, the mass mwith the spin inertia ms, the magnetic charge qwith the spin\namplitudeS, and the electric ﬁeld Eon the particle with the magnetic ﬁeld SBon spin (we\nuse calligraphic letters for electromagnetic ﬁelds on the particle).\nBeing restricted on the unit sphere, the dynamical degrees of fre edom of the particle are\nthe spherical angles θandφ, and the equation of motion is, in vectorial form,\neθ/parenleftbiggd\ndt∂L\n∂˙θ−∂L\n∂θ/parenrightbigg\n+eφ\nsinθ/parenleftbiggd\ndt∂L\n∂˙φ−∂L\n∂φ/parenrightbigg\n= 0\n11orm¨x=qx×˙x+E−(E·x+m˙x2)x, (3.13)\nwhich is identical with (2.9) (with ∂in= 0). The last term proportional to xin the right\nhand side of the second line is the constraint force, which keeps |x|= 1. The particle is on a\nunit sphere and is subject to the monopole magnetic ﬁeld Bm=qx(on the sphere|x|= 1)\nand the electric ﬁeld E. The general motion of the particle is, in the absence of the electric\nﬁeldE, the cyclotron motion, and, in the presence of the electric ﬁeld E, theE×Bmdrift\nmotion3.\nIn view of the equivalence between the dynamics of spin with inertia an d of the charged\nparticle on a sphere, the free precession of spin described in the las t subsection corresponds\nto the cyclotron motion of the particle with the frequency ω0∼q/m=S/ms, and the\nLarmor precession of spin accompanied by nutation corresponds t o theE×Bmdrift motion.\nWe summarize here the three pictures for the classical dynamics de scribed by (2.7).\npicture inertia intrinsic magnetic ﬁeld\nspin the inertia ms the amplitude S\ncharged particle the massm the magnetic charge q\nsymmetric spinning top the moment of inertia I1the angular momentum M3\nWe have seen in this section that spin with ﬁnite inertia has an intrinsic p recession mode\n(i.e. free precession or nutation). The magnetic ﬁeld causing this pr ecession is the monopole\nmagnetic ﬁeld intrinsic to spin, i.e., the spin Berry curvature. The fre quency of the intrinsic\nprecession mode, ω0∼S/ms, is inﬁnite when the inertia of spin msis zero, but goes down\nto ﬁnite value when msbecomes non-zero.\nAs related works, we mention refs.[17, 18, 19, 20, 21, 22] where n utational motion of spin\ninJosephsonjunctionsortunneljunctionsbetween ferromagne tswasdiscussed byexamining\nthe Landau-Lifshitz-Gilbert equation with time-dependent Gilbert d amping coeﬃcient.\n4 Behavior of spin with spatial inhomogeneity and un-\nder time-dependent magnetic ﬁeld\nWehave so farassumed spatiallyhomogeneous conﬁgurationof spin under time-independent\nmagnetic ﬁeld. Let us include spatial variation of spin and time-depen dence of applied\n3Our convention for the sign of magnetic ﬁeld is such that the equatio n of motion of a particle is m¨x=\nB×˙x+E, which yields drift motion of the particle in B×Edirection, rather than E×Bdirection.\n12magnetic ﬁeld. The most typical behavior of spin under time-depend ent magnetic ﬁeld is\nthe resonance phenomenon. In ref.[8], it is shown that spin with ﬁnite inertiamshas a\nresonance peak at the intrinsic frequency ω0∼S/ms. We discuss other behaviors of spin,\nsuch as spin wave, domain wall motion. We also discuss an unusual beh avior of spin under\na large and time-dependent magnetic ﬁeld.\n4.1 Spin waves\nLet us consider spin wave solutions of spin with inertia, and see that t here exists a new\ngapful spin wave mode with unusual handedness. When we assume n= (nx,ny,1) with\nnx,ny≪1, the equation of motion is4, from the action (2.7),\nms¨n=Sn×˙n+JS2∂2\nin (4.1)\nwhere we have set B= 0. Since the equation of motion is of second order in the time\nderivative, we have now two spin wave modes. When we substitute a p lane wavenx+iny=\nAe−i(ωt−kx)to the equation of motion, with Aan arbitrary small constant, we obtain the\ndispersion relation (Fig.3),\nω=1\n2S\nms/parenleftig\n−1±/radicalbig\n1+4Jmsk2/parenrightig\n∼\n\nJSk2+O(k4)\n−/parenleftig\nS\nms+JSk2/parenrightig\n+O(k4).(4.2)\nOne is the usual spin wave mode and the other is the new spin wave mod e, the latter of\nwhich is essentially the free precession of spin mediated in space by th e interaction Jwith\nneighboring spins. We have expanded the dispersion relation (4.2) in p owers ofkand cut\nhigher order terms because we are in the framework of the derivat ive expansion in deriving\neq.(2.7).\nThe usual spin wave mode is clockwise seen from the positive z-direction, while the\nnew spin wave mode is counterclockwise. In fact, to the leading orde r of the dispersion\nrelation (4.2), the former obeys the equation of motion S˙n=−J∂2\nin×ez= +Jk2n×ez,\nwhile the latter ms¨n=−S˙n×ez, whereezis the unit vector in z-direction. Therefore,\ntheir handednesses are opposite. The total angular momentum j0in (2.10) also rotates in\ncounterclockwise for the new spin wave mode.\nWhenthespin andtheangularmomentum aligninthesamedirection, j0=Sn, theﬂuc-\ntuation of spin around the stable conﬁguration is always clockwise. T his can be understood\n4The assumption nx,ny≪1 means that we consider the problem in the tangent plane on nz= 1, so the\nconstraint force in (2.9) vanishes.\n13/g17439/g17440/g664\n/g17439\n/g17440\n/g364/g894/g258/g895 /g894/g271/g895\n/g454\n/g1004/g400/g393/g349/g374/g94/g460/g1089/g94\u0001\nFigure 3: (a) The trajectory of a spin, which ﬂuctuates around Sz=Spoint (this ﬂuctuation propagates\nin space as spin wave). In the ﬁgure, two ﬂuctuation modes are sup erimposed. One is the usual gapless\nmode rotating in clockwise seen from the positive z-direction (denoted by 2/circleco√yrt), while the other is a new\ngapful mode rotating counterclockwise (denoted by 1/circleco√yrt). (b) The dispersion relation for the two modes of\nspin wave. Besides the usual gapless spin wave mode (in green color a nd denoted by 2/circleco√yrt), spin with inertia\nhas a gapful spin wave mode (in red color and denoted by 1/circleco√yrt), which is nutation of spin propagating in\nspace by the spin-spin exchange. See eq.(4.2).\nas follows (see e.g. ref.[23]). Let the stable conﬁguration be nz= 1. The Poisson commuta-\ntion relation of the angular momenta, [ j0\nx,j0\ny] =j0\nz∼=S, leads to that of spin, [ nx,ny] = 1/S.\nTherefore, the xandycomponents of spin are canonically conjugate to each other. More -\nover, the Hamiltonian of spin expanded around the stable point is H∼=D[(nx)2+ (ny)2]\nwithD >0. These two facts, i.e. the canonical commutation relation of nxandny, and\nthe form of the Hamiltonian, indicate that the ﬂuctuation of spin aro undnz= 1 has the\nsame dynamical structure as that of a one-dimensional harmonic o scillator with coordinate\nxand its canonical momentum pgoverned by the commutation relation [ x,p] = 1 and the\nHamiltonian H=D(x2+p2). The trajectory of the harmonic oscillator is always clockwise\nin the phase space. Therefore, the ﬂuctuation of spin is also always clockwise. However,\nwhen the inertia of spin is ﬁnite, the spin and the angular momentum ge nerally do not align\ndue to the additional contribution msn×˙ninj0(2.10), and moreover the Hamiltonian of\nspin such as eq.(3.7) does not consist only of the potential term but also of the kinetic term.\nTherefore, the ﬂuctuation of spin with inertia is not restricted to c lockwise motion.\n4.2 Domain wall\nLet us see that a domain wall also has the free precession mode, cor responding to that of\neach spin. When easy-axis anisotropy ( KS2/2)cos2θis included to the action (2.7), the\nsystem has a static domain wall solution, cos θ= tanh[(z−X)/λ],φ=φ0(we takez-axis as\n14the wall normal) where Xandφ0are integration constants, and λ=/radicalbig\nJ/K. Promoting X\nandφ0to dynamical variables, substituting the domain wall solution into the action (2.7)\nand integrating it over space, we obtain the Lagrangian of Xandφ0:\nL[X,φ0] =Mw\n2(˙X2+λ2˙φ2\n0)+SNw\n2λ(˙Xφ0−X˙φ0) (4.3)\nwithMw=msNw/λ2andNw= 2λA/a3whereAis the cross sectional area of the domain\nwall (we discarded irrelevant constant terms in the Lagrangian). W e are going to show that\nthisLagrangianisofthesameformasthatofachargedparticleona cylinder withaconstant\nmagnetic ﬁeldB=SNw/λ2perpendicularly penetrating the surface of the cylinder. Take an\northogonalcoordinateframe x= (x,y)onacylinder, with y∼y+2πλtheperiodicdirection.\n(The directions x,yand the direction outward normal to the surface of the cylinder ma ke an\northogonalright-handedframe.) Then, thegaugepotential Afortheconstantmagneticﬁeld\nBin the outward normal direction of the surface can be taken as Ax=By/2,Ay=−Bx/2,\nand the Lagrangian of the particle is\nL=m\n2˙x2−A·˙x=m\n2(˙x2+ ˙y2)+B\n2(˙xy−x˙y) (4.4)\nTherefore, when we identify ( X,λφ0) with (x,y), the Lagrangian (4.3) is that of a charged\nparticle under a perpendicular magnetic ﬁeld of ﬂux B=SNw/λ2(Fig.4). The magnetic\nﬁeldBfor the particle originates in the spin Berry phase, and the mass Mwof the particle\ncomes from the inertia of spin in the action (2.7).\nAs a result, the particle undergoes cyclotron motion (in other word s, the domain wall un-\ndergoestheintrinsicfreeprecession mode), where Xandλφ0oscillate,X=rcos(ω0t),λφ0=\nrsin(ω0t) withra constant, at frequency ω0=B/Mw=SNw/(λ2Mw) =S/ms. This fre-\nquency corresponds to that of spin free precession discussed in s ection 3. When an external\nﬁeld is applied on this domain wall, the particle on a cylinder ( X,λφ0) feels the correspond-\ning force on it, and it moves in the direction perpendicular to the forc e, accompanied by the\ncyclotron motion (nutation). That is, the particle undergoes E×Bdrift motion, where E\nis the force and Bis the magnetic ﬁeld penetrating the cylinder ( X,λφ0) perpendicularly.\nThe discussion above is well valid when the inertia msis so large that the intrinsic\nfrequencyω0∼S/msis less than the frequency of spin waves: S/ms<KS2//planckover2pi1. WhenS/ms\nexceeds thisfrequencyscale, thedynamicsofthedomainwallcanb enolongerdescribedonly\nby the collective coordinates Xandφ0, and spin wave excitations should also be included\ninto the dynamics.\n15/g121/g644/g660/g1004/g393/g258/g396/g410/g349/g272/g367/g286\nFigure 4: The collective coordinates ( X,λφ0) of a domain wall can be regarded as coordinates of a particle\non a cylinder. This particle is subject to a magnetic ﬁeld ﬂux B=SNw/λ2penetrating the cylinder\nperpendicularly. The inertia of spin introduces the mass Mw=msNw/λ2into this particle.\n4.3 Parallel shift of spin under time-dependent magnetic ﬁe ld\nInthespinnutationsuchasinFig.1,spinmovesupanddownparallellyto thegivenmagnetic\nﬁeld, but this motion is oscillatory and the guiding center (i.e. the cent er of the nutation)\nmoves only perpendicularly to the magnetic ﬁeld, just as the usual L armor precession. Is it\npossible that the guiding center motion becomes diﬀerent from the u sual Larmor precession\nand has a ﬁnite component of velocity parallel to the magnetic ﬁeld, c onsidering that the\nspin with inertia does not obey the usual Landau-Lifshitz equation? In this subsection, we\nsee that when the magnetic ﬁeld is time-dependent, the guiding cent er motion has ﬁnite\ncomponent of velocity parallel to the time-derivative of the magnet ic ﬁeld.\nBearing the particle analogy discussed in section 3 in mind, this parallel motion is most\ndirectly seen by looking at the behavior of a charged particle, locate d on a plane ( x,y) under\na constant perpendicular magnetic ﬁeld Band a time-dependent in-plane electric ﬁeld E(t)\n(recall that an electric ﬁeld Eon the particle corresponds to a magnetic ﬁeld Bon spin).\nThe general solutions of the equation of motion is given by ( x,y) = (ξ,η) + (X,Y), such\nthat\nm/parenleftigg¨ξ\n¨η/parenrightigg\n=B/parenleftigg\n0−1\n1 0/parenrightigg/parenleftigg˙ξ\n˙η/parenrightigg\n. (4.5)\nand\n/parenleftigg˙X\n˙Y/parenrightigg\n=1\nB1\n1+ω−2c(d2/dt2)/parenleftigg\nω−1\ncd/dt−1\n1ω−1\ncd/dt/parenrightigg/parenleftigg\nEx\nEy/parenrightigg\n, ωc=B\nm.(4.6)\nThat is, we can divide the motion into the cyclotron motion ( ξ,η) and the guiding center\nmotion (X,Y) even when the electric ﬁeld is time-dependent5. We can see that the guiding\n5As long asω−1\ncd/dt<1 and the operator (1+ ω−2\ncd2/dt2)−1is well-deﬁned.\n16center motion( X,Y) hasaﬁnitecomponent ofvelocity paralleltotheelectric ﬁeld, when the\nmass is ﬁnite and the electric ﬁeld is time-dependent. As we saw in sect ion 3, the classical\ndynamics of spin with inertia is a spherical version of this planar proble m, with the role of an\nelectric ﬁeld for the particle played by a magnetic ﬁeld for spin, and th erefore the essential\nfeatures will be the same (especially when the nutation radius is small and the curvature\nof the sphere can be neglected): (i) the spin nutation and the guidin g center motion are\ndecoupled; (ii) the eﬀect of the inertia on the guiding center motion is signiﬁcant when the\ntime rate of change of the applied ﬁeld is comparable to the intrinsic fr equencyS/ms.\nTherefore, the eﬀect of the inertia is not only the superimposition o f the nutational\nmotion to the Larmor precession. The inertia of spin gives an indepen dent eﬀect on the\ntrajectory of the Larmor precession itself, cooperatively with th e time-dependence of the\napplied ﬁeld. Even when the nutation radius is small and there are no s igns of nutation in\nexperimental data about the time-proﬁle of magnetization, it neve r indicates that the inertia\nhas no eﬀects on the dynamics, especially when the applied magnetic ﬁ eld has very rapid\ntime-dependence, such as ultrafast magnetization.\nWe illustrate this fact by numerical calculation (Fig.5). We ﬁx a magne tic ﬁeld in the\nz-direction. First, we make the magnetic ﬁeld in the negative z-direction, then change it\ninto the positive z-direction. We can see that the spin moves parallelly (i.e. in the positive\nz-direction) to the time derivative of the magnetic ﬁeld when the magn etic ﬁeld is changing6.\nThis phenomenon is similar to usual spin ﬂip by the Gilbert damping, but is diﬀerent in that\nwe do not have to vary the direction of the magnetic ﬁeld and moreov er the time change\nrate of the magnetic ﬁeld can be arbitrary (as long as our derivative expansion is valid) and\nis irrelevant to the damping coeﬃcient.\nThis unusual behavior of spin can be simply understood when we look a t the angular\nmomentum rather than spin itself. As in (2.10), the angular momentu m of spin with ﬁnite\ninertia is no longer Snbut has additional non-adiabatic contribution msn×˙n. When\nthe Larmor frequency Bis much smaller than the intrinsic frequency (nutation frequency),\nB≪S/ms, then the vector msn×˙npoints inward to the center of the nutation circle, and\nthe trajectory of the angular momentum of spin is overlapped with t hat of spin (Fig.6(a)).\nOn the other hand, when B∼S/ms, nutation does not form circles but rather ripples on\nthe trajectory of spin. In this case, the vector msn×˙npoints in almost the same direction\nat each time. As the result, the trajectory of the angular moment um deviates from that\n6The switching of the sign of the magnetic ﬁeld from negative to positiv e is not essential for the shift of\nthe spin in z-direction. Only the time-dependence of the magnetic ﬁeld is essent ial.\n17-1-0.500.51\n-1-0.500.51-1-0.500.5B\nSz\nS\nSy\nS\nSx\nS(III)\n(II)(I)\n-0.6-0.4-0.200.20.40.6\n0 0.5 1 1.5 2-100-50050100\nt (ps)Sz\nSB\n(THz)\n(I) (II) (III)(a) (b)\nspin\nmagneticfieldtrajectoryofspin\nFigure 5: Response of spin with ﬁnite inertia to a time-dependent magnetic ﬁeld B(t) =B0tanh(f(t−t0))\ninz-direction with B0= 100THz and f= 2πTHz = 6.3THz. We take the spin amplitude as S=/planckover2pi1and the\ninertia asms//planckover2pi1= 0.1ps/(2π) = 0.016ps. (a) Time proﬁle of Sz/S(=nz) of spin and applied magnetic ﬁeld\nB. (b) The trajectory of spin. The time is divided into three domains I- III. In I, the magnetic ﬁeld points\nin the negative z-direction and spin precesses counterclockwise seen from the pos itivez-direction. In II, the\nmagnetic ﬁeld switches to positive z-direction and the z-component of spin, Sz/S, also changes accordingly.\nIn III, the magnetic ﬁeld points in positive z-direction and spin precesses clockwise.\n18of spin (Fig.6(b)). Fig.7 is a replot of the trajectory of spin in Fig.5 tog ether with that\nof the angular momentum. The vector msn×˙npoints in the positive z-direction in the\ntime domain I (described in Fig.5), and in the negative z-direction in the time domain III,\nresulting in the conservation of the z-component of the angular momentum.\nThis parallel shift of spin in the direction of the time derivative of the m agnetic ﬁeld can\nbe understood also from a viewpoint of a spinning top, where the mag netic ﬁeld on spin\ncorresponds tothegravitationalﬁeldonthetop. Thetime-variat ionofthegravitationalﬁeld\ncan be realized ﬁctitiously by putting the top in an elevator accelerat ed in the negative z-\ndirection. There, the top feels, besides the real gravitational ﬁe ld in the negative z-direction,\naﬁctitious inertial forceinthepositive z-directionduetotheacceleration. This inertialforce\nin the positive z-direction makes the top stand up in the positive z-direction, which explains\nthe shift of spin in z-direction. Thus, the inertia of spin yields an inertial force, literally.\nIn this respect, the connection between the inertia of spin and the inertial eﬀect on spin\n[24, 25] will also be interesting.\n5 Summary and discussion\nWe examined the inertia of spin in metallic ferromagnets, which arises in the derivative\nexpansion of spin eﬀective action and represents non-adiabatic co ntribution from environ-\nmental degrees of freedom. We derived an explicit expression of sp in inertia in an sdmodel.\nThe equivalence between the dynamics of spin with inertia, spinning to p and a charged\nparticle on a sphere was explained. The behavior of spin with spatially in homogeneous con-\nﬁguration or under time-dependent magnetic ﬁeld is studied. The ﬁn ite inertia has mainly\ntwo eﬀects: the superimposition of the nutation on the usual Larm or precession and the\nparallel shift of the Larmor precession.\nAs explained several times in this paper, most of unusual behavior o f spin with ﬁnite\ninertia originates in the non-adiabatic component msn×˙nin the relation between spin\nand the angular momentum, j0=Sn+msn×˙n. Due to this diﬀerence between them,\nspin acquires a kind of separate freedom from the angular momentu m, whose behavior is\nrelativelyrestrictedduetotheconservationlaw. Astheresult, sp inundergoesfreeprecession\nor nutation discussed in section 3, and furthermore parallel shift a long the direction of the\ntime derivative of applied magnetic ﬁelds, discussed in section 4. Since one of the main\npurposes of spintronics is understanding and application of diverse cooperative phenomena\nbetween localized spins and their environmental degrees of freedo m, researches beyond the\n19/g400/g393/g349/g374/g3/g410/g396/g258/g361/g286/g272/g410/g381/g396/g455/g410/g396/g258/g361/g286/g272/g410/g381/g396/g455/g3/g381/g296\n/g258/g374/g336/g437/g367/g258/g396/g3/g373/g381/g373/g286/g374/g410/g437/g373\n/g894/g258/g895\n/g894/g271/g895\nFigure6: Trajectoriesofspin Snandangularmo-\nmentum j0=Sn+msn×˙n. (a)When B≪S/ms,\nspin nutation forms circle shape. The additional\ncomponent msn×˙npoints inward to the center\nof the nutation, and the trajectory of the angu-\nlar momentum sits within that of spin. (b) When\nB∼S/ms, spin nutation forms ripple shape. The\nadditional component msn×˙npoints to one side\nof spin trajectory. As the result, the trajectories of\nspin and the angular momentum deviate.-1.5 -1 -0.5  0  0.5  1  1.5\n-1.5-1-0.5 0 0.5 1 1.5-1-0.5 0 0.5trajectory of angular momentum\ntrajectory of spin\nFigure 7: Replot of the trajectory of spin in Fig.5\n(from slightlya diﬀerent view angle), togetherwith\nthat of the angularmomentum. The magnetic ﬁeld\nisB∼100THz and S/ms∼20πTHz = 62.8THz,\ncorresponding to the case (b) in Fig.6. Although\nthez-component of spin changes in time, that of\nthe angular momentum is constant in motion. The\naxes are in unit of S.\n20adiabatic limit will be well-motivated for this spirit of spintronics; in the a diabatic limit, the\nenvironmental degreesoffreedomareratherobedient totheloc alizedspins. Inthispaper, we\nestimated that in metallic ferromagnets such non-adiabaticity is eﬀe ctive for sub-picosecond\ndynamics, i.e. ultrafast magnetization. For broader applications, p hysical situations where\nnon-adiabaticity is eﬀective for sub-nanosecond dynamics are also desirable.\nApart from the context of eﬀective dynamics (i.e. the dynamics inclu ding the eﬀects of\nenvironmental degrees of freedom), the inertial term, togethe r with the spin Berry phase\nterm, can be present in the dynamics of the order parameters in sy stems whose ground\nstates can be intermediate states between ferromagnet and ant iferromagnet (see e.g.[26]).\nThe discussion of this paper will be also applicable to such systems.\n6 Acknowledgment\nT. K. would like to thank Aron Beekman, Yan-Ting Chen, Hideo Kawagu chi, Se Kwon Kim,\nNguyen Thanh Phuc, Henri Saarikoski, members in Eiji Saitoh group in Tohoku university,\nand other people for helpful comments.\nA spin eﬀective action\nIn this appendix, we calculate the spin eﬀective action by integrating out a conduction\nelectron coupled to spin through the sdinteraction, and derive the expression of spin inertia\n(2.4). We always make summation over repeated indices unless other wise stated.\nThe action of a conduction electron, represented by ﬁeld operato rscand ¯c, coupled to\nlocalized spin nis\nSe[c,¯c,n] =/integraldisplay\nd3xdt¯c/parenleftbigg\ni/planckover2pi1∂t+/planckover2pi12∂2\ni\n2m+ǫF+gsdn·σ/parenrightbigg\nc. (A.1)\nwithgsdthesdcoupling constant. What we calculate in this appendix is the contribut ion\nto spin eﬀective action from the conduction electron, ∆ Seﬀ[n], deﬁned as\nexp/parenleftbiggi\n/planckover2pi1∆Seﬀ[n]/parenrightbigg\n≡/integraldisplay\nD¯cDcexp/parenleftbiggi\n/planckover2pi1Se[c,¯c,n]/parenrightbigg\n. (A.2)\nAdding this ∆Seﬀ[n] to the original spin action\nSs[n] =/integraldisplayd3x\na3/bracketleftig\nSl˙φ(cosθ−1)−JlS2\nl(∂in)2/bracketrightig\n, (A.3)\n21whereθandφare the polar angles of the unit vector n, we obtain the total spin eﬀective\nactionSeﬀ[n] =Ss[n]+∆Seﬀ[n].\nFirst, we diagonalize the sdcoupling in (A.1). In order for that purpose, it is conve-\nnient and essential to employ the unitary-transformed electron ﬁ eldasuch thatc(x,t) =\nU(x,t)a(x,t). Here, the SU(2) matrixUis deﬁned so that\nU†(n·σ)U=σ3. (A.4)\nAssuch,Uhasindeﬁnitenessabouttheright U(1)gaugetransformation, U→Uexp(iχ(x,t)σ3)\nwithχ(x,t) an arbitrary function. [Due to the deﬁning relation c=Uaofa, this unitary\ntransformation corresponds to that of a, i.e.a→exp(−iχ(x,t)σ3)a. The original electron\nﬁeldcremainsinvariant.] Onechoiceof UisU=m·σwithm= (sinθ\n2cosφ,sinθ\n2sinφ,cosθ\n2).\nGeometrically, Urotates the electron spin in the positive z-direction,|↑/angbracketright, to the direction\nofn. For the choice U=m·σabove,U|↑/angbracketright=|n/angbracketrightwith|n/angbracketright≡cos(θ/2)|↑/angbracketright+eiφsin(θ/2)|↓/angbracketright.\nThe indeﬁniteness of U,U→Uexp(iχσ3), corresponds to the phase rotation of |n/angbracketright, i.e.\n|n/angbracketright→exp(iχ)|n/angbracketright. Thesdcoupling becomes diagonal in the a-frame, ¯c(n·σ)c= ¯aσ3abut,\nin compensation for the SU(2) gauge transformation by U, there appears a spin gauge ﬁeld\nAµ≡−iU†∂µU= (m×∂µm)·σin the derivative terms, ∂µc=U(∂µ+iAµ)a.\nThen, the action (A.1) becomes [27]\nSe=/integraldisplay\nd3xdt¯a/parenleftbigg\ni/planckover2pi1(∂t+iAt)+/planckover2pi12(∂i+iAi)2\n2m+ǫF+gsdσ3/parenrightbigg\na\n=S0[a,¯a]+/integraldisplay\nd3xdt/bracketleftbigg\nAa\nt(−/planckover2pi1¯aσaa)+Aa\ni/parenleftbiggi/planckover2pi12\n2m¯aσa← →∂ia/parenrightbigg\n+Aa\niAa\ni/parenleftbigg\n−/planckover2pi12\n2m¯aa/parenrightbigg/bracketrightbigg\n.(A.5)\nHereS0[a,¯a] is\nS0[a,¯a]≡/integraldisplay\nd3xdt¯a/parenleftbigg\ni/planckover2pi1∂t+/planckover2pi12∂2\ni\n2m+ǫF+gsdσ3/parenrightbigg\na, (A.6)\nandAµ=Aa\nµσa(a= 1,2,3) is the spin gauge ﬁeld (make distinction between the electron\nﬁeldaand theSU(2) indexa), which is, in terms of the polar angles of n,\nAµ=1\n2\n−∂µθsinφ−sinθcosφ∂µφ\n∂µθcosφ−sinθsinφ∂µφ\n(1−cosθ)∂µφ\n(A.7)\nwhere (Aµ)a=Aa\nµ. Because Aµis proportional to the ﬁrst order derivative of n, the\nderivative expansion of ∆ Seﬀin powers of ∂µncorresponds to that of Aµ:\ni\n/planckover2pi1∆Seﬀ[n] =i\n/planckover2pi1/integraldisplay\nd4x/bracketleftbigg\nAa\nt(−/planckover2pi1/angbracketleft¯aσaa/angbracketright)+Aa\nii/planckover2pi12\n2m/angbracketleft¯aσa← →∂ia/angbracketright+Aa\niAa\ni/parenleftbigg\n−/planckover2pi12\n2m/angbracketleft¯aa/angbracketright/parenrightbigg/bracketrightbigg\n22+1\n2/parenleftbiggi\n/planckover2pi1/parenrightbigg2/integraldisplay\nd4xd4x′/bracketleftig\nAa\nt(x)Ab\nt(x′)/planckover2pi12/angbracketleft¯aσaa(x)¯aσba(x′)/angbracketright\n+Aa\ni(x)Ab\nj(x′)/parenleftbiggi/planckover2pi12\n2m/parenrightbigg2\n/angbracketleft¯aσa← →∂ia(x)¯aσb← →∂ja(x′)/angbracketright\n+Aa\nt(x)Ab\ni(x′)/parenleftbigg\n−i/planckover2pi13\nm/parenrightbigg\n/angbracketleft¯aσaa(x)¯aσb← →∂ia(x′)/angbracketright/bracketrightig\n+O(A3\nµ) (A.8)\nwhere time-ordered quantum expectation values are taken about S0,\n/angbracketleft···/angbracketright=/integraldisplay\nD¯aDa(···)ei\n/planckover2pi1S0[a,¯a], (A.9)\nand we take only the connected diagrams to evaluate the quantum e xpectation values. Be-\ncausethenon-perturbedaction S0(A.6)isisotropicinspace, /angbracketleft¯aσa← →∂ia/angbracketrightand/angbracketleft¯aσaa(x)¯aσb← →∂ia(x′)/angbracketright\nvanish. We deﬁne the notation for correlation functions as\nρab(x,x′)≡i/angbracketleft¯aσaa(x)¯aσba(x′)/angbracketright, χab\nij(x,x′)≡i/angbracketleft¯aσa← →∂ia(x)¯aσb← →∂ja(x′)/angbracketright.(A.10)\nUsing the Fourier components, we expand these non-local quantit ies. For example,\nρab(x,x′) =/integraldisplayd4q\n(2π)4eiq·(x−x′)ρab(q)\n=δ(4)(x−x′)ρab|q=0+1\n2i∂µδ(4)(x−x′)∂ρab\n∂qµ|q=0+···.(A.11)\nWhen we substitute this into the expansion (A.8), the terms other t han the ﬁrst term in\n(A.11)increasetheorderofthederivativeon nafterpartialintegration,e.g. Aa\nt(x)Ab\nt(x′)∂µδ(4)(x−\nx′)→−∂µ[Aa\nt(x)Ab\nt(x′)]δ(4)(x−x′). Therefore, we take only the ﬁrst term in (A.11) as long\nas we are interested in at most O(∂µn)2-terms. After all, ∆ Seﬀ[n] is, to (∂µn)2-order,\n∆Seﬀ[n] =/integraldisplay\nd4x/bracketleftbigg\n(−/planckover2pi1/angbracketleft¯aσaa/angbracketright)Aa\nt+/parenleftbigg\n−/planckover2pi12\n2m/angbracketleft¯aa/angbracketrightδabδij−/planckover2pi13\n8m2Reχab\nij|q=0/parenrightbigg\nAa\niAb\nj+/parenleftbigg/planckover2pi1\n2Reρab|q=0/parenrightbigg\nAa\ntAb\nt/bracketrightbigg\n(A.12)\n(the imaginary parts of χab\nijandρabcontribute to damping and are neglected here). We\ncalculate each term in the following.\nFirstly, we calculate /angbracketleft¯aσaa/angbracketright. The time-ordered Green’s function for S0(A.6) is\n−i/angbracketleftaα(x)¯aβ(x′)/angbracketright=δαβ/integraldisplayd3kdω\n(2π)4eik·(x−x′)−iω(t−t′)gk,ω,α(no summation over α),\ngk,ω,α≡1\nω−ωkα+iδα, (A.13)\n(the indices α,β,···=±correspond to spin up or down) with\n/planckover2pi1ωk±≡/planckover2pi12k2\n2m−ǫF∓gsd, δα= sgn(ωkα)×0, (A.14)\n23where 0 denotes a positive inﬁnitesimal. We deﬁne a 2 ×2 matrixgk,ωby (gk,ω)αβ≡δαβgk,ω,α\n(no summation over α), or,\ngk,ω≡1\n2(gk,ω,++gk,ω,−+σ3(gk,ω,+−gk,ω,−)). (A.15)\nThen,\n/angbracketleft¯aσaa/angbracketright=−i/integraldisplayd3kdω\n(2π)4eiω0tr[σagk,ω] =−iδa\n3/summationdisplay\n±(±)/integraldisplayd3kdω\n(2π)4eiω0\nω−ωk±+iδ±\n=δa\n3/summationdisplay\n±(±)/integraldisplayd3k\n(2π)3θ(−ωk±) =k3\nF+−k3\nF−\n6π2δa\n3,/planckover2pi12k2\nF±\n2m≡ǫF±gsd,(A.16)\nwhereθ(x) = 1 forx≥0 and zero otherwise.\nSecondly, the number density of the conducting electrons n≡/angbracketleft¯aa/angbracketrightis given by n=\n(k3\nF++k3\nF−)/(6π2).\nThirdly, we calculate Re χab\nij|q=0. Using\ntr[σagk,ωσbgk+q,ω+Ω] =/summationdisplay\n±[(δab−δa3δb3∓iεab3)gk,ω,±gk+q,ω+Ω,∓+δa3δb3gk,ω,±gk+q,ω+Ω,±]\n(A.17)\nand\n−i/integraldisplaydω\n2πgk,ω,αgk+q,ω+Ω,β=θ(ωkα)θ(−ωk+qβ)\nωk+qβ−ωkα−Ω+i0−θ(−ωkα)θ(ωk+qβ)\nωk+qβ−ωkα−Ω−i0,(A.18)\nwe obtain\nχab\nij(q,Ω) =−i/integraldisplayd3kdω\n(2π)4(2ki+qi)(2kj+qj)tr[σagk,ωσbgk+q,ω+Ω]\n=/summationdisplay\n±/integraldisplayd3k\n(2π)3(2ki+qi)(2kj+qj)\n×/braceleftig\n(δab−δa3δb3∓iεab3)/bracketleftbiggθ(ωk±)θ(−ωk+q∓)\nωk+q∓−ωk±−Ω+i0−θ(−ωk±)θ(ωk+q∓)\nωk+q∓−ωk±−Ω−i0/bracketrightbigg\n+δa3δb3/bracketleftbiggθ(ωk±)θ(−ωk+q±)\nωk+q±−ωk±−Ω+i0−θ(−ωk±)θ(ωk+q±)\nωk+q±−ωk±−Ω−i0/bracketrightbigg/bracerightig\n.(A.19)\nWe are interested here only in the real part, Re χab\nij. Using\n1\nx±i0= P1\nx∓iπδ(x) andθ(x)θ(−y)−θ(−x)θ(y) =θ(x)−θ(y) =θ(−y)−θ(−x),(A.20)\nthen,\nReχab\nij(q,Ω) =/summationdisplay\n±/integraldisplayd3k\n(2π)3(2ki+qi)(2kj+qj)\n24×/braceleftig\n(δab−δa3δb3)Pθ(−ωk+q∓)−θ(−ωk±)\nωk+q∓−ωk±−Ω+δa3δb3Pθ(−ωk+q±)−θ(−ωk±)\nωk+q±−ωk±−Ω\n±εab3π[θ(−ωk+q∓)−θ(−ωk±)]δ(ωk+q∓−ωk±−Ω)/bracerightig\n. (A.21)\nLet us calculate the value of each component of Re χab\nij|q=0. First,\nReχ11\nij|q=0= Reχ22\nij|q=0=/summationdisplay\n±/integraldisplayd3k\n(2π)34kikj(±)/planckover2pi1\n2gsd[θ(−ωk∓)−θ(−ωk±)]\n=−/planckover2pi1\ngsd2(k5\nF+−k5\nF−)\n15π2δij. (A.22)\nThese components do not depend on the order of taking the q→0 and Ω→0 limit. On\nthe other hand, the value of the component Re χ33\nijdepends on the order: From eq.(A.21),\nit is clear that when we take the q→0 limit ﬁrst, Re χ33\nijvanishes, while when we take the\nΩ→0 limit ﬁrst and then take the q→0 limit, it gives 0 /0 and, using the L’Hopital’s rule,\nlim\nq→0lim\nΩ→0Reχ33\nij=/summationdisplay\n±/integraldisplayd3k\n(2π)34kikj(−1)δ(−ωk±) =−4mn\n/planckover2pi1δij. (A.23)\nThe order of the q→0,Ω→0 limits can be simply determined by the isotropy of the\ninternal space (i.e. the space where the vector nis deﬁned). For example, the combination\n(A1\ni)2+(A2\ni)2= (1/4)(∂in)2(see eq.(A.7)) is isotropic (i.e. does not depend on the rotation\nn→Rnwith a constant SO(3) matrix R), but (A3\ni)2= (1/4)(1−cosθ)2(∂iφ)2is not an\nisotropic quantity. When we employ lim Ω→0limq→0Reχ33\nij= 0 as the value of Re χ33\nij|q=0,\nthen the (A3\ni)2-term appears in ∆ Seﬀ(A.12) due to the nonzero /angbracketleft¯aa/angbracketright-term. On the other\nhand, when we employ eq.(A.23) as the value of Re χ33\nij|q=0, then the ( A3\ni)2-term vanishes in\n∆Seﬀdue to the cancellation with the /angbracketleft¯aa/angbracketright-term. Therefore, we should employ eq.(A.23) as\nthe value of Re χ33\nij|q=0.7\nFor non-diagonal elements, it is clear that Re χ12\nij|q=0= 0 regardless of the order of the\nq→0 and Ω→0 limits.\n7The order of q→0,Ω→0 limits can be determined also by the U(1) gauge symmetry U→\nUexp(iχσ3), a→exp(−iχσ3)adescribed below eq.(A.4). Under this gauge transformation, the sp in gauge\nﬁeldAµ=Aa\nµσachanges as\n/parenleftigg\nA1\nµ\nA2\nµ/parenrightigg\n→/parenleftigg\ncos2χ−sin2χ\nsin2χcos2χ/parenrightigg/parenleftigg\nA1\nµ\nA2\nµ/parenrightigg\n, A3\nµ→A3\nµ+∂µχ. (A.24)\nThe eﬀective action ∆ Seﬀ(A.12) should be invariant under this gauge transformation. There fore, (A1\ni)2+\n(A2\ni)2-term is allowed but ( A3\ni)2-term is not.\n25Finally,letuscalculateRe ρab|q=0. ThecalculationisalmostthesameasthatofRe χab\nij|q=0:\njust drop the factor −(2ki+qi)(2kj+qj) from eq.(A.19). Then,\nReρ11|q=0= Reρ22|q=0=−/summationdisplay\n±/integraldisplayd3k\n(2π)3(±)/planckover2pi1\n2gsd[θ(−ωk∓)−θ(−ωk±)]\n=/planckover2pi1\ngsdk3\nF+−k3\nF−\n6π2. (A.25)\nTheorderoftakingthe q,Ω→0limitsmattersagainforRe ρ33. Thistime,lim q→0limΩ→0Reρ33/negationslash=\n0 while lim Ω→0limq→0Reρ33= 0. Again, we determine the order of the limits by the\nisotropy about n. When Reρ33|q=0does not vanish, A3\ntA3\ntappears in ∆Seﬀ, which breaks the\nisotropy. Therefore, contrary to the case of Re χ33\nij|q=0, we employ lim Ω→0limq→0Reρ33= 0\nas Reρ33|q=0. Finally, Re ρ12|q=0obviously vanishes.\nAfter all, the expression of the contribution fromthe conduction e lectron to spin eﬀective\naction is\n∆Seﬀ=/integraldisplay\nd4x/bracketleftig\n−/planckover2pi1k3\nF+−k3\nF−\n6π2A3\nt+/planckover2pi12\n2m/parenleftbigg/planckover2pi12\n2m1\n∆sdk5\nF+−k5\nF−\n15π2−n/parenrightbigg\n[(A1\ni)2+(A2\ni)2]\n+/planckover2pi12\n2∆sdk3\nF+−k3\nF−\n6π2[(A1\nt)2+(A2\nt)2]/bracketrightig\n=/integraldisplayd3x\na3dt/bracketleftbigg\nSc˙φ(cosθ−1)−JcS2\nc\n2(∂in)2+ms\n2˙n2/bracketrightbigg\n(A.26)\nwhereais thelattice constant andwe have deﬁned thespin polarization Scof the conduction\nelectron per lattice site, the spin-spin exchange coupling Jcand the inertia of spin msas\nSc≡a3/planckover2pi1\n2k3\nF+−k3\nF−\n6π2, Jc≡a3\nS2c/planckover2pi12\n4m/parenleftbigg\nn−/planckover2pi12\nm1\ngsdk5\nF+−k5\nF−\n30π2/parenrightbigg\n, ms≡/planckover2pi1Sc\n2gsd,(A.27)\nrespectively. It is easy to see that Jc>0 by rewriting it as\nJc=a3\nS2c/planckover2pi12\n120π2m(kF+−kF−)2\nkF++kF−(k2\nF++3kF+kF−+k2\nF−). (A.28)\nWe may safely neglect the eﬀect of impurity, because the lifetime τof the conduction elec-\ntrons will enter only as /planckover2pi1/(ǫFτ) or/planckover2pi1/(gsdτ), both of which we assume much smaller than\none.\nAdding this ∆Seﬀ[n] to the action of localized spin (2.6), we obtain the total eﬀective\nactionSeﬀ[n],\nSeﬀ[n] =/integraldisplayd3x\na3dt/bracketleftbigg\nS˙φ(cosθ−1)−JS2\n2(∂in)2+ms\n2˙n2/bracketrightbigg\n. (A.29)\nwithS≡Sc+SlandJ≡(JcS2\nc+JlS2\nl)/S2.\n26B Derivation of the angular momentum (2.10)\nIn this appendix, we describe the derivation of the angular momentu m current ( j0,ji) (2.10)\nfrom the eﬀective action Seﬀ(2.7), for the readers who are not familiar with such derivation.\nConsider a SO(3) rotation in internal space (spin space)8. Under this rotation, a vector v\nin internal space, such as spin nand magnetic ﬁeld B, is rotated9. When the rotation is\ninﬁnitesimally small, it is given by δva(x,t) =εabcθbvc(x,t), whereθaare constant inﬁnites-\nimal transformation parameters. The eﬀective action (2.7) is invar iant under this rotation.\nThe Noether current corresponding to this invariance, i.e. the ang ular momentum current,\ncan be obtained by the following trick. Make the transformation par ametersθabe depen-\ndent on the position in the real space and time, θa(x,t). Then, the variation of the action\nunder this space-time dependent internal rotation does not vanis h, but is given in the form,\nδSeﬀ=/integraldisplayd3x\na3dt(∂0θ·j0+∂iθ·ji) =−/integraldisplayd3x\na3dtθ·(∂0j0+∂iji).(B.1)\nHere, terms without any time- and spatial-derivatives on θaare absent, because the rotation\nwould be a symmetry of the action if θawere constants. We have performed the partial\nintegration in the last equality in eq.(B.1). When nsatisﬁes its equation of motion, then\nδSeﬀ= 0 because such ngives a saddle point of Seﬀ. SinceδSeﬀ= 0 for arbitrarily position-\ndependentθa(x,t), it follows that ∂0j0+∂iji= 0 when nsatisﬁes its equation of motion.\nThus, (j0,ji) is the conserved current corresponding to the symmetry under internal rota-\ntions, i.e. the angular momentum current. [In fact, since nis a unit vector, an arbitrary\nvariation of nis given by a rotation in internal space. Therefore, from the variat ion (B.1),\nthe conservation law ∂0j0+∂iji= 0 is the equation of motion itself.]\nConcretely, the variation of each term in the eﬀective action (2.7) is given as follows. For\nthe spin Berry phase term,\nδ/integraldisplayd3x\na3dt S˙φ(cosθ−1) =δ/integraldisplayd3x\na3dt/integraldisplay1\n0du S˜n·(∂u˜n×∂t˜n)\n=/integraldisplayd3x\na3dt/integraldisplay1\n0duS[∂u(˜θ·∂t˜n)−∂t(˜θ·∂u˜n)]\n8An element RofSO(3) group and an element UofSU(2) group are related by Rabσb=U†σaUwith\nσathe Pauli matrices. We are interested in the classical dynamics of a s pin vector, that is, the behavior of\nn≡/angbracketleftψ|σ|ψ/angbracketright, where|ψ/angbracketrightis a quantum spin state. A SU(2) rotation on|ψ/angbracketrightcauses aSO(3) rotation on n,\ni.e./angbracketleftψ|U†σaU|ψ/angbracketright=Rabnb.\n9The magnetic ﬁeld Bforms a scalar product with nin the Zeeman term, which means that Bis also a\nvector in internal space with the same transformation property a sn, within the framework presented here.\n27=/integraldisplayd3x\na3dt˙θ·Sn, (B.2)\nwhere we have temporarily used ˜n(x,t,u)which is deﬁned over extended space-time ( x,t,u)\nwith 0≤u≤1 a dummy direction and with boundary conditions ˜n(x,t,u= 0) =n(x,t)\nand˜n(x,t,u= 1) = const .The transformation parameters θare also extended to ˜θ(x,t,u)\nsimilarly. In the last equality in eq.(B.2), we have performed the partia l integration with\nrespect tot. For the inertial term and the exchange coupling term,\nδ/integraldisplayd3x\na3dtms\n2˙n2=/integraldisplayd3x\na3dt˙θ·(msn×˙n),\nδ/integraldisplayd3x\na3dt−JS2\n2(∂in)2=/integraldisplayd3x\na3dt ∂iθ·(JS2∂in×n). (B.3)\nThe Zeeman term is irrelevant for deriving the angular momentum bec ause it does not\ninvolve the derivative on n, which means that it does not yield terms proportional to ∂0θ\nor∂iθin eq.(B.1) under the rotational transformation. In all, from eq.(B.1 ), the angular\nmomentum current can be read as\nj0=Sn+msn×˙n,ji=JS2∂in×n. (B.4)\nC Derivation of higher order terms in spin eﬀective\naction\nIn this appendix, we derive higher order terms in spin eﬀective action ∆Seﬀ(2.15) in the\nway discussed in the last part of section 2. We concentrate only on t ime-derivative terms,\nbut derivation of higher order terms with spatial-derivatives is esse ntially the same. First,\nall possible terms in spin eﬀective action to the fourth-order are\n∆Seﬀ=/integraldisplayd3x\na3dt/bracketleftig\nSc˙φ(cosθ−1)+ms\n2˙n2+c3n·(˙n×¨n)+c4¨n2+˜c4(˙n2)2/bracketrightig\n.(C.1)\nThe angular momentum ∆ j0can be derived from this action in the way described in Ap-\npendix B, as\n∆j0=Scn+msn×˙n+c3(2¨n+3˙n2n)+2c4(˙n×¨n−n×...n)+4˜c4˙n2n×˙n.(C.2)\nThisj0must satisfy ∂0∆j0+ (2gsd//planckover2pi1)n×∆j0= 0. Substitution of eq.(C.2) makes, with\nωsd≡gsd//planckover2pi1,\n∂0∆j0+(2gsd//planckover2pi1)n×∆j0\n28=(Sc−2ωsdms)˙n+(ms+4ωsdc3)n×¨n\n+(2c3+4ωsdc4)...n+(6c3+12ωsdc4)(˙n·¨n)n+(3c3−4ωsdc4−8ωsd˜c4)˙n2˙n.(C.3)\nThe unique solution in order that each coeﬃcient vanishes is that\nms=Sc\n2ωsd, c3=−ms\n4ωsd, c4=−c3\n2ωsd,˜c4=5c3\n8ωsd. (C.4)\nReferences\n[1] section 3.3 in A. Altland and B. Simons, “Condensed Matter Field The ory,” Cambridge\nUniversity Press (2006).\n[2] H. Kohno, G. Tatara, J. Shibata, “Microscopic Calculation of Spin Torques in Disor-\ndered Ferromagnets,” J. Phys. Soc. Jpn. 75, 113706 (2006).\n[3] T. L. Gilbert, “A phenomenological Theory of Damping in Ferromag netic Materials,”\nIEEE Trans. Mag. 40, 3443 (2004).\n[4] H. Suhl, “Theory of the magnetic damping constant,” IEEE Trans . Magn. 34, 1834\n(1998).\n[5] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y. Bigot, “Ultrafast Spin Dynamics in\nFerromagnetic Nickel,” Phys. Rev. Lett. 76, 4250 (1996).\n[6] A. Kirilyuk, A. V. Kimel, Th. Rasing, “Ultrafast optical manipulation of magnetic\norder,” Rev. Mod. Phys. 82, 2731 (2010).\n[7] M. C. Ciornei, J. M. Rubi, J. E. Wegrowe, “Magnetization dynamics in the inertial\nregime: Nutation predicted at short time scales,” Phys. Rev. B 83, 0 20410 (2011).\n[8] E. Olive, Y. Lansac, J. E. Wegrowe, “Beyond ferromagnetic res onance: The inertial\nregime of the magnetization,” App. Phys. Lett. 100, 192407 (2012 ).\n[9] W. F. Brown, Jr., “Thermal Fluctuations of a Single-Domain Partic le,” Phys. Rev. 130,\n1677 (1963).\n[10] J. E. Wegrowe, M. C. Ciornei “Magnetization Dynamics, Gyroma gnetic Relation, and\nInertial Eﬀects,” Am. J. Phys. 80, 607 (2012).\n29[11] M. Fahnle, D. Steiauf, C. Illg “Generalized Gilbert equation includin g inertial damping:\nDerivation from an extended breathing Fermi surface model,” Phys . Rev. B 84, 172403\n(2011).\n[12] S. Bhattacharjee, L. Nordstrom, J. Fransson, “Atomistic S pin Dynamic Method with\nboth Damping and Moment of Inertia Eﬀects Included from First Prin ciples,” Phys.\nRev. Lett. 108, 057204 (2012).\n[13] G. Tatara, H. Kohno, J. Shibata, “Microscopic approach to cu rrent-driven domain wall\ndynamics,” Phys. Rep. 468, 213-301 (2008).\n[14] H. Kawaguchi, G. Tatara, “Coupling Theory of Emergent Spin Ele ctromagnetic Field\nand Electromagnetic Field,” J. Phys. Soc. Jpn. 83, 074710 (2014).\n[15] section 10.4 in A. H. Morrish, “The Physical Principles of Magnetis m,” Wiley-IEEE\nPress (2001).\n[16] L. D. Landau, E. M. Lifshitz, “Mechanics,” Butterworth-Heine mann, 3rd ed. (1982).\n[17] J. X. Zhu, Z. Nussinov, A. Shnirman, A. V. Balatsky, “Novel Sp in Dynamics in a\nJosephson Junction,” Phys. Rev. Lett. 92, 107001 (2004).\n[18] Z.Nussinov, A. Shnirman, D.P.Arovas, A.V. Balatsky, J. X.Z hu, “Spinandspin-wave\ndynamics in Josephson junctions,” Phys. Rev. B71, 214520 (2005) .\n[19] J. X. Zhu, J. Fransson, “Electric ﬁeld control of spin dynamics in a magnetically active\ntunnel junction,” J. Phys. Condens. Matter 18, 9929 (2009).\n[20] J. Fransson, “Detection of spin reversal and nutations thro ugh current measurements,”\nNanotechnology 19, 285714 (2008).\n[21] J. Fransson, J. X. Zhu, “Spin dynamics in a tunnel junction bet ween ferromagnets,”\nNew J. Phys. 10, 013017 (2008).\n[22] J.Fransson,“Subnanosecondswitchingoflocalspin-exchan gecoupledtoferromagnets,”\nPhys. Rev. B 77, 205316 (2008).\n[23] M. Onoda A. S. Mishchenko, N. Nagaosa, “Left-Handed Spin Wa ve Excitation in Fer-\nromagnet,” J. Phys. Soc. Jpn. 77, 013702 (2008).\n30[24] F. W. Hehl, W.-T. Ni, “Inertial eﬀects of a Dirac particle,” Phys. R ev. D 42, 2045,\n(1990).\n[25] M. Matsuo, J. Ieda, E. Saitoh, S. Maekawa, “Eﬀects of mecha nical rotation on spin\ncurrents,” Phys. Rev. Lett. 106, 076601 (2011).\n[26] Y. Kawaguchi, M. Ueda, “Spinor Bose-Einstein condensates,” P hys. Rep. 520, 253\n(2012).\n[27] G. Tatara, H. Fukuyama, “Macroscopic Quantum Tunneling of a Domain Wall in a\nFerromagnetic Metal,” J. Phys. Soc. Jpn. 63, 2538 (1994).\n31"    },    {        "title": "1306.5889v2.Dynamical_spin_spin_coupling_of_quantum_dots.pdf",        "content": "arXiv:1306.5889v2  [cond-mat.mes-hall]  28 Aug 2013Dynamical spin-spin coupling of quantum dots\nVahram L. Grigoryan and Jiang Xiao ( 萧江)∗\nDepartment of Physics and State Key Laboratory of Surface Ph ysics, Fudan University, Shanghai 200433, China\nWe carried out a nested Schrieﬀer-Wolﬀ transformation of an Anderson two-impurity Hamiltonian\nto study the spin-spin coupling between two dynamical quant um dots under the inﬂuence of rotating\ntransverse magnetic ﬁeld. As a result of the rotating ﬁeld, w e predict a novel Ising type spin-spin\ncoupling mechanism between quantum dots, whose strength is tunable via the magnitude of the\nrotating ﬁeld. The strength of the new coupling could be comp arable to the strength of the RKKY\ncoupling. The dynamical coupling with the intristic RKKY co upling enables to construct a four\nlevel system of maximally entangled Bell states in a control lable manner.\nPACS numbers: 75.30.Hx 75.40.Gb 73.40.Gk\nQuantum control of electron spins in semiconduc-\ntor nanostructures is a central issue in the emerging\nﬁelds of spintronics and quantum information process-\ning. Electron spins conﬁned in a semiconductor quan-\ntum dot (QD) was proposed1as a qubit for the real-\nization of scalable quantum computers. In the context\nof quantum-computational applications it is necessary to\ncouple qubits which are not nearest neighbors. The long-\nrange type spin-spin interactions include the Ruderman-\nKittel-Kasuya-Yosida (RKKY) interaction,2Anderson’s\nsuperexchange interaction3and couplings mediated by\ncavity photons,4etc.\nIn all of the above mentioned systems the localized\nspins in the QDs are assumed to be static. The dynam-\nics of mesoscopic systems have been discussed in context\nof charge pumping.5This mechanism is called quantum\npumping and was experimentally realized ﬁrst by Poth-\nieret. al.using QD in 1990.6The pumping mechanism\nis also suitable for producing spin currents, an essential\ningredient for spintronics.7,8A pure spin pump was the-\noretically proposed by Mucciolo et al9in 2001 and was\nexperimentally realized by the Marcus group.10\nThespin-pumping induced dynamicexchangecoupling\nbetween ferromagnetic ﬁlms separated by normal-metal\nspacers is reported by experiments with suﬃciently large\nnormal spacers.11The dynamical coupling was ﬁrst dis-\ncussedinthecontextofelectronspinresonancebyBarnes\nin 1974,12who pointed out its long range nature as com-\npared to static coupling. In the context of ferromag-\nnetic resonance experiments, dynamic exchange coupling\nhas been widely studied by diﬀerent authors.11–15It was\nshown that in multilayersand superlattices, on top of the\nequilibrium spin currents that communicate the nonlocal\nstaticexchangecoupling, adynamicexchangeinteraction\nwith a much longer range becomes important.\nDespite the extensive study of dynamical coupling in\nferromagneticnanostructures,11–15thequantum counter-\npart of the phenomenon has not been discussed yet. In\nthis paper we study the dynamical coupling of two QDs\nin Fermi sea. We ﬁnd a new type of coupling Hamilto-\nnian between the spins in two QDs, which is due to the\ndynamical cotunneling process induced by the rotating\ntransverse magnetic ﬁeld applied. The strength of this\nnew dynamical coupling is tunable via the magnitude ofthe transverse magnetic ﬁeld, and found to be compa-\nrable to the static RKKY coupling. The eigen energies\nwith this dynamical coupling is diﬀerent from that of the\nRKKY coupling, and can be used identifying the dynam-\nical coupling strength. The corresponding eigen states, a\nset of Bell states, can be used for quantum computation.\nWe consider two singly occupied QDs residing in an\nelectron bath and are exposed to a magnetic ﬁeld with\nweak DC component B/bardblalongZaxis (which breaks the\nspin degeneracy by a Zeeman splitting) and a strong AC\ncomponent B⊥(t) in theXYplane whose frequency sat-\nisﬁes the resonance condition of the QD.16The Hamilto-\nnian of the 2-QD system can be written as\nH=H0+Ht+HB(t), (1)\nwhere\nH0=/summationdisplay\ni,σεσ\ninσ\ni+/summationdisplay\niUin↑\nin↓\ni+/summationdisplay\nk,σǫkcσ†\nkcσ\nk,(2a)\nHt=/summationdisplay\ni,k,σ/bracketleftBig\nTi,k(Ri)dσ†\nicσ\nk+T∗\ni,k(Ri)cσ†\nkdσ\ni/bracketrightBig\n,(2b)\nHB(t) =/summationdisplay\ni/planckover2pi1ω⊥/parenleftBig\nd↑†\nid↓\nie−iωt+d↓†\nid↑\nieiωt/parenrightBig\n.(2c)\nH0is the energy of non-interacting quantum dots and\nconduction electrons where dσ†\ni(dσ\ni) creates (annihilates)\nan electron with spin σ=±1(↑,↓) in QD- i(i= 1,2).\nεσ\ni=εi+σ/planckover2pi1ω/bardbl/2 is the energy for spin- σin QD-i\nwith/planckover2pi1ω/bardbl=gµBB/bardblthe Zeeman splitting due to exter-\nnal magnetic ﬁeld B/bardbland gyromagnetic factor gand\nBohr magneton µB.nσ\ni≡dσ†\nidσ\niis the number opera-\ntor of QD- i.Uiis the Coulomb interaction energy on\nQD-i. The third term in Eq. (2 a) stands for the ki-\nnetic energy for the noninteracting electrons in the bath\nwithcσ\nkbeing the annihilation operator of a conduc-\ntion electron with momentum kand spin σwith energy\nǫk. Htis the tunneling Hamiltonian between the local-\nized electrons in QD- iand the conduction electrons with\nthe tunneling rates at the QD position Ri, Ti,k(Ri) =\nTi,ke−ik·Ri.17,18The eﬀect of a rotating transverse mag-\nnetic ﬁeld B⊥(t) =B⊥[icos(ωt)+jsin(ωt)] is in2\nHB(t) with/planckover2pi1ω⊥=gµBB⊥and driving frequency ω.\nWe assume that the QDs are in the Kondo regime, i.e.\nεi< ǫk< Uiand the transfer matrix elements Ti,k(Ri)\nbetween the dots and the continuum are small compared\nwithǫkandUi,i.e.Ti,k≪Ui,Ui−ǫk,ǫk. Under this\nconditions the number of electrons on the dot is a well-\ndeﬁned quantity. To eliminate the explicit time depen-\ndence of the Hamiltonian (1) we make a unitary tran-\nfromtaion to the frame of reference, rotating with fre-\nquencyω.\nFollowing Refs. 19–22, we use a two-stage or nested\nSchrieﬀer-Wolﬀ (SW)23transformation to derive an ef-\nfective spin Hamiltonian to obtain the low-energy spin\ninteractions of the system. The perturbative tunneling\nHamiltonian Ht24enables us to apply the SW transfor-\nmation to the total Hamiltonian Hto eliminate Ht:\nH′=eSHe−S=H+[S,H]+1\n2[S,[S,H]]+···,(3)\nwherethe generatoroperator Sisrequiredto satisfy Ht+\n[S,H0] = 0 :\nS=/summationdisplay\ni,k,σ/parenleftbiggn¯σ\ni\nEki−Ui+1−n¯σ\ni\nEki/parenrightbigg\nT∗\ni,kcσ†\nkdσ\ni−H.c.(4)\nwithEki≡ǫk−εσ\ni. In the absence of magnetic ﬁeld, the\nabove transformation reduces to two impurity Anderson\nmodel.25We are interested in the subspace of single occu-\npancyoftheQDs, i.e.onerequires ni=/summationtext\nσnσ\ni= 1.This\nconstraint can be established by using the Gutzwiller op-\neratorP=/producttext\ni/parenleftBig\nn↑\ni−n↓\ni/parenrightBig2\n,26and we retain only the con-\ntributions in H′that survive under the projection with\nPH′P/negationslash= 0. Up to second order of tunneling rate after\nGutzwiller projection\nH′=H0+HB+1\n2[S,Ht]+1\n2[S,[S,HB]]+···,(5)\nwhere the third term is second order in tunneling rate\nand includes the Kondo (also called “s-d”) Hamiltonian\nplus a potential scattering term.23\nTo study the low-energy spin interactions of\nthe two-impurity Anderson model, a nested19–22or\ngeneralized17,18,27SW transformation has been used to\nderive an eﬀective spin interaction Hamiltonian. The\npurpose of the second transformation is to remove (at\nleast partially) the contribution in second-order in tun-\nneling rate, i.e.the third term in Eq. (5) .19The second\nSW transformation should be done with the generator\noperator St\n2fulﬁlling [ St\n2,H0] =−1\n2[S,Ht]. Note that\nSt\n2∝T2\ni,k.This transformation reveals 4-th order inter-\nactions such as RKKY and a correlatedKondo term.17,19\nUsing the same idea as above, we perform a diﬀerent\nsecond SW transformation using generator operator SB\n2\nH′′=eSB\n2H′e−SB\n2 (6)to eliminate the second order interaction term\n1\n2[S,[S,HB]] that corresponds to the rotat-\ning magnetic ﬁeld in Eq. (5) ,which requires/bracketleftbig\nSB\n2,H0/bracketrightbig\n=−1\n2[S,[S,HB]],/parenleftBig\nSB\n2∝T2\ni,k/parenrightBig\n.The pur-\npose of this transformation is to extract the interactions\ninduced by the external rotating magnetic ﬁeld. The\ngenerator operator SB\n2fulﬁlling the above condition in\nthe rotating frame of reference with frequency ωis\nSB\n2=/planckover2pi1ω⊥\n2/summationdisplay\ni,k,qPi,k,q\nǫk−ǫq\n×/bracketleftBig\nnk,q/parenleftbig\nS+\ni+S−\ni/parenrightbig\n−ni/parenleftBig\nS+\nk,q+S−\nk,q/parenrightBig/bracketrightBig\n.(7)\nwithS+(−)\nk,q=c↑(↓)†\nkc↓(↑)\nq,S+(−)\ni=d↑(↓)†\nid↓(↑)\ni, nk,q=/summationtext\nσcσ†\nkcσ\nqand\nPi,k,q=T∗\ni,kTi,q/bracketleftbigg1\n(Eki−Ui)(Eqi−Ui)−1\nEkiEqi/bracketrightbigg\n.(8)\nNote that for small Zeeman ﬁeld the expressions are sim-\npliﬁed by assuming that the band energy levels are not\nspin-dependent εσ\ni=εi.After the transformation, the\neﬀective Hamiltonian becomes\nH′′=H0+HB+1\n2[S,Ht]\n+1\n2/bracketleftbig\nSB\n2,[S,Ht]/bracketrightbig\n+1\n4/bracketleftbig\nSB\n2,[S,[S,HB]]/bracketrightbig\n.(9)\nThe aim of SW transformations is to eliminate perturba-\ntively ”old” terms from Hamiltonians in favor of ”novel”\ninteractions. The ﬁrst SW transformation in Eq. (5)\ngives rise to the second order (in tunneling rates) pro-\ncesses, including the standard Kondo interaction, and an\nanalog of the spin pumping and spin torque interaction\nbetween the QDs and the continuum. The second SW\ntransformation in Eq. (9) gives rise to the fourth order\nprocesses resulting from the rotating magnetic ﬁeld. We\ndiscuss the second order and forth order processes below.\nIn addition to the standard Kondo Hamiltonian, the\nﬁrst SW transformation yields the spin pumping induced\nby the rotating ﬁeld and spin torque due to the absorp-\ntion of the pumped spins (the last term in Eq. (5)). In\nthe rotating frame\n1\n2[S,[S,HB]] =/planckover2pi1ω⊥\n2/summationdisplay\ni,k,qPi,k,qei(k−q)·Ri(10)\n×/bracketleftBig\nni/parenleftBig\nS+\nk,q+S−\nk,q/parenrightBig\n−nk,q/parenleftbig\nS+\ni+S−\ni/parenrightbig/bracketrightBig\n.\nThe ﬁrst term in Eq. (10) is responsible for cotunnel-\ning induced spin pumping from QD to the continuum.\nThe second term gives the torque acting the QD by the\npumped spins in the continuum. The physics of the pre-\ncession induced spin pumping from the QD is depicted\nin Figure 1. Due to the cotunneling processes, one spin- ↓\nelectron with momentum qand energy ǫqin the contin-\nuum tunnels into the dot and occupies energy level ε↓,3\nfollowed by another spin- ↑electron on ε↑tunneling out\nto the continuum with momentum kand energy ǫk.Af-\nterwards, due to the rotating ﬁeld the electron in the\nQD absorbs a ”photon” and transists to spin- ↑.With\nthese Kondo-type cotunneling processes, the spin in QD\nremains unchanged, but one spin down is ﬂipped to spin\nup in the continuum via QD. Such processes keep trans-\nferring electrons in the continuum from spin- ↓to spin-↑.\nThe reverse process also happens if a spin- ↓electron re-\nsides in the QD (not shown in the ﬁgure): spin- ↑elec-\ntron tunnels into the QD followed by the spin- ↓electron\ntunneling out to the continuum, emits a ”photon” and\nﬂips to spin- ↓.The average outcome is that the elec-\ntrons from continuum ﬂows toward the scattering region,\nchange their spins and ﬂow away from it. It should be\nnoted, that as the average number of pumped electrons\nwith opposite spins are equal and thus no spin or charge\ncurrent is induced. But we are, in the end, interested in\nfourth order processes, when spins pumped from one QD\ncan be absorbed by the second one and thus mediate new\ncoupling mechanism between two dynamic QDs.\nAfter two-step SW transformation, the Hamiltonian in\nEq. (9) contains two fourth order terms, i.e.the last two\nterms. The explicit form of fourth term is:\n1\n2/bracketleftbig\nSB\n2,[S,Ht]/bracketrightbig\n=/planckover2pi1ω⊥J1\n2/bracketleftbig\nn1/parenleftbig\nS+\n2+S−\n2/parenrightbig\n+n2/parenleftbig\nS+\n1+S−\n1/parenrightbig/bracketrightbig\n,\n(11)\nwith\nJ1=1\n4/summationdisplay\nk,q/summationdisplay\n/angbracketlefti,j/angbracketrightnk−nq\nǫk−ǫqPi,k,qJj,q,kei(k−q)·Ri,j,\nJi,k,q=T∗\ni,kTi,q/parenleftbigg1\nEki+1\nEqi−1\nEki−Ui−1\nEqi−Ui/parenrightbigg\n.\n(12)\nandRi,j=Ri−Rj. The Eq. (11) represents a new\neﬀective coupling of QDs induced by the spin ﬂip scat-\ntering of electrons. The physics of new coupling can be\nunderstood as follows: the conduction electron tunnels\ninto one QD, ﬂips its spin, and then tunnels out back\ninto the continuum. Afterwards the ﬂipped spin ﬂows\ntoward the other QD and exchanges with the electron\nspin in it. The exchange of electron spins between two\nQDs is via the RKKY-type coupling where the energy of\nthe intermediate excitation is given by ǫq−ǫk. It can be\n(1) \nǫq ǫk\nε↑(2) \nǫq ǫk\nε↑\nε↓(3) \nε↓(4) \nFIG. 1. (Color online) Schematic plot of the cotunneling in-\nduced spin pumping. The arrows correspond to the electrons\nwith spin up and down.seen from Eq. (11) that spin-ﬂip scattering induced cou-\npling reduces to an eﬀective magnetic ﬁeld from adjacent\nquantum dot.\nThe last term of Eq. (9)\n1\n4/bracketleftbig\nSB\n2,[S,[S,HB]]/bracketrightbig\n=(/planckover2pi1ω⊥)2J2\n4\n×/parenleftbig\nn1n2+S+\n1S+\n2+S−\n1S+\n2+S+\n1S−\n2+S−\n1S−\n2/parenrightbig\n,(13)\nwith\nJ2≡1\n2/summationdisplay\nk,q/summationdisplay\n/angbracketlefti,j/angbracketrightnk−nq\nǫk−ǫqPi,k,qPj,q,kei(k−q)·Ri,j.\nThe Hamiltonian Eq. (13) describes the processes when\nelectrons after spin-ﬂipping in one QD tunnel into the\nsecond one and ﬂip again.\nSince we are interested in the subspace of single occu-\npancyoftheQDs( ni= 1)wecanextracttheinteractions\nthat survive with Gutzwiller projection. The eﬀective\nHamiltonian in the laboratory frame becomes\nH′′\nB(t) =/summationdisplay\ni/planckover2pi1ω/bardblSi,z+/summationdisplay\nigµB(2+J1)B⊥(t)·Si\n+(gµB)2J2[B⊥(t)·S1][B⊥(t)·S2].(14)\nThis eﬀective coupling between QDs due to the rotat-\ning magnetic ﬁeld is the main result of this paper. The\nJ1part in the second term in Eq. (14) represents that\nthe spin S1of th QD-1 feels the magnetic ﬁeld acting\nonS2on QD-2, and vice versa. While the third term of\nEq. (14) represents the Ising-type coupling between the\nspins on QD-1 and QD-2 induced by the rotating mag-\nnetic ﬁeld B⊥(t), and it magnitude is tunable by chang-\ning the magnitude of B⊥(t). We can perform a unitary\ntransformation ˆU(t) =e−i/summationtext\niωSi,zthat removesthe time\ndependence:\nH′′\nB=ˆU(t)†(H′′\nB(t)−i/planckover2pi1∂t)ˆU(t) =/summationdisplay\ni/planckover2pi1/parenleftbig\nω/bardbl−ω/parenrightbig\nSi,z\n+/summationdisplay\ni/planckover2pi1ω⊥(2+J1)Si,x+(/planckover2pi1ω⊥)2J2S1,xS2,x.(15)\nWhen the resonance condition/parenleftbig\nω/bardbl=ω/parenrightbig\nis fulﬁlled,\nEq. (15) becomes\nH′′\nB=/summationdisplay\ni/planckover2pi1ω⊥(2+J1)Si,x+(/planckover2pi1ω⊥)2J2S1,xS2,x,(16)\nwhere the ﬁrst is the sum of Eq. (11) and the external\ntransverse magnetic ﬁeld HB=/summationtext\ni2/planckover2pi1ω⊥Si,xin rotating\nreference frame and the second term corresponds to Eq.\n(13) with Si,x=1\n2/parenleftbig\nS+\ni+S−\ni/parenrightbig\n.\nIn Ref. 28 Coqblin and Schrieﬀer presented their\nwidely used approach29–32to the two-impurity An-\nderson model. After a single SW transformation\nand treating that Hamiltonian in second order pertur-\nbation theory, they compute a RKKY-like spin-spin4\nStateWave function Energy\n|1/angbracketrightΦ++Ψ+J0+2/planckover2pi1ω⊥(2+J1)+(/planckover2pi1ω⊥)2J2\n|2/angbracketrightΦ+−Ψ+J0−2/planckover2pi1ω⊥(2+J1)+(/planckover2pi1ω⊥)2J2\n|3/angbracketright Φ−J0−(/planckover2pi1ω⊥)2J2\n|4/angbracketright Ψ−−3J0−(/planckover2pi1ω⊥)2J2\nTABLE I. Eigen-energies and eigen-states for Heﬀin Eq. (17).\n/VertBar11/RAngleBracket1\n/VertBar12/RAngleBracket1\n/VertBar13/RAngleBracket1\n/VertBar14/RAngleBracket1\n0/LParen12/PlusJ1/RParen1/Slash1J2/Minus3J0J0\n/HBarΩ/UpTeeE\nFIG. 2. (Color online) The eigenenergies of the eigenstates\n|1/angbracketright,|2/angbracketright,|3/angbracketrightand|4/angbracketrightas afunctionofthemagnitudeofthetrans-\nverse magnetic ﬁeld in units of J0.\ninteraction2of the form J0S1·S2with coupling con-\nstantJ0=/summationtext\nk,qJ1,k,qJ2,q,knk−nq\nǫk−ǫq.We evaluate /planckover2pi1ω⊥J1\nand (/planckover2pi1ω⊥)2J2in Eq. (14) comparing them with RKKY\ncoupling constant. For simplicity we assume identical\nQDs (ε1=ε2≡ε, U1=U2≡U).In symmetric Kondo\nregime the typical energy of QDs ( ε= 0) is|ǫk,q| ≡ǫ=\nU/2≃0.1 meV.33By setting the rotating ﬁeld strength\nB⊥= 1 T and g= 2,we have from Eqs. (8) and\n(12)|/planckover2pi1ω⊥J1|/J0≃/planckover2pi1ω⊥/4ǫ= 0.28 and ( /planckover2pi1ω⊥)2J2/J0≃\n(/planckover2pi1ω⊥)2/4ǫ2= 0.32. This means the magnitude of the\nnew coupling terms in Eq. (16) is comparable to the\nRKKY coupling.\nWe study the eﬀective Hamiltonian (Eq. 14) in the ro-\ntating reference frame in the presence of RKKY coupling\nJ0S1·S2, which always exists in the system. The total\neﬀective Hamiltonian becomes\nHeﬀ=H′′\nB+J0S1·S2. (17)\nUsing the four Bell states: Φ±=1√\n2(|↓↓/angbracketright±|↑↑/angbracketright ) andΨ±=1√\n2(|↓↑/angbracketright±|↑↓/angbracketright ) as basis, the eigen-energies and\nthe eigen-states are listed in Table I. In Figure 2, we\nplot the eigenenergies of the eﬀective Hamiltonian Heﬀ\nas a function of the magnitude of the transverse mag-\nnetic ﬁeld B⊥=/planckover2pi1ω⊥/gµB. It is seen that in the limit\nof no transverse ﬁeld, we have singlet and (threefold de-\ngenerate) triplet states. The transverse magnetic ﬁeld\nremoves the degeneracy of the triplet states and enables\nto construct a four level system in a controllable manner,\nin which two of the states ( |2/angbracketrightand|3/angbracketright) swap positions at\n/planckover2pi1ω⊥= (2+J1)/J2. This feature canbe used forquantum\ncomputation in desirable set of maximally entangled Bell\nstates. Finally, the transverse magnetic ﬁeld dependence\nof the eigen energies in Figure 2 can be measured in ex-\nperiment, and the coupling strengths J1andJ2can be\ninferred from the slope (at zero ﬁeld) and curvature of\nthe lines, respectively.\nIn conclusion, using a 2-stage SW transformation, we\ntransform a two-impurity Anderson model into an ef-\nfective spin Hamiltonian. Without the rotating mag-\nnetic ﬁeld the second orderexpansionyields the standard\nKondo Hamiltonian for two impurities with additional\nscattering terms.17–19,23,26The introduction of the rotat-\ning magnetic ﬁeld gives rise to a magnetic ﬁeld-induced\nspin pumping from QD and the torque that experiences\nthe QD from the continuum via the cotunneling pro-\ncesses. These cotunneling processes yield to two addi-\ntional QD coupling mechanisms: 1) the QD feels a non-\nlocal magnetic ﬁeld acting on the neighboring QD; 2) the\nQDs are coupled via an Ising-like coupling. Because of\nits dynamical origin from the rotating ﬁeld, the new cou-\npling mechanism is intrinsically diﬀerent from all exist-\ning static coupling mechanisms such as RKKY coupling.\nMore importantly, the strength of the new dynamical\ncoupling is tunable via the magnitude of the transverse\nmagneticﬁeld, andfoundtobe comparabletothe RKKY\ncoupling strength at reasonably large rotating magnetic\nﬁeld. 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Vignale\nDepartment of Physics and Astronomy, University of Missour i, Columbia, Missouri 65211, USA\n(Dated: June 25, 2018)\nWe show that an electric ﬁeld parallel to the wavefronts of an electron-hole grating in a GaAs\nquantum well generates, via the electronic spin Hall eﬀect, a spin grating of the same wave vector\nand with an amplitude that can exceed 1% of the amplitude of th e initial density grating. We refer\nto this phenomenon as “collective spin Hall eﬀect”. A detail ed study of the coupled-spin charge\ndynamics for quantum wells grown in diﬀerent directions rev eals rich features in the time evolution\nof the induced spin density, including the possibility of ge nerating a helical spin grating.\nThe spin Hall eﬀect (SHE), i.e., the generation of a\ntransverse spin current from a charge current and vice\nversa,hasattractedmuchattentioninthepastdecade[1–\n10], and has now become one of the standard tools for\nthe generationand detection ofspin currentsin magneto-\nelectronic devices [11–15]. Theoretically, both intrinsic\nand extrinsic mechanisms have been shown to contribute\nto the SHE in semiconductors. While the intrinsic mech-\nanism originates from the spin-orbit coupling (SOC) in\nthe band structure [4, 5], the extrinsic one results from\nthe SOC with impurities [1–3]. Experimentally, the ﬁrst\nevidence of SHE in semiconductors was the observation\nof a spin accumulation at the edges of n-doped GaAs [6].\nThis is clearly a single-particle eﬀect taking place in a\nmacroscopically homogeneous sample. Recently, Ander-\nson et al. [16] have proposed an interesting collective\nmanifestation of the SHE in a periodically modulated\nelectron gas. They suggested that an optically induced\nspin density wave (transient spin grating [17–19]) in a\ntwo-dimensionalelectrongascouldbepartiallyconverted\ninto a density wave when an electric ﬁeld perpendicular\nto the grating wave vector is applied.\nThere are some diﬃculties with the implementation of\nthis idea. First of all, the electric ﬁeld due to the in-\nduced charge density, when properly taken into account,\neﬀectively prevents the accumulation of charge. Second,\nthe SOC considered in that work comes solely from band\nstructure (i.e., it is purely “intrinsic”) and, for this rea-\nson, the spin-chargecouplingis found to be ofthird order\nin the, presumably small, strength of the SOC.\nIn the present work, we re-examine the coupled spin-\ndensity transport in a periodically modulated electron\ngasin a novel set-up which is free ofthe above-mentioned\ndiﬃculties. Diﬀering from Ref.16, we start from an elec-\ntrically neutral electron-hole grating (uniform spin den-\nsity) in an n-type semiconductor quantum well and show\nthat an electric ﬁeld parallel to the wavefronts of the\ngratinggenerates, viaSHE,a periodicspin modulationof\nthe same wave vector as the initial electron-hole grating\n(see Fig.1). Since any local charge imbalance is screened\nquickly by the background electrons, we can safely as-\nsume that the system remains charge-neutralthroughout\nitsevolutionand,inparticular,noadditionalelectricﬁeld\nis created. Furthermore, going beyond the treatment ofinitial e-h grating\ninduced spin grating (Sz)E\nxyinitial e-h grating\ninduced spin grating (Sz)E\nxy\nFIG. 1: (Color online) Collective spin Hall density proﬁle\ninduced by a transverse electric ﬁeld ( E) in a periodically\nmodulated electron/hole gas.\nRef.16, we include not only the intrinsic but also the ex-\ntrinsic SHE. We conﬁrm that the spin-density coupling\ndue to the intrinsic SHE is an eﬀect of third-order in\nthe SOC strength [20], which is consistent with previous\nworks [21–24]. However, we also ﬁnd that the dominant\nextrinsic spin Hall mechanism, skew scattering [25], leads\nto an inhomogenous spin-charge coupling that is of ﬁrst\norderin the SOC strength.\nIn the light of this analysis, the chances for the ob-\nservation of the collective SHE appear much better than\npreviously thought. For electron-hole density gratings\ninduced by optical means in an n-type GaAs quantum\nwell, we predict the amplitude of the induced spin grat-\ning to be larger than 1% of the amplitude of the original\ngrating at an electric ﬁeld of ∼105V/m. These num-\nbers are within the reach of contemporary experimental\ntechniques [17–19].\nTheoretical framework – Up to the linear order in mo-\nmentum, the Hamiltonian of electrons in an n-GaAs QW\ncan be written as\nH0=k2\n2me+1\nmk·A, (1)\nwhereA=m(λ1σy+γyσz,λ2σx−γxσz) is the spin-\ndependent vector potential that describes SOC. Speciﬁ-\ncally, ifαandβdenote the Rashba [26, 27] and Dressel-\nhaus coeﬃcients [28], we have λ1=β+α,λ2=β−α2\nandγi=λ2\ne\n4eEiin a (001) QW (the xandyaxes are\nin the [110] and [ ¯110] directions, forming 45◦angles with\nthe cubic axes), while λ1=α,λ2=−α,γx=λ2\ne\n4eEx\nandγy=λ2\ne\n4eEy−βin a (110) QW . The terms con-\ntaining the eﬀective Compton wavelength λe(∼4.6˚A in\nGaAs) describe the SOC from the applied electric ﬁeld.\nThe Hamiltonian for the (heavy) holes has a similar form\nwith a diﬀerent eﬀective mass mh. In this case, however,\nwe assume that the spin polarization is quenched, due\nto strong spin-orbit interaction in the valence band, on\na time scale that is shorter than that of the diﬀusion\nprocess. For this reason, no spin-dependent terms are\nincluded for the holes.\nOur analysis is based on the quantum kinetic equation\nfor the density matrix ρk(r) of electrons [14, 22, 23]:\n∂tρk+1\n2{∇kH0,˜∇rρk}+i[H0,ρk] =∂tρk|scat,(2)\nwhere˜∇r=∇r+eE∂ǫk. In the relaxation time approx-\nimation, the scattering term on the right-hand side is\n∂tρk|scat=−ρk\nτ+ρk\nτ+1\n2mτ{k·A,∂ǫkρk}\n−1\n2αss/summationdisplay\n|k′|=|k|{k×k′·σ,ρk′},(3)\nwhereρk=/an}b∇acketle{tρk/an}b∇acket∇i}htis the momentum-space angular average\nof the density matrix. The last term on the right-hand\nside of Eq. (3) is the skew scattering term [29, 30] with\nthe coeﬃcient αss=/planckover2pi1\n8πmniλ2\ne/parenleftbigmui\n/planckover2pi12/parenrightbig3, whereniandui\nare the density and the scattering potential ofimpurities,\nrespectively [31]. The third term on the right hand side\nof Eq.(3), which eﬀectively amounts to shifting the ar-\ngument of ρkfromktok+A, is critically important to\nensure the vanishing of the spin-charge coupling to lin-\near order in SOC [32]. From Eq. (2) we derive coupled\nequations of motion for the inhomogeneous density and\nspin density of electrons and the density of holes. The\nspin density of the holes is assumed to be zero.\n(110) quantum well – For orientation, let us begin\nwith the simplest case, namely a symmetric (110) GaAs\nQW. What makes this system most interesting from\nour perspective is that the Dresselhaus eﬀective mag-\nnetic is along z-direction, and therefore preserves the z-\ncomponent of the electron spin, Sz. The intrinsic SHE is\ncompletely absent. The extrinsic SHE, embodied in the\nskew-scatteringterm, is present and clearlyconserves Sz.\nTherefore, we can write down separate kinetic equations\nfor spin-up and spin-down electrons:\n∂tnσk+1\nmk·∇rnσk+eE·∇knσk\n=−nσk−nσk\nτ−σαss/summationdisplay\n|k′|=|k|(k×k′·ˆz)nσk′,(4)Sz/A0 (10-2)\n(a) 0  1  2  3\nt (ns)-0.5-0.25 0 0.25 0.5x (L)\n-1.5-1-0.5 0 0.5 1 1.5\n∆N/A0\n(b) 0  1  2  3\nt (ns)-0.5-0.25 0 0.25 0.5x (L)\n-1-0.5 0 0.5 1\nFIG. 2: (Color online) Time evolution of the electric-ﬁeld-\ninduced spin grating for electrons (a) (normalized by ampli -\ntude of the initial spin grating) as function of position (no r-\nmalized by the wave length) with q= 0.3µm−1in sym-\nmetric (110) QWs. (b) Time evolution of the e-h density\ngrating. Here, we take E= 1 kV/cm, Da= 20 cm2/s,\nDs= 200 cm2/s,τ= 1 ps and Γ = 1 ns−1.\nwithσ= +,−representing spin up and down with re-\nspect toz-direction, respectively. Following the standard\nprocedure we substitute the “ﬁrst-order solution”\nnσk≈¯nσk−σαss/summationdisplay\n|k′|=|k|(k×k′·ˆz)¯nσk′,(5)\nwhere ¯nσk≡/parenleftbig\n1−τ\nmk·∇r−eτE·∇k/parenrightbig\nnσk, into Eq.(4),\nand sum over kto obtain the diﬀusion equation\n∂tnσ−D∇2\nrnσ+vd·∇rnσ−σvss·∇rnσ= 0,(6)\nwherenσ=/summationtext\nknσkis the total density of electron with\nspinσandD=/an}b∇acketle{tk2\n2m2τ/an}b∇acket∇i}htis the diﬀusion constant. The\ndrift velocity and spin-Hall drift velocity are given by\nvd=τeE\nmandvss= 2αssτeDm(E×ˆz), respectively. We\nthen combine the two equations of diﬀerent spins and\nget coupled kinetic equations for the total density ( N=\nn++n−) and the total spin polarization ( Sz=n+−n−):\n(∂t−D∇2\nr+vd·∇r)N−vss·∇rSz= 0,(7)\n(∂t−D∇2\nr+vd·∇r)Sz−vss·∇rN= 0.(8)\nNotice the appearance of a spin-density coupling, which\noccurs only in a non-uniform system and is proportional\nto the skew-scattering drift velocity – a quantity of ﬁrst\norder in the SOC strength. The equation for the hole\ndensity is similar to Eq.(7), with Dandvdreplaced by3\nthe corresponding quantities for the holes, but without\nthe last term, because the spin polarization of the holes\nis neglected. In fact, the last term can also be neglected\non the left-hand side of Eq.(7) for the electrons, since it\nleads to minute corrections to the evolution of the den-\nsity. By imposing the local neutrality condition, that is,\nassuming that the electron density is always equal to the\nhole density, we combine the diﬀusion equations for elec-\ntrons and holes into ambipolar diﬀusion and spin-density\ntransport equations\n(∂t−Da∇2\nr+Γ)N= 0, (9)\n(∂t−Ds∇2\nr)Sz−vss·∇rN= 0,(10)\nwhereDaandDsrepresent the ambipolar and spin dif-\nfusion constants, respectively. Here, we have introduced\nthe rate Γ of electron-hole recombination. The solution\nof these equations is\n∆N=A0cos(qx)e−(Daq2+Γ)t, (11)\nSz=−A0sin(qx)vssq\n(Ds−Da)q2−Γ[e−(Daq2+Γ)t−e−Dsq2t].(12)\nIn Fig.2, we plot the time evolution of the induced-spin\ngrating as well as the density grating. One can see that\nthe amplitude of the spin grating initially increases and\nthen begins to decrease after a maximum around 1% the\namplitudeoftheinitialdensitygrating. Theinducedspin\ngrating shows aπ\n2phase shift from the density grating.\nFrom Eq.(12), we see that, for a given q,Szreaches the\nmaximal value\nAmax\nSz(q)\nA0=vssq\nDaq2+Γ/parenleftbiggDsq2\nDaq2+Γ/parenrightbiggDsq2\nDaq2+Γ−Dsq2\n,(13)att= (Daq2+Γ−Dsq2)−1ln/parenleftig\nDaq2+Γ\nDsq2/parenrightig\n. Noting that\nthe quantity within the round brackets is of order 1, we\nsee that the amplitude ratio is roughly the fraction of\nthe grating wavelength covered by an electron that trav-\nels at the skew-scattering drift velocity ( vss) during the\ndiﬀusion lifetime of the grating (1 /Daq2). Not surpris-\ningly, this ratio shows a non-monotonic dependence on\nq, reaching a maximumAmax\nSz(qopt)\nA0∼1.4×10−2at the\noptimal wave vector qopt∼0.2µm−1, with the material\nparameters listed in the caption of Fig.2.\n(001) quantum well – In a (001) QW, the presence\nof the in-plane eﬀective magnetic ﬁeld due to band SOC\nand the non-conservation of Szlead to more complex\nscenarios. To begin with, the coupling of longitudinal\nand transverse spin ﬂuctuations leads to a set of drift-\ndiﬀusion equations of the form\n∂t(∆N,Sx,Sy,Sz)T=−D(˜q)(∆N,Sx,Sy,Sz)T,(14)\nwhereD(˜q)isthe4×4drift-diﬀusionmatrixactingonthe\ncolumn vector of the Fourier amplitudes of the density at\nwave vector q. Here˜q:≡q−ieE∂ǫkis a momentum-\nspace operator, which takes into account drift under the\naction of the electric ﬁeld E. Without going into tech-\nnical details we only summarize the salient results (for\ndetails, see Ref. 34). Taking q=qˆ xandE=Eˆ yand\nassumingkFq\nm≪1 and|α±β|kF≪EF(conditions that\ndeﬁne the diﬀusive regime) we ﬁnd\nD(˜q) =\nDq2−1\n2τλ2q2vd−4iτDλ1˜λ2\n2q+iτ˜λ2γyqvd\n−iqvssD˜λ2/vd−4iτD˜λ2\n2γyq−iτλ1˜λ2qvd\n−iqvss\n−1\n2τλ2q2vd+4τλ2(˜λ2\n1+ ˜γ2\ny)vd\n+2˜λ1vssDq2+1\nτsx−4iD˜γyq 4iD˜λ1q\n−4iτDλ1˜λ2\n2q+iτ˜λ2γyqvd\n−iqvssD˜λ2/vd4iD˜γyq Dq2+1\nτsy−4D˜λ1˜γy+2˜λ2vd\n−4iτD˜λ2\n2γyq−iτλ1˜λ2qvd\n−iqvss−4iD˜λ1q−4D˜λ1˜γy−2˜λ2vd Dq2+1\nτsz\n\n(15)\nwherevd=τeE\nm. Here, ˜λi=mλi, ˜γi=mγi,1\nτsx=\n4D(˜λ2\n1+˜γ2\ny),1\nτsy= 4D(˜λ2\n2+˜γ2\ny) and1\nτsz= 4D(˜λ2\n1+˜λ2\n2).\nWe note that our diﬀusion matrix diﬀers from the\none reported in Ref.16 in two ways: (i) in addition to\nthe “standard” terms linear in ˜qand cubic in the SOC\nstrength, we include terms of second order in both ˜qand\nthe SOC strength as well as terms of third order in ˜qand\nﬁrst order in SOC. All these terms can be of comparablemagnitude in real systems. (ii) At variance with Ref.16,\nour diﬀusion matrix is non-symmetric: Di1/ne}ationslash=D1i. This\nlack of symmetry comes from a careful consideration of\nthe operatorial character of ˜ q, whereby ˜ qǫk/ne}ationslash=ǫk˜ q, as ex-\nplained in the supplemental material [34]. Eqs. (14-15)\nare our main theoretical result: they combine extrinsic\nand intrinsic contributions to the SHE as well as spin\nprecession, and reduce to the results of the previous sec-4\ntion if the intrinsic SOCs appropriate for (110) QW are\nused.\n(001) quantum well with balanced SOC – The case of\na (001) QWs with identical Dresselhaus and Rashba co-\neﬃcients, α=β(corresponding to the condition λ2= 0)\nwithqoriented along the [110] direction gives us the op-\nportunity to demonstrate a particularly interesting ap-\nplication of Eqs. (14-15). Just as in a symmetric (110)\nQW, only the skew scattering contributes to the collec-\ntive SHE, but now Szis not conserved. Since γyis negli-\ngibly small (two orders smaller than the band SOC), the\nSycomponent decouples from the SxandSzcomponents\nand the diﬀusion matrix reduces to\nD(q) =\nDq20 0 −iqvss\nq0vssD(q2+q2\n0) 0 2 iDqq0\n0 0 Dq20\n−iqvss−2iDqq00D(q2+q2\n0)\n,\n(16)\nwithq0=4mβ\n/planckover2pi12≃3.5µm−1. After imposing the charge-\nneutrality condition, the diﬀusion equations for the den-\nsity and the two helical components of the spin density\nS±=1√\n2(Sx±iSz) are found to be\n∂t∆N=−Daq2∆N−1√\n2qvssS−+1√\n2qvssS+,(17)\n∂tS−=1√\n2(q+q0)vssN−Ds(q+q0)2S−,(18)\n∂tS+=−1√\n2(q−q0)vssN−Ds(q−q0)2S+.(19)\nAs in the previous calculations, we neglect the last two\nterms on the right-hand side of Eq.(17). Then the solu-\ntion for the density reduces to a simple diﬀusion process,\nand the solution for the two helical modes is given by\nS∓=±1√\n2q±A0eiqxvss\nDsq2\n±−Daq2[e−(Daq2−Dsq2\n±)t−1]e−Dsq2\n±t,(20)\nwhich yields the spin-polarization\nSx=/summationdisplay\n±±1\n2q±A0cos(qx)vss\nDsq2\n±−Daq2[e−Daq2t−e−Dsq2\n±t],(21)\nSz=−/summationdisplay\n±1\n2q±A0sin(qx)vss\nDsq2\n±−Daq2[e−Daq2t−e−Dsq2\n±t].(22)\nThese amplitudes show a strong dependence on the wave\nvector. One can see that the contributions from the S−\nmode is proportional to q+=q+q0while the contribu-\ntion from the S+mode is proportional to q−=q−q0\n(the correspondence is reversed if we switch the sign of\nβ). Further, the S+mode is long-lived, due to the slowly\ndecaying term e−Dsq2\n−t, while the S−mode is short-\nlived [19, 23, 35, 36, 39]. The long-time behavior of Sz,\nbeing dominated by the S+component, is positive for\nq > q0and negative for q < q0. In the special case q=q0\n– a practically realizable case – S+vanishes identically,\nand the amplitude of Szdecays to zero most rapidly. In\nthis case, the skew scattering converts the initial den-\nsity grating into a helical wave of wave vector q0! Fur-\nther interpretation of this intriguing eﬀect, based on an-1.5-1-0.5 0 0.5 1 1.5\n 0  50  100  150  200Sz(-0.25L)/A0 (10-3)\nt (ps)q/q0 = 1.2\n1.5\n0.9\n0.21.0\n0.5\nFIG. 3: (Color online) Time evolution of the spin component,\nSz, from the density grating with diﬀerent values of q/q0in\nthe case of α=β,q0=4mβ\n/planckover2pi12. In the calculation, we take the\nDresselhaus coeﬃcient β= 10 meV ˚A(corresponding to 10 nm\nGaAs QW) and q0≃3.5µm−1. Other parameters are taken\nto be the same as Fig.2.\nSU(2) gauge transformation that eliminates the intrinsic\nSOC [23, 39], is given in Ref. 34.\nIn Fig. 3, we plot the amplitude of the z-component\nof the electron spin, Sz, as function of time at a distance\nx=−0.25Lfrom a peak of the density grating. Notice\nthe reversal of sign of the long-time behavior and the\nquick decay of the signal at q=q0, due to the vanishing\nof theS+mode.\nIn summary, we have studied the collective spin Hall\neﬀectinaperiodicallymodulatedelectrongasinthepres-\nenceofanin-planeelectricﬁeldperpendiculartothewave\nvector of the initial density modulation. In the symmet-\nric (110) quantum well the amplitude of the induced spin\ndensity is controlled solely by skew scattering and can\nbe as large as 1% of that of the initial density modula-\ntion. This shouldbe observablein state-of-theartexperi-\nments [17–19]. 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Sherman, arXiv:1302.2121.arXiv:1306.0889v1  [cond-mat.mes-hall]  4 Jun 2013Supplemental material ﬁle for “Collective spin-Hall eﬀect for electron-hole gratings”\nKa Shen and G. Vignale\nDepartment of Physics and Astronomy, University of Missour i, Columbia, Missouri 65211, USA\nThis material includes the detailed derivation of the\ndrift-diﬀusion equation in (001) GaAs [Eqs. (14) and\n(15) of main text] and further discussions ofits structure,\nin particular the asymmetry of the diﬀusion matrix. We\nalso supply the details of the SU(2) transformation of\nthe skew scattering for the case of identical Rashba and\nDresselhaus coupling strengths.\n1. DERIVATION OF DRIFT-DIFFUSION\nEQUATION IN (001) GAAS QUANTUM WELLS\nBy deﬁning the xandyaxes in the [110] and [ ¯110]\ndirection separately, we express the Hamiltonian of elec-\ntron gas in (001) GaAs QWs as\nH0=k2\n2me+kx(λ1σy+γyσz)+ky(λ2σx−γxσz),(1)\nwhereλ1=β+αandλ2=β−αwithαandβbe-\ning the Rashba and Dresselhaus coeﬃcients. γi=λ2\ne\n4eEi\ndescribes the SOC due to the in-plane electric ﬁeld with\nλecorresponding to the eﬀective Compton wavelength.\nIn the relaxation time approximation, the kinetic equa-\ntion for the local electron density matrix ρk(r) is given\nby [1, 2]\n∂tρk+1\n2{∇kH0,˜∇rρk}+i[H0,ρk]\n=−ρk\nτ+ρk\nτ+1\n2mτ{k·A,∂ǫkρk}\n−1\n2αss/summationdisplay\n|k′|=|k|{k×k′·σ,ρk′}, (2)\nwhere˜∇r=∇r+eE∂ǫk. The last term on the right-hand\nside is the skew scattering term from the second-order\nBorn approximation [3, 4]. With Fourier transform with\nrespect to tandr, we rewrite Eq.(2) as [5]\n(I+Kk)\ng0\nk\ngx\nk\ngy\nk\ngz\nk\n= (I+Tk)\ng0\nk\ngx\nk\ngy\nk\ngz\nk\n+Mk,k′\ng0\nk′\ngx\nk′\ngy\nk′\ngz\nk′\n,\n(3)\nwithρk=/summationtext\nigi\nkσiandρk=/summationtext\nigi\nkσi. HereIrepresents\nthe 4×4 identical matrix, and TandKare given by\nTk=\n0−Bx∂ǫk−By∂ǫk−Bz∂ǫk\n−Bx∂ǫk0 0 0\n−By∂ǫk0 0 0\n−Bz∂ǫk0 0 0\n,(4)and\nKk=\nΩ iλ2τ˜qyiλ1τ˜qxiτ(γy˜qx−γx˜qy)\niλ2τ˜qy Ω−2Bzτ2Byτ\niλ1τ˜qx2BzτΩ −2Bxτ\niτ(γy˜qx−γx˜qy)−2Byτ2Bxτ Ω\n,\n(5)\nrespectively, where Bx=−kyλ2,By=−kxλ1,Bz=\n−(kxγy−kyγx), and Ω = −iωτ+iτ\nmk·˜q. The eﬀective\nwave vector is given by ˜q=q−ieE∂ǫk. The matrix, M,\ndescribing skew scattering, is given by\nMk,k′=−αssτ\n0 0 0 kxk′\ny−kyk′\nx\n0 0 0 0\n0 0 0 0\nkxk′\ny−kyk′\nx0 0 0\n.\n(6)\nBymultiplying( I+Kk)−1toEq.(3)andsubstituting the\ngi\nk=/summationtext\nj[(I+Kk)−1(I+Tk)]ijgj\nkinto the skewscattering\nterm, we obtain\n[I−/an}b∇acketle{t(I+1\nI+Kk′Mk′,k)1\nI+Kk(I+Tk)/an}b∇acket∇i}ht]\ng0\nk\ngx\nk\ngy\nk\ngz\nk\n= 0.\n(7)\nHere,/an}b∇acketle{t.../an}b∇acket∇i}htrepresents the average over the angular depen-\ndence of the momentum. By further averaging over the\ndistribution function with respect to energy, retaining ω\nup to the ﬁrst order under the condition ωτ≪1, and\nFourier transforming back to the time domain we get the\nﬁnal diﬀusion equation in the form\n∂t(∆N,Sx,Sy,Sz)T=−D(˜q)(∆N,Sx,Sy,Sz)T.(8)\nwith\nD(˜q) =I −/an}b∇acketle{t(I+1\nI+Kk′Mk′,k)1\nI+Kk(I+Tk)/an}b∇acket∇i}ht|ω=0.\n(9)\nForkFq\nmand|α±β|kF≪EF, we do perturbation\nexpansion with respect to TandK. By substituting\n˜q:=qˆx−ieEˆy∂ǫk, we ﬁnally obtain the diﬀusion matrix,\nwritten up to the third order of K, (including Tk,Kk,\nKkTk,K2\nk,K2\nkTk,K3\nk,K3\nkTk,Kk′Mk′,kKk,Kk′Mk′,kTk):2\nD(˜q) =\nDq2−1\n2τλ2q2vd−4iτDλ1˜λ2\n2q+iτ˜λ2γyqvd\n−iqvssD˜λ2/vd−4iτD˜λ2\n2γyq−iτλ1˜λ2qvd\n−iqvss\n−1\n2τλ2q2vd+4τλ2(˜λ2\n1+ ˜γ2\ny)vd\n+2˜λ1vssDq2+1\nτsx−4iD˜γyq 4iD˜λ1q\n−4iτDλ1˜λ2\n2q+iτ˜λ2γyqvd\n−iqvssD˜λ2/vd4iD˜γyq Dq2+1\nτsy−4D˜λ1˜γy+2˜λ2vd\n−4iτD˜λ2\n2γyq−iτλ1˜λ2qvd\n−iqvss−4iD˜λ1q−4D˜λ1˜γy−2˜λ2vd Dq2+1\nτsz\n,\n(10)\nwithvd=τeE\nmandvss= 2αssτeDmE. Here,˜λi=mλi,\n˜γi=mγi,1\nτsx= 4D(˜λ2\n1+ ˜γ2\ny),1\nτsy= 4D(˜λ2\n2+ ˜γ2\ny) and\n1\nτsz= 4D(˜λ2\n1+˜λ2\n2).\n2. ASYMMETRY OF THE DIFFUSION MATRIX\nThe above diﬀusion matrix, Eq.(10) [i.e., Eq.(15) in\nmain text], clearly shows that the spin-charge coupling\nis non-symmetric, i.e., D1i/ne}ationslash=Di1. This comes from our\ncareful consideration of the order of the quantity ǫkand\nthe operator ∂ǫkin ˜qy, e.g.,\n/an}b∇acketle{t˜qyǫk/an}b∇acket∇i}ht=−i1\nNeE/summationdisplay\nk∂ǫk(ǫkρk) = 0,(11)\n/an}b∇acketle{tǫk˜qy/an}b∇acket∇i}ht=−i1\nNeE/summationdisplay\nkǫk∂ǫkρk=im\nτvd.(12)Notice that the asymmetry here is created by the exter-\nnal electric ﬁeld: no violation of Onsager’s reciprocity\nrelations is implied. If one erroneously assumed /an}b∇acketle{t˜qyǫk/an}b∇acket∇i}ht=\n/an}b∇acketle{tǫk˜qy/an}b∇acket∇i}ht /ne}ationslash= 0, as was apparently done in Ref.6, the diﬀu-\nsion matrix would be symmetric and both spin-density\nand density-spin couplings would diﬀer from zero in a\nhomogeneous system ( q= 0). To better understand the\nimplicationsofthisfact, observethatinthehomogeneous\ncase our diﬀusion equation reduces to\n∂t\nN\nSx\nSy\nSz\n=−\n0 0 0 0\n4τλ2(˜λ2\n1+ ˜γ2\ny)vd+2˜λ1vss1\nτsx0 0\n0 01\nτsy−4D˜λ1˜γy+2˜λ2vd\n0 0 −4D˜λ1˜γy−2˜λ2vd1\nτsz\n\nN\nSx\nSy\nSz\n.(13)\nObviously, the complete vanishing of the D1iguarantees\nthe conservation of the particle number (i.e., the q= 0\ncomponent of the particle density) – an exact physical\nconstraint. However, a non-zero D12arising from the in-\ncorrect treatment of the ordering of the operators would\nviolate this constraint when a uniform spin polarization\nis present. On the other hand, the non-zero matrix ele-\nmentD21in Eq.(13) predicts the generation of a uniform\nin-plane spin polarization by a steady electric current, in\nthe presence of SOC. This is a well-known eﬀect – the\nso-called Edelstein eﬀect [7, 8] – and has been observed\nin experiments [9]. For example, for a pure Rashba SOC\n(˜λ1=−˜λ2=mα) in the steady state we obtain, to linearorder in the electric ﬁeld,\nP=Sx\nN=ατ\nDvd−vss\n2Dαm. (14)\nThe ﬁrst term is solely due to Rashba SOC [7–9], while\nthe second term describes the correction due to the skew\nscattering [3, 4]. We should point out that in our deriva-\ntion so far the spin relaxation has been assumed to be\ndominated by the D’yakonov-Perel’ (DP) mechanism:\ntherefore, αshould be ﬁnite. In the limit α→0, dif-\nferent spin relaxation mechanisms, e.g., the Elliott-Yafet\n(EY) mechanism, become dominant. By replacing1\nτsby\n1\nτDPs+1\nτEYsin our theory, we see that the spin polarization\nvanishes in the α→0 limit as physically expected, and\nin agreement with previous work [4].3\n3. ANALYSIS OF THE BALANCED CASE: α=β\nTo better understand the generation of helical spin\nmodes by skew scattering in the case of balanced SOC\n(α=β) in a (001) GaAs quantum well we recall that in\nthisspecialcasetheSOCcanbecompletelyeliminatedby\nan SU(2) gauge transformation, which is actually a non-\nuniform rotation in spin space [2, 10]. The spin dynam-\nics in the rotated reference frame is simply determined\nby the competition of the spin-conserving drift-diﬀusion\nprocess and the gauge-transformed skew-scattering pro-\ncess. The gauge-transformed skew-scattering term has\nthe form\n∂t˜ρk|ss\nscat=−αss\n2/summationdisplay\n|k′|=|k|(k×k′)z{cos(q0x)σz,˜ρk′}\n+(k×k′)z{sin(q0x)σx,˜ρk′},(15)\nwhich suggests that the spin Hall velocity in the rotated\nframe has a sinusoidal variation in space, with the wave\nvectorq0=4mβ\n/planckover2pi12. More precisely, the transverse drift\nvelocity for the ˜Szcomponent is vz\nss(x) =vsscos(q0x),\nwhile that of the ˜Sxcomponent is vx\nss(x) =vsssin(q0x).\nSince the band SOC is completely gauged away, one has\nthe carrier conservation equation\nd˜ρz(x)\nσ(x,t)\ndt=∂t˜ρz(x)\nσ(x,t)−σ∂x[˜ρz(x)\nσ(x,t)vz(x)\nss(x)]≡0,\n(16)\nwhere the diﬀusion process has been neglected for sim-\nplicity. By substituting ˜ ρz(x)\nσ(x,t)≃1\n2A0cos(qx), we ob-\ntain the equations for ˜Szand˜Sxin the following form:\n∂t˜Sz(x) =−1\n2A0vss[q−sin(q−x)+q+sin(q+x)],\nand\n∂t˜Sx(x) =−1\n2A0vss[q−cos(q−x)−q+cos(q+x)].\nThe component with wave vector q−corresponds to the\nslowlydecayinghelicalmode S+, whoseamplitude is pro-portional to q−. Similarly, the component with wavevec-\ntorq+corresponds to the rapidly decaying helical mode\nS−, with amplitude proportional to q+. By transforming\nback to the original, unrotated spin space, we obtain\n∂tSx= cos(q0x)∂t˜Sx−sin(q0x)∂t˜Sz\n=/summationdisplay\n±±1\n2q±A0cos(qx)vss, (17)\n∂tSz= sin(q0x)∂t˜Sx+cos(q0x)∂t˜Sz\n=−/summationdisplay\n±1\n2q±A0sin(qx)vss, (18)\nwhichareconsistentwithEqs.(21-22)inmaintext. From\nthis analysis one can see that in the SU(2)-rotated frame\nthe skew scattering cannot create a uniform spin polar-\nization, which would correspond to the S+mode in the\noriginal frame. This is the reason for the vanishing am-\nplitude of the ˜S+mode at q=q0.\n[1] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin,\nPhys. Rev. Lett. 93, 226602 (2004).\n[2] B. A.Bernevig, J. Orensteinand S.C. Zhang, Phys. Lett.\nPhys.97, 236601 (2006).\n[3] J. L. Cheng and M. W. Wu, J. Phys.: Condens. Matter\n20, 085209 (2008).\n[4] R. Raimondi, P.Schwab, C. Gorini, andG. Vignale, Ann.\nPhys.524, 153 (2012).\n[5] X. Liu and J. Sinova, Phys. Rev. B 86, 174301 (2012).\n[6] B. Anderson, T. D. Stanescu, and V. Galitski, Phys. Rev.\nB81, 121304R (2010).\n[7] A. G. Aronov and Y. B. Lyanda-Geller, JETP Lett. 50,\n431 (1989).\n[8] V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n[9] R. H. Silsbee, J. Phys.: Condens. Matter 16, R179\n(2004).\n[10] I. V. Tokatly and E. Ya. Sherman, arXiv:1302.2121."    },    {        "title": "1903.01424v1.Spin_Phonon_Relaxation_in_Molecular_Qubits_from_First_Principles_Spin_Dynamics.pdf",        "content": "Spin-Phonon Relaxation in Molecular Qubits from First Principles Spin Dynamics\n1Alessandro Lunghi∗and1Stefano Sanvito\n1School of Physics, AMBER and CRANN, Trinity College, Dublin 2, Ireland\nThe coupling between electronic spins and lattice vibrations is fundamental for driv-\ning relaxation in magnetic materials. The debate over the nature of spin-phonon\ncoupling dates back to the 40’s, but the role of spin-spin, spin-orbit and hyperﬁne in-\nteractions, has never been fully established. Here we present a comprehensive study of\nthe spin dynamics of a crystal of Vanadyl-based molecular qubits by means of ﬁrst-order\nperturbation theory and ﬁrst-principles calculations. We quantitatively determine the\nrole of the Zeeman, hyperﬁne and electronic spin dipolar interactions in the direct\nmechanism of spin relaxation. We show that, in a high magnetic ﬁeld regime, the mod-\nulation of the Zeeman Hamiltonian by the intra-molecular components of the acoustic\nphonons dominates the relaxation mechanism. In low ﬁelds, hyperﬁne coupling takes\nover, with the role of spin-spin dipolar interaction remaining the less important for\nthe spin relaxation.\nSpin 1/2 systems represent the fundamental prototype\nof magnetic materials and the understanding of their\nproperties is pivotal for the rationalization of any more\ncomplex magnetic compound. Their study has deep roots\nin the early days of quantum mechanics and, despite\ntheir basic nature, they still represent a very rich quan-\ntum playground with non-trivial and elusive dynamical\nproperties. The spin life-time in insulating materials is\nessentially limited by the interaction between spins and\nlattice vibrations, namely the spin-phonon coupling. At\nﬁnite temperature, spins can absorb/emit one or multi-\nple phonons from/into the lattice and relax to an equilib-\nrium state. The detailed understanding of this process at\nthe ﬁrst-principles level represents a long-standing ques-\ntion in physics and chemistry and goes well beyond the\nfundamental-theory aspect.\nSpin-lattice relaxation is key in several ﬁelds. For in-\nstance, the eﬃciency of magnetic-resonance-imaging con-\ntrast agents [1] and the spin life-time of single molecule\nmagnets [2] is determined by the magnetization decay\nrate of paramagnetic elements. Turning to the main fo-\ncus of this work, in both molecular and solid-state qubits,\nspin-lattice relaxation sets the upper limit for the coher-\nence time [3–6] and the engineering of this interaction is a\nprimary challenge in the spin-based quantum computing\nﬁeld. The synthetic versatility of molecular compounds\noﬀers an intriguing route to the tailoring of spin-phonon\ncoupling but needs to be supported by a rational under-\nstanding of the physical principles governing spin relax-\nation.\nThe debate over the role of phonons in the relax-\nation of electronic spins can be traced back to the 40’s,\nwhen it was discussed for the ﬁrst time in the context of\ntransition-metal chemistry. Relaxation pathways occur-\nring through the modulation of spin dipolar interactions\nor the modulation of the d-electrons’ crystal ﬁeld were\nﬁrstly pointed out by Waller [7] and Van Vleck [8, 9],\n∗lunghia@tcd.ierespectively. More recently, the role of hyperﬁne inter-\naction has also been discussed [10]. Van Vleck’s mech-\nanism remains to date the most commonly accepted ex-\nplanation for the microscopic origin of spin-lattice re-\nlaxation of electronic spins. Several questions, however,\nremain unanswered. In particular, early models, being\nphenomenological, could not entirely address the nature\nof the coupling between phonons and spins and simply\nascribe acoustic vibrations as responsible for this interac-\ntion. Molecular compounds are inherently complex and a\nrational description of spin relaxation in terms of molec-\nular motions is still to be developed.\nFirst-principles calculations represent the perfect\nground to provide unbiased answers to such fundamental\nquestions. The possibility to predict the spin-relaxation\ntimes in magnetic materials without introducing any\nphenomenological parameter is also of fundamental im-\nportance from a materials-design perspective. Quan-\ntum chemistry and related disciplines have been already\nproved to be invaluable tools for providing insights into\nthe physics of new chemical systems [11] and they have\nbeen used as a screening method to predict new ma-\nterials with tailored properties [12]. These computa-\ntional strategies represent a rich opportunity for the ﬁeld\nof quantum informatics. The development of a ﬁrst-\nprinciple framework able to predict the spin-relaxation\ntime is the ﬁrst step in that direction. In this work we\noﬀer a theoretical and computational explanation of the\nnature of the atomistic processes that lead to the relax-\nation of a molecular electronic S= 1/2 system.\nOur approach builds on previous contributions from\nourselves [13, 14] and others [15–18] and signiﬁcantly\nextends the state of the art in the ﬁeld. For the\nﬁrst time we include in the formulation the contribu-\ntion of phonons from the entire Brillouin zone and the\nhyperﬁne and dipolar spin-spin interactions, both at a\nstatic and dynamical level. The description of all the\nfundamental spin-relaxation channels occurring at the\nﬁrst-order of perturbation theory is therefore here com-\nplete. Our method is purely ﬁrst-principle and pro-\nvides a rationalization of spin-lattice relaxation with-arXiv:1903.01424v1  [quant-ph]  4 Mar 20192\nout any previous knowledge of the system’s properties\nother than its crystal structure. The S= 1/2 sys-\ntem investigated is the crystal of the Vanadyl complex\nVO(acac) 2, being acac=acetylacetonate [19]. Vanadyl-\nbased molecular qubits represent archetypal systems for\nroom-temperature quantum-computing applications [20].\nWe demonstrate that Van Vleck’s mechanism is the\ndominant ﬁrst-order relaxation channel for S= 1/2\nmolecular spins in a high magnetic ﬁeld, while hyper-\nﬁne contributions become dominant in a low-ﬁeld regime.\nMost importantly, our calculations show that acoustic\nphonons are not rigid molecular translations, shedding\nlight on the origin of spin-phonon coupling in solid-state\nspin 1/2 compounds, and thus solving a eighty-year-old\ncontroversy.\nI. RESULTS\nA. First-Principles Spin Dynamics\nThe quantum dynamics of even a single spin is, in prin-\nciple, entangled with the dynamics of all the other spin\nand lattice degrees of freedom that it is interacting with.\nMore explicitly, it is driven by the total Hamiltonian\nH=Hs+Hph+Hsph, where the three terms are, re-\nspectively, the spins and phonons Hamiltonian, and the\nspin-phonon coupling. It is convenient to think at the\nproblem as composed of two parts: the simulation of the\nsole spin degrees of freedom as an isolated system and the\ninteraction of this ensemble with a thermal bath, namely\nwith the phonons.\nThe spin Hamiltonian of Eq. (1) describes the funda-\nmental interactions taking place within an ensemble of\nNsspins,\nˆHs=Ns/summationdisplay\niβi/vectorB·g(i)·/vectorS(i) +1\n2Ns/summationdisplay\nij/vectorS(i)·D(ij)·/vectorS(j),(1)\nwhere thei-th spin, S(i), interacts with an external mag-\nnetic ﬁeld /vectorBthrough the gyromagnetic tensor, βg(i).\nEq. (1) can account for both electronic and nuclear spins\non the same footing. Thus the spin-spin interaction ten-\nsor,D, may coincide with the point-dipole interaction,\nDdip, or the hyperﬁne tensor, A, depending on whether\nthe interaction is among electronic and/or nuclear spins.\nThe state of the spin system can be described in term of\nthe spin density matrix, ˆ ρ, whose dynamics is regulated\nby the Liouville equation,\ndˆρ\ndt=−i\n/planckover2pi1[ˆHs,ˆρ]. (2)\nOnce the eigenstates and eigenvalues, {Ea}, of the spin\nHamiltonian are known, it is possible to integrate exactly\nEq. (2) and obtain an expression for the time evolution\nofρI\nabin the so-called conﬁguration interaction,\nρI\nab(t) =e−iωab(t−t0)ρab(t0), (3)whereωab= (Ea−Eb)//planckover2pi1. Despite its simple form, Eq. (3)\nhides a high level of complexity, which originates from the\nhigh dimension of the Hilbert space it acts on. Several nu-\nmerical schemes exist to reduce the computational costs\nassociated to this problem [21, 22] and these will be the\nsubject of future investigations. Here we have decided to\nretain the full Hilbert space description and restrict the\nstudy to a small number of interacting spins.\nThe basic theory for phonon-driven spin relaxation has\nbeen derived before [13, 14]. Here we extend it by includ-\ning acoustic phonons and the phonon reciprocal space\ndispersion. For a periodic crystal, deﬁned by a set of\nreciprocal lattice vectors, q, and byNatoms in the unit-\ncell, the lattice’s dynamics can be described in terms of\nperiodic displacement waves (phonons), Qαq, with fre-\nquencyωαqand obeying to an harmonic Hamiltonian\nˆHph=/summationdisplay\nαq/planckover2pi1ωαq(ˆnαq+ 1). (4)\nHere ˆnαqis the phonon’s number operator. The spin-\nphonon coupling Hamiltonian, responsible for the energy\nexchange between the spins and the lattice, writes\nHsph=/summationdisplay\nαq/parenleftBig∂Hs\n∂Qαq/parenrightBig\nQαq. (5)\nUsually, phonons dynamics develops at a time scale much\nshorter than that of the spin relaxation. Therefore, if no\nphonon bottleneck is at play, the Born-Markov approx-\nimation will be valid and the phonons dynamics can be\nconsidered to be always at the thermal equilibrium. Un-\nder these circumstances the full set of Redﬁeld equations\ncan be used to study the spin dynamics under the ef-\nfect of a phonon bath [23]. In this framework, the spin\ndensity matrix ρI\nab(t) evolves according to the equation,\ndρI\nab\ndt=/summationdisplay\ncd/summationdisplay\nαqRαq\nab,cdρI\ncd(t). (6)\nThe transition rates between the elements of the density\nmatrix are of the form\nRαq\nab,cd∝π\n2/planckover2pi12Vαq\nacVαq\ndb/parenleftBig\nG(ωdb,ωαq) +G(ωcd,ωαq)/parenrightBig\n,(7)\nwhereVαq\nab=/angbracketlefta|∂Hs\n∂Qαq|b/angbracketrightis the matrix element of the\nspin-phonon Hamiltonian in the spin Hamiltonian eigen-\nfunctions basis and G(ωij,ωαq) is the single phonon cor-\nrelation function at ﬁnite temperature. For harmonic\nlattices,G(ωij,ωαq) is deﬁned as\nG(ωij,ωαq) = ¯nαqδ(ωαq−ωij) + (¯nαq+ 1)δ(ωαq+ωij),\n(8)\nwhere ¯nαqis the Bose-Einstein population at a tempera-\ntureT. Once Eq. (6) is solved, the magnetization dynam-\nics for thei-th spin can the computed by the canonical\nexpression /vectorM(i) = Tr{ˆρ(t)/vectorS(i)}.3\nFIG. 1. VO(acac) 2structure and spin phonon coupling distributions. Panel (a) shows the geometrical structure of the\ntwo VO(acac) 2molecular units inside the crystal’s unit-cell. Vanadium atoms are represented in pink, oxygen in red, carbon in\ngreen and hydrogen in white. Panel (b) shows the spin-phonon coupling distribution relative to the Zeeman energy as function\nof the phonons’ frequency. Panel (c) shows the spin-phonon coupling distribution relative to the dipolar spin-spin energy as\nfunction of the phonons’ frequency. Panel (d) shows the spin-phonon coupling distribution relative to the hyperﬁne energy as\nfunction of the phonons’ frequency.\nB. Spin Phonon Coupling in Molecular Qubits\nIn order to make the physics of Eq. (7) more trans-\nparent, a study of the spin-phonon coupling terms, Vab,\nfor the molecular qubit VO(acac) 2is hereafter provided.\nThis V4+complex crystallises with a triclinic primitive\ncell containing two inversion-symmetry-related molecular\nunits [19], as reported in Fig. 1. Each molecule bears a\nsingle electronic S= 1/2 spin in addition to the I= 7/2\nnuclear spin of51V. Let us consider one electronic spin\n/vectorSiinteracting with the rest of the crystal electronic spins\n/vectorSjand the local vanadium nuclear spin /vectorIi. Given the\ndeﬁnition in Eq. (5), the spin-phonon coupling Hamilto-\nnian will contain three distinguished contributions: a ﬁrst\nintra-molecular spin-phonon coupling due to the modu-\nlation of the Land´ e tensor, g; a second intra-molecular\ncoupling coming from the modulation of the hyperﬁne\ninteraction, A; and an inter-electronic spins interaction\noriginating from the modulation of the dipolar terms,\nDdip, namely\n∂Hs(i)\n∂Qαq=βi/vectorB·∂g(i)\n∂Qαq·/vectorS(i) +/vectorS(i)·∂A(ii)\n∂Qαq·/vectorI(i)+\n+Ns/summationdisplay\nj/vectorS(i)·∂Ddip(ij)\n∂Qαq·/vectorS(j).(9)\nA 3×3×3 super-cell containing 1620 atoms is optimised\nat the density functional theory (DFT) level and it is\nused to compute the lattice vibrational properties. The\nmolecular optimised structure has been further employed\nfor the calculation of all the spin-phonon coupling co-\neﬃcients appearing in Eq. (9). The details about the\nprotocol used for the phonons and spin-phonon calcu-\nlations can be found in the Method Section. All theseinteractions break the single-spin time-reversal symme-\ntry and are potentially active in intra-Kramer-doublet\nspin relaxation, but the physics beyond the three pro-\ncesses is quite diﬀerent. In order to make these diﬀer-\nences more evident we have calculated the spin-phonon\ncoupling squared norms V2\nsph(deﬁned in the Methods sec-\ntion) associated to each phonon modes and their corre-\nsponding distributions. The results, reported in panels\n(b) through (d) of Fig. 1, show striking diﬀerences both\nin qualitative and quantitative terms.\nThe anisotropy of the spin Hamiltonian is the ﬁnger-\nprint of the dependence of spin degrees of freedom on\natomic positions and the same interactions contribut-\ning to magnetic anisotropy are also contributing to the\nderivatives of Eq. 9. The tensors gandAhave\nanisotropic components coming from spin-orbit coupling\nand other interactions localized on the Vanadium centre.\nSuch short-ranged interactions are only inﬂuenced by lo-\ncalized intra-molecular vibrations and local rotations af-\nfecting the ﬁrst coordination sphere. In this case, high-\nenergy phonons are still operative. Conversely, dipo-\nlar interactions act between diﬀerent molecules and are\nboth non-local and long-wavelength in nature. These\nare expected to be prominently modulated by molecu-\nlar translations. Their interaction thus vanishes at high\nfrequency.\nC. VO(acac) 2Crystal Spin Dynamics\nThe spin-phonon coupling coeﬃcients discussed in the\nprevious section have been used together with Eq. (6)\nto simulate the spin dynamics of three diﬀerent systems:\none single electronic spin, one electronic spin interacting\nwith the V nuclear spin and two interacting electronic4\nspins. Tests including more that two coupled electronic\nspins are reported in the supplementary information (SI)\nand show a very small dependence of the relaxation time,\nτ, on spins farther than the ﬁrst-neighbour ones. In all\nsimulations, spins are interacting with all the phonons of\nthe periodic crystal, calculated by integrating the Bril-\nlouin zone with homogeneous grids up to 64 ×64×64k-\npoints.\nFIG. 2. Spin Relaxation time as function of the B ﬁeld\nand the temperature for one electronic spin coupled\nto one nuclear spin. The relaxation time, τ, in ms as a\nfunction of the external ﬁeld in Tesla is reported for the sim-\nulations of one electronic spin coupled to one nuclear spin\nand relaxing due to the phonons modulation of the Zeeman\nand hyperﬁne energies (black line and dots). The contribution\ncoming from the sole hyperﬁne energy modulation is reported\nwith a red line and dots, while the sole Zeeman contribution\nis reported by the green line and dots. The experimental re-\nlaxation time as extracted from AC magnetometry [19] is also\nreported (blue dots and line). The inset describes the simu-\nlated temperature dependence of spin relaxation at 5 Tesla.\nFig. 2 shows the spin relaxation time as function of the\nexternal magnetic ﬁeld /vectorBat 20 K for a single electronic\nspin coupled to the V nuclear spin. The combined ef-\nfect of the Zeeman and hyperﬁne modulation, described\nby the black line and dots, shows a ﬁeld dependence ap-\nproaching|/vectorB|4in the high ﬁeld limit ( B > 5 T), where\nτstarts to converge to the experimental value [19]. The\ngreen curve in Fig. 2, relative to the single electronic spin\ncontribution, shows that in this regime the leading mech-\nanism is the modulation of the Zeeman energy. In the\nlow-ﬁeld regime the relaxation time as a function of ﬁeld\napproaches a plateau where it is about three orders of\nmagnitude longer than the experimental value. The red\ncurve in Fig. 2 shows that the rate-determining mecha-\nnism in this regime is the modulation of the hyperﬁne\nHamiltonian.\nNumerical tests concerning the eﬀects of the numer-\nical noise in the calculation of the frequencies and the\nspin-phonon coupling parameters are reported in the SI\nand prove the robustness of our results. However, it isimportant to remark that the energies at play in the low-\nﬁeld regime ( B < 1 T) are extremely small, if compared\nwith the phonons frequencies and the simulations. Thus,\na more sophisticated Brillouin-zone integration scheme\nmight be needed to obtain a more robust estimate of the\nrelaxation times in this ﬁeld regime. Nonetheless, the\ncalculated relaxation time for low ﬁelds is orders of mag-\nnitude longer than the experimental one, suggesting the\npresence of higher-order relaxation mechanisms at play\nin this regime.\nThe inset of Fig. 2 displays the temperature depen-\ndence of the spin relaxation time in a ﬁeld of 5 T. In con-\ntrast to the experimental results, which show τ∝T−n\n(withn >2) [19], in this ﬁeld range we simulate a T−1\nbehaviour. Our result is in agreement with what ex-\npected from a ﬁrst-order approximation to spin-phonon\ncoupling and a harmonic lattice dynamics. Deviations\nfrom the T−1power law can be considered as ﬁngerprints\nof higher-order processes taking place. Interestingly, our\nsimulations also show a residual T-independent process\nin theT→0 limit. In this regime, due to the absence\nof populated phonons states, the only possible relaxation\npathway is provided by the T-independent spontaneous\nphonons emission from a spin excited state. This phe-\nnomenon has been recently observed in N-V centres [17].\nFIG. 3. Spin relaxation time as function of external\nﬁeld for two coupled electronic spins. The relaxation\ntime, τ, in ms as a function of the external ﬁeld in Tesla is\nreported for the simulations of two electronic spins relaxing\ndue to the phonons modulation of the Zeeman and dipolar en-\nergies (black line and dots). Green line and dots represent the\nrelaxation time of two isolated spins, where only the Zeeman\nenergy is modulated by phonons. The contribution coming\nfrom the sole dipolar energy modulation is reported with red\nline and dots. The experimental relaxation time as extracted\nfrom AC magnetometry [19] is also reported (blue dots and\nline).\nFig. 3 presents the spin-relaxation time as function of\nthe external magnetic ﬁeld at 20 K for two coupled elec-\ntronic spins. At high ﬁelds ( B < 1 T) the simulated re-\nlaxation dynamics, including both inter- and intra-spin5\ndirect relaxation mechanisms (black curve and dots in\nFig. 3), shows no signiﬁcant diﬀerence from that of two\nisolated spins relaxing through the modulation of the\nZeeman interaction (green curve and dots in Fig. 3). The\nrelaxation time due to the sole dipolar contribution (red\ncurve and dots in Fig. 3) becomes predominant only at\nlow ﬁelds and it is found to be two orders of magnitude\nslower than that associated to the hyperﬁne coupling.\nIn order to understand the nature of the interaction\nbetweenS= 1/2 spins and the lattice, it is now neces-\nsary to look at the nature of the phonons involved in the\nprocess. It has been shown that, in the absence of spin-\nspin interactions, spin relaxation can only occur through\nthe modulation of the spin Hamiltonian by local rota-\ntions and intra-molecular distortions [14]. However, for\na spin 1/2 in reasonable external ﬁelds, the Zeeman spin\nsplitting (up to a few cm−1) is much smaller than the\nﬁrst Γ-point optical mode, here around 50 cm−1. Energy\nconservation [see Eq. (8)] leaves only acoustic phonons\nas candidates for the spin-phonon interaction, suggest-\ning that no energy exchange between lattice and spin is\npossible under these conditions. The solution of this co-\nnundrum is provided by the analysis of the phonons na-\nture by means of their decomposition into local molecular\ntranslation, local molecular rotations and intra-molecular\ndistortions [14]. The results of this analysis, carried out\nover the phonons in the entire Brillouin zone, are summa-\nrized in Fig. 4, where the total and decomposed phonons\ndensity of states are reported.\nFIG. 4. Phonons density of states. In purple it is re-\nported the total phonons density of states as a function of the\nfrequency. The total phonons density of states has been also\ndecomposed in a pure translational contribution (green line),\na rotational contribution (blue line) and an intra-molecular\ncontribution (yellow line), all relative to a single molecule in-\nside the unit-cell. The inset shows the details of the density of\nstates in the low energy part of the spectrum. The Brillouin\nzone has been integrated with a uniform mesh of 643points.\nThe reported density has been smeared with a Gaussian func-\ntion with breadth 1.0 cm−1.\nThe total phonons density of states, shows the typ-ical∼ω2dependence at low frequency, where acoustic\nphonons dominate the vibrational spectra. However, the\ndecomposition of these modes shows that at low fre-\nquency the nature of the acoustic modes is far from be-\ning that of a pure molecular translation. A signiﬁcant\nrotational and intra-molecular contribution is present at\nfrequencies corresponding to the energy levels’ Zeeman\nsplitting considered in this work. These contributions\nprovide an eﬃcient relaxation pathway even for a single\nspin isolated from other magnetic centres.\nII. DISCUSSION\nThe results of the ﬁrst-principle calculations presented\nhere are in agreement with Van Vleck’s interpretation of\nthe spin relaxation in high ﬁelds, where a direct relax-\nation mechanism due to the Land´ e tensor modulation is\nthe relevant relaxation pathway. Most importantly we\nhave here demonstrated the mechanism underlying the\nenergy transfer between phonons and spins. The presence\nof an intra-molecular contribution inside the low-energy\nacoustic phonons is of fundamental importance in order\nto open a relaxation channel, that otherwise would be\ncompletely inactive due to the translational invariance of\nthe spin-orbit coupling interaction.\nThe presence of intra-molecular components in acous-\ntic modes can be understood as a mixing between rigid\nreticular translations and soft molecular modes and it\ncan be used as a rationale to engineer solid-state qubits.\nMore rigid molecular modes are expected to diminish sen-\nsibly the contamination of low-energy modes, therefore,\nextending the spin lifetime. Such a synthetic strategy has\nbeen recently attempted [25, 26] on the basis of similar\nobservations for high-spin molecular magnets [14].\nTo conclude, we have presented a general and fully\nﬁrst-principles method to study spin-lattice dynamics in\nmagnetic materials. We have predicted the correct spin\nrelaxation time and ﬁeld dependence for a solid-state\nmolecular qubit in high external ﬁelds. In particular,\nwe have given, for the ﬁrst time, a full microscopic ratio-\nnale of the spin relaxation in solid-state molecular qubits\nby ranking the three fundamental interactions at play\namong electronic and nuclear spins at the ﬁrst order of\nperturbation theory. Details concerning the spin dynam-\nics at low ﬁelds and the nature of the polynomial depen-\ndence of the relaxation rate at high- T[19] remain elusive,\nsuggesting the presence of higher-order processes taking\nplace. The method presented here can readily been ex-\ntended to include higher-order processes such as Raman\nrelaxation mechanism and phonon-phonon interactions.\nThis will be the subject of future work.6\nIII. METHODS\nA. Lattice Dynamics\nAll the structural optimisation and Hessian calcula-\ntions have been performed with the CP2K software [27]\nat the level of density functional theory (DFT) with the\nPBE functional including Grimme’s D3 van der Waals\ncorrections [28, 29]. A double-zeta polarised (DZVP)\nMOLOPT basis set and a 600 Ry of plane-wave cutoﬀ\nhave been used for all the atomic species. The compari-\nson between simulated and experimental lattice parame-\nters is available in ESI. All the translational symmetry in-\ndependent force constants have been computed by ﬁnite\ndiﬀerence approach with a 0.01 ˚A step. Being Φ ij(lm)\nthe force constant, coupling the i-th atomic degrees of\nfreedom in the lattice cell at the position Rland thej-th\natomic degrees of freedom in the lattice cell of position\nRm, the dynamical matrix, D(q), at the q-point, is built\nas\nDij(q) =/summationdisplay\nlΦl0\nijeiq·Rl. (10)\nThe eigenvalues of D(q) areω2(q), while the eigenvectors\nL(q) deﬁnes the normal modes of vibration. The calcu-\nlated phonons are in good agreement with those previ-\nously performed at the sole Γ-point[19].\nB. Spin-Phonon Coupling Coeﬃcients\nThe ORCA software [30] has been employed for the\ncomputation of the gandAtensors for both equilibrium\nand distorted geometries. We have used the basis sets\ndef2-TZVP for V and O, def2-SVP for C and H and a\ndef2-TZVP/C auxiliary basis set for all the elements. For\nthe calculation of the Atensors the entire basis set have\nbeen de-contracted. The calculations of the gtensors\nhave been carried out at the CASSCF+NEVPT2 level of\ntheory, with a (1,5) active space and spin-orbit contri-\nbutions included through quasi-degenerate perturbation\ntheory. The calculations of the Atensors have been per-\nformed at the DFT level with the PBE functional [28].\nThe spin phonon coeﬃcients relative to the gandA\ntensors have been calculated as numerical derivatives.\nTen Cartesian displacements ranging from ±0.01˚A. have\nbeen used to estimate ∂g/∂Xisand∂A/∂Xis, whereXis\nrefers to the sCartesian component of the i-th atom in\nthe DFT optimized unit-cell. The gvsXisandAvsXis\nproﬁles have been ﬁtted with a fourth order polynomial\nexpression and set to zero if the ﬁtting error on the lin-\near term exceeded 7%. The spin-phonon coupling coeﬃ-\ncients relative to the point-dipole-dipole interaction have\nbeen obtained by analytical diﬀerentiation. The Carte-\nsian derivatives ∂ˆHs/∂Xishave then projected onto thenormal modes by means of the expression\n/parenleftBig∂Hs\n∂Qαq/parenrightBig\n=Ncells/summationdisplay\nlN,3/summationdisplay\nis/radicalBigg\n/planckover2pi1\nNqωαqmieiq·RlLαq\nis/parenleftBig∂Hs\n∂Xl\nis/parenrightBig\n,\n(11)\nwhereXl\nisis thesCartesian coordinate of the i-th atom\nofNwith massmi, inside the unit-cell replica at position\nRl, andNqis the number of q-points used. All the data\nregarding spin Hamiltonian parameters and their diﬀer-\nentiation are reported in ESI. The spin-phonon coupling\ndistributions have been calculated starting from the spin-\nphonon coupling coeﬃcients squared norm, deﬁned as\nV2\nsph(ωα) =1\nNq/summationdisplay\nq/summationdisplay\nij/parenleftBig∂gij\n∂Qαq/parenrightBig2\n, (12)\nand analogously for the AandDDiptensors. The\nDirac’s Delta function appearing in Eq. (8) has been\nevaluated as a Gaussian function in the limit for inﬁnite\nq-points and vanishing Gaussian breadth. A grid of\n643q-points and a Gaussian breadth of 1 cm−1was\nestimated to accurately reproduce this limit for all\nthe temperature and ﬁeld values investigated. Some\nresults about these convergence tests are reported in ESI.\nSupplementary Information\nSupplementary information is available: the entire\nform of the non-secular Redﬁeld equations, comparison\nbetween experimental and calculated spin Hamiltonian\nparameters and crystallographic parameters, the spin\nHamiltonain parameters for all molecular distortions\nand relaxation time convergence tests.\nData Availability\nAll the relevant data discussed in the present paper are\navailable from the authors upon request.\nAcknowledgments\nThis work has been sponsored by Science Foundation\nIreland (grant 14/IA/2624). Computational resources\nwere provided by the Trinity Centre for High Perfor-\nmance Computing (TCHPC) and the Irish Centre for\nHigh-End Computing (ICHEC). 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(2)\nIn our implementation we only considered the same-index terms. This correspond to assuming that the modulation\nof each term occurs independently from the others.\nSUPPLEMENTARY NOTE 2. LATTICE PARAMETERS AND SPIN HAMILTONIAN.\nSupplementary Table 1 reports the calculated and experimental lattice parameters. Taking into account an expected\nvolume reduction due to the absence of temperature expansion in the simulations, the agreement between experimental\nand calculated ones is excellent.\nModel a (˚A) b ( ˚A) c ( ˚A)α(◦)β(◦)γ(◦)\nExpa7.346 8.149 11.207 72.86 72.20 66.94\nSimulated 7.060 7.935 11.091 74.467 72.622 68.216\naTaken from Ref. [1].\nSupplementary Table 1. Lattice parameters for the VO (acac )2molecular crystal. The X-ray experimental lattice\nparameters are compared with those obtained from the periodic DFT optimization of 3x3x3 super-cell.\nSupplementary Table 2 reports the comparison between simulated and experimental values of the spin Hamiltonain\nparamenters. Both symmetry and magnitude of the eigenvalues of Land` e and Hyperﬁne tensors are well reproduced.\nThegtensor anisotropy is slightly overestimated. On the contrary, the eigenvalues of the Atensor are slightly\nunderestimated. The order of magnitude of these interactions is however well reproduced and we believe that these\ndiﬀerences between calculated and experimental values cannot signiﬁcantly bias our results.\nModel gx gy gz |Ax|(cm−1)|Ay|(cm−1)|Az|(cm−1)\nExpa1.9845 1.981 1.9477 0.00580 0.00627 0.01712\nSimulated 1.9830 1.9814 1.9274 0.00354 0.00396 0.01396\naTaken from Ref. [1].\nSupplementary Table 2. Spin Hamiltonian parameters for VO (acac )2.The spin Hamiltonian experimental parameters\nare compared with ﬁrst-principles ones.arXiv:1903.01424v1  [quant-ph]  4 Mar 20192\nSUPPLEMENTARY NOTE 4. K-POINTS AND PHONON’S LINE-WIDTH CONVERGENCE.\nIn this work we investigated the harmonic phonons limit of the spin dynamics. In this limit, the phonons’s green\nfunction contains a Dirac’s delta distribution δ(ω−ωαq) that enforces energy conservation. The evaluation of the\nphonons’ Green function requires two numerical approximations. One concerns the use of a discrete number of k-\npoints, in principles inﬁnite due to the inﬁnite extension of a crystalline solid, and the approximation of the Dirac’s\ndelta distribution in terms of a regular function.\nA Gaussian function converges to a Dirac’s delta for a Gaussian breadth approaching 0.\n1\nσ√πe−(ω−ωαq)2\nσ2σ→0−−−−→δ(ω−ωαq) (3)\nHowever, this is only true assuming an inﬁnitely dense number of states as function of phonons’ frequency. Numeri-\ncally, this requires the number of k-points to be large enough to guarantee an approximately inﬁnite number of states\ninside the energy window selected by σ. Therefore, for each value of σ, spin relaxation time must be ﬁrst converged\nwith respect to the number of k-points. Then, sigma is gradually reduced until relaxation time is converged with\nrespect to this quantity as well.\nSupplementary Figure 1. Spin relaxation time as function of external ﬁelds for diﬀerent values of k-points. The\nrelaxation time, τ, in ms as function of the Gaussian smearing σfor diﬀerent values of external ﬁeld magnitude is reported.\nThe simulation corresponds to the dynamics of one electronic spin relaxing due to the phonons modulation of the Zeeman term\nof the spin Hamiltonian. The left panel correspond to simulations with σ= 1 cm−1and the right Panel to simulations with\nσ= 4 cm−1.\nThe left panel of Supplementary Figure 1 shows the convergence of τwith respect to the number of k-points when\nσis ﬁxed to 1 cm−1. For small Brillouin zone integration grids the spin life-time is largely overestimated. This is\nbecause no phonons are available in the near-Γ part of the spectra. Increasing the ﬁness of the integration grid the\nvalue converges. For larger values of σ, as reported in the right Panel of Supplementary Figure 1, the convergence is\nreached for smaller integration grids.\nSupplementary Figure 2 reports the dependence of the relaxation time τwith respect to the value σ. For each value\nofσthe convergence with respect to the number of k-points has been achieved as described above. Relaxation time\nis nicely converged for σ= 1 cm−1. For smaller values of σit was not possible to reach convergence on k-points with\na reasonable ﬁness of the Brillouin zone integration grid. This is probably due to the fact that an energy window of\nless than 1 cm−1is small compared to the numerical noise aﬀecting the phonons calculations.3\nSupplementary Figure 2. Spin relaxation time as function of the Gaussian breadth for diﬀerent external ﬁelds.\nThe relaxation time, τ, in ms as function of the Gaussian smearing σfor diﬀerent values of external ﬁeld magnitude is reported.\nThe simulation correspond to the dynamics of one electronic spin relaxing due to the phonons modulation of the Zeeman term\nof the spin Hamiltonian.4\nSUPPLEMENTARY NOTE 5. NUMBER OF SPINS CONVERGENCE.\nDue to the exponential increase of the dimension of the master matrix in Eq. 1, the study of relaxation time as\nfunction of the number of interacting electronic spins has to be restricted to a maximum of six spins. Within this\nconstraint, six diﬀerent systems can been studied: two and three unit-cells replicated along x,y and z, respectively.\nThe results are summarized in Supplementary Figure 3. The inclusion of multiple electronic spins beyond the ﬁrst-\nneighbour one does not dramatically aﬀect the spin dynamics and the estimations of relaxation time made with just\ntwo spins correctly capture the essential physics.\nSupplementary Figure 3. Spin relaxation time as function of the number of coupled spins. The relaxation time, τ,\nin ms as function of the external ﬁeld magnitude for diﬀerent sets of spins is reported: the two spins as in the crystal unit-cell,\nfour spins as obtained by alying two unit-cells and six spins as obtained by alinging three unit-cells. The top-left Panel shows\nthe results for the reticular direction a, top-right Panel shows the results for the reticular direction band the bottom Panel\nshows the results for the reticular direction c.5\nSUPPLEMENTARY NOTE 6. EFFECT OF NUMERICAL NOISE.\nWe investigated the eﬀect of numerical noise on the two most important computed quantities: spin-phonon coupling\nparameters and phonons’ frequencies. For the former we studied the eﬀect of multiplying by a factor two all the spin-\nphonon coupling coeﬃcients relatives to the modulation of the hyperﬁne Hamiltonian, while for the latter we applied\na 20% reduction of all the frequencies. The magnitude of these perturbations has been chosen accordingly to the error\nthese quantities are aﬀected by. In Supplementary Note 2 we have shown that hyperﬁne parameters are underestimated\nby a factor of almost two. Comparing our phonons simulations with THz spectroscopy[2] we observe an error on the\nﬁrst frequency at the Γ-point, here calculated at ∼50 cm−1, of no more than 20%.\nSupplementary Figure 4. Eﬀect of numerical noise on spin relaxation time. The relaxation time, τ, in ms as function of\nthe external ﬁeld magnitude in Tesla for the unbiased case (Green line and dots), for a double hyperﬁne spin-phonon coupling\n(Black line and dots) and for a 20% rescaled frequencies (Red line and dots).\nSupplementary Figure 4 reports the comparison between the unbiased dynamics (Green line and dots) and the\ndynamics in presence of a doubled hyperﬁne interaction (Black line and dots) and the dynamics obtained after\nfrequencies rescaling (Red line and dots). The eﬀect of both the perturbations investigated show that numerical\nerrors in the determination of spin-phonon coupling coeﬃcients and phonons’ frequencies do not signiﬁcantly aﬀect\nour results and the orders of magnitude of our estimations are well converged.6\nSUPPLEMENTARY REFERENCES\n[1] Lorenzo Tesi, Alessandro Lunghi, Matteo Atzori, Eva Lucaccini, Lorenzo Sorace, Federico Totti, and Roberta Sessoli,\n“Giant spin-phonon bottleneck eﬀects in evaporable vanadyl-based molecules with long spin coherence,” Dalt. Trans. 45,\n16635–16645 (2016).\n[2] Matteo Atzori, Lorenzo Tesi, Stefano Benci, Alessandro Lunghi, Roberto Righini, Andrea Taschin, Renato Torre, Lorenzo\nSorace, and Roberta Sessoli, “Spin Dynamics and Low Energy Vibrations: Insights from Vanadyl-Based Potential Molecular\nQubits,” J. Am. Chem. Soc. 139, 4338–4341 (2017)."    },    {        "title": "2103.08517v3.Entanglement_Dynamics_between_Ising_Spins_and_a_Central_Ancilla.pdf",        "content": "Entanglement Dynamics between Ising Spins and a Central Ancilla\nJoseph C. Szabo1and Nandini Trivedi1\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n(Dated: November 5, 2021)\nWe investigate competing entanglement dynamics in an open Ising-spin chain coupled to an external\ncentral ancilla qudit. In studying the real-time behavior following a quench from an unentangled spin-\nancilla state, we ﬁnd that the ancilla entanglement entropy SvN;Atracks the dynamical phase transi-\ntion in the underlying spin system. In this composite setting, purely spin-spin entanglement metrics\nsuch as mutual information and quantum Fisher information (QFI) decay as the ancilla entanglement\nentropy grows. We deﬁne multipartite entanglement loss (MEL) as the difference between collective\nmagnetic ﬂuctuations and QFI, which is zero in the pure spin chain limit. MEL directly quantiﬁes\nthe ancilla’s effect on the development of spin-spin entanglement. One of our central results is that\nMEL(t)∝eSvN;A(t). Our results provide a platform for exploring composite system entanglement dy-\nnamics and suggest that MEL serves as a quantitative estimate of information entropy shared between\ncollective spins and the ancilla qudit. Our results present a new framework that connects physical\nspin-ﬂuctuations, QFI, and bipartite entanglement entropy between collective quantum systems.\nI. INTRODUCTION\nEntanglement is the most fascinating and perplexing\nfeature of composite, many-particle quantum systems.\nUnderstanding its origin in physical platforms; whether\ndue to particle statistics, correlations, and/or dynamical\ninteractions, lies at the heart of quantum matter, sens-\ning, and algorithm development [1–3]. This pursuit\nis being driven by technological advances in cold-atomic\ncondensates [4–8], trapped-ion platforms [9–18], cav-\nity QED [19–22], and superconducting circuits [23, 24],\nwhich have brought dynamical quantum systems to the\nforefront of experimental and theoretical research. These\nexperiments allow us to study how entanglement devel-\nops from purely classical initial states in the presence of\nquenched interactions, dissipation and driving, and deco-\nherence. Here, many fundamental questions regarding\nentanglement remain unanswered such as determining\nit’s relationship to physical properties as well as under-\nstanding the connections among the various methods for\nmeasuring entanglement.\nTo quantify entanglement, we work along two fronts:\n(i) through a partitioning structure that measures entan-\nglement entropy, providing an information theoretic char-\nacterization of the number of entangled degrees of free-\ndom (DoF) shared between subsystems; and (ii) through\nholistic multipartite entanglement measures that capture\ncollective quantum correlations. Entanglement entropies\nhave proven to be an essential theoretical diagnostic for\ncharacterizing quantum phases [25, 26], detecting topo-\nlogical order [27, 28], and understanding nonequilibrium\nquantum dynamics and thermalization in pure quantum\nsystems [29–31]. From the experimental side, entangle-\nment entropy remains one of the most challenging mea-\nsures, as entropy does not remain an entanglement mono-\ntone when subject to loss, dissipation, and interaction\nwith an environment. Few or multi-particle correlators\naccessible to realistic experimental platforms can only be\nrelated to entropy in ﬁne-tuned integrable models [32–34], otherwise entropy calculations require full-state to-\nmography, robust multi-correlations measures that pro-\nvide a lower bound estimate [35, 36], or many-body in-\nterference [37, 38]. On the other hand, multipartite en-\ntanglement witnessed through Quantum Fisher informa-\ntion (QFI) provides a true entanglement monotone re-\ngardless of classical entropy contributions, and in pure\nquantum systems can be measured through collective sin-\ngle particle operators [6, 39]. In thermal equilibrium, QFI\nis directly related to the dynamical response function of\nfew particle operators [40–42]. Though entropy and QFI\nprovide qualitatively different perspectives on entangle-\nment, they are essential parts of the same phenomenol-\nogy: phase transitions, quantum thermalization, many-\nbody localization.\nQFI and entropy have been studied extensively with\nregards to quantum information dynamics in nonequi-\nlibrium and open environments. Each provide a unique\nperspective regarding the transition from semiclassical to\nquantum behavior as well as establishing myriad ther-\nmalization physics of quantum systems. Despite their\npopular theoretical investigations, few direct connections\nbetween the two have been established.\nRecent theoretical works have revealed qualitative re-\nlations connecting correlations, entanglement entropy,\nand QFI in a purely semiclassical perspective for collec-\ntive spin and light-matter systems [43]. In such sys-\ntems, it is well-known that collective light-matter entan-\nglement is necessary for generating effective long-range\nspin Hamiltonians and achieving highly squeezed, mul-\ntipartite entangled spin states [20, 22, 44–46]. Though\nwe have this heuristic understanding, no numerical con-\nnection between QFI and entropy has been interrogated.\nTheoretical studies examining the fate of entanglement\nsubject to classical entropy, suggest that strong connec-\ntions between QFI and entropy exist and their dynamics\nare intimately related [47–50]. Beyond these, few works\ndirectly investigate the dynamical connection between en-\ntropy and multipartite entanglement precisely in fullyarXiv:2103.08517v3  [quant-ph]  4 Nov 20212\nquantum systems away from integrability and semiclas-\nsical approximations. The questions that remain to be\nanswered are how do composite systems share quantum\ninformation, what are the implications for subsystem en-\ntanglement content, and what are the roles of various\nquantum information metrics?\nIn this paper, we provide general insights with regards\nto these questions by studying local quantum systems\nbeyond integrability. We use numerically exact meth-\nods to study the entanglement dynamics of a quenched\nIsing spin-chain with weak exchange with a multilevel\nancilla qudit. The Ising model provides a highly stud-\nied, integrable starting point for understanding nonequi-\nlibrium entanglement dynamics, where entropy and QFI\nhave been thoroughly characterized in various limits. The\nancilla then provides an environment through which the\nspin ensemble can undergo loss and decoherence while\nat the same time allowing us to interrogate the ancilla\ndegrees of freedom (DoF) as it relates to the reduced,\nspin-subsystem. Interestingly, such a central mode re-\nlates to cavity quantum electrodynamics (cQED) exper-\niments [51, 52], central spin or dressed cavity in cold-\natoms [53–56] or trapped ions [57], nuclear magnetic res-\nonance (NMR), and is a common element of quantum com-\nputing codes employing an oracle vertex in a qubit net-\nwork [57–59]. In what follows, we brieﬂy outline the key\nresults of our numerical investigation, review the entan-\nglement metrics employed in this paper, and ﬁnely dis-\ncuss the microscopic Hamiltonian and resulting dynam-\nics.\nII. SUMMARY\nIn this paper we analyze the subsystem entanglement\ndynamics of a quenched Ising spin-system with each site\nuniformly coupled to a qudit ancilla. We speciﬁcally focus\non\ni Ancilla entanglement: SvN;A\nii Mutual information: MI=SvN;1/slash.left2−1\n2SvN;A\niii QFI (F)and spin ﬂuctuations: F(ρ,ˆSµ)≤∆ˆS2\nµ\nThis work represents one of the ﬁrst studies characteriz-\ning the fate of entanglement in quenched quantum mat-\nter coupled to a controllable environment, here a bosonic\nmode. We work in the large bosonic-limit where results\nare converged with respect to the bosonic dimension ( q>>\nL). Without any interaction between spins and ancilla, we\nobserve the dynamical quantum phase transition associ-\nated with GGE description of the interacting spins and\nthe transition from area to volume-scaling 1 /slash.left2-chain en-\ntanglement entropy, SvN∼O(L), where Lis the number\nof spins in the 1D-chain. We similarly ﬁnd that long-time\nsaturation value of the transverse and longitudinal spin-\nﬂuctuations agree with previous analytic results.\nWe ﬁnd surprising physics away from integrability in\nthe full spin-ancilla model, which admits simple intuitive\nFIG. 1. Spin-ancilla schematic with BandArepresenting the\n1D spin-chain and multi-level ancilla, respectively. Ising spin\ninteractions given by J, magnetic-ﬁeld h, and spin-ancilla cou-\npling λ. (b) Cartoon representation of the dynamical quench\nfrom an initially polarized spin state. Entangled region across\nthe center boundary (green) grows in time as quasiparticles\nwithin a distance t/slash.leftvBare transmitted between the two bipar-\ntite regions. (c) Schematic representation of our key ﬁnding in\nthe interacting spin-boson composite system; operator ﬂuctua-\ntions∆ˆOB(t)and the corresponding quantum Fisher informa-\ntion f(ρB(t),ˆOB)within the spin subsystem begin to deviate in\ntime as external entropy SvN;Agrows (red). This discrepancy\ndeﬁned at MEL (green) develops proportionally to the eSvN;A(t).\nThe auxiliary bosonic system entanglement oscillates like the\ncharacteristic level spacing ωc.\nextensions from the isolated Ising spin chain limit. For\nweak-coupling between spins and ancilla we observe:\ni Ancilla entanglement entropy SvN;Atracks the un-\nderlying spin DQPT, which persists under nonlocal\nancilla coupling.\niiSvN;Ascales like ∝logL.\niii Mutual Information (MI) and QFI within spins de-\ncreases monotonically with coupling as expected be-\ntween system-environment.\niv We deﬁne this discrepancy between the variance\nin spin-spin correlations (ﬂuctuations) and spin-\nspin multipartite entanglement (QFI) as multipar-\ntite entanglement loss (MEL).\nv Entanglement loss witnessed by MEL (t)develops\nproportionally to eSvN;A(t).3\nOur most signiﬁcant ﬁnding is that the corresponding\ngrowth and decay of entanglement within the ancilla and\nspins respectively are intimately connected. We ﬁnd that\nthe discrepancy between spin-spin ﬂuctuations and QFI\ncaptures the entanglement proﬁle of the ancilla and pro-\nvides a strong estimation on information transfer. This\nformulation comes from a complementary approach to\nprevious results, where we ﬁnd an identical log relation-\nship that connects correlations and entanglement [43].\nThis relationship not only provides a strong estimate in\nthe long-time limit, but accurately captures the magni-\ntude of real-time dynamics. We expect that this small,\nﬁnite ancilla behavior captures essentially semiclassical\nentanglement features of the collective interacting spin-\nsystem and provides an exciting frame to understand\nthermalization of many-body systems with their environ-\nment.\nIII. ENTANGLEMENT METRICS\nThe key measures of entanglement in this work are en-\ntanglement entropies (von Neumann and mutual infor-\nmation) and multipartite entanglement (spin-ﬂuctuations\nand quantum Fisher information). The former two rely\non bipartitions of the Hilbert space to calculate entropy,\nwhile the latter characterizes holistic entanglement con-\ntent shared between eigenvalues of collective few-body op-\nerators.\nThe von Neumann entanglement entropy taken be-\ntween two regions ( A,B) is calculated as\nSvN=/summation.disp\nkλklog(λk). (1)\nWhere λkare eigenvalues of the reduced density ma-\ntrix (RDM) obtained by integrating out either subsystem\nAorB. Though insightful in theoretical investigations,\nSvNis not an entanglement monotone for open quan-\ntum systems. In this scenario where we employ a spin-\nchain and an ancilla, the ancilla serves as an environ-\nment when considering the entropy in the spin-chain.\nTherefore, we calculate the1\n2-chain mutual information\n(MI). For a 3-body example deﬁned by total Hilbert space\nH=Ha⊗Hb⊗Hcthe mutual information between abis\nI(ab;c)=SvN(a/divides.alt0bc)+SvN(b/divides.alt0ac)−SvN(ab/divides.alt0c),(2)\nwhere SvN(i/divides.alt0j)is the standard bipartite entanglement\nentropy of reduced space i, explicitly tracing out j. Mu-\ntual information serves as a useful rectiﬁcation to entan-\nglement entropies’ failure in open quantum systems and\nis the subject of recent theoretical investigations [50].\nQFI on the other hand, is an entanglement monotone\nregardless of purity and environment [60–62]. In a pure\nquantum state it is precisely equal to collective ﬂuctua-\ntions witnessed by operator ˆOF(ˆO,ρ)=2/summation.disp\nα,βvα−vβ\nvα+vβ/divides.alt0/uni27E8uα/divides.alt0ˆO/divides.alt0uβ/uni27E9/divides.alt02≤4/uni27E8∆ˆO2/uni27E9⋅(3)\nuα,vαare the eigenvectors and eigenvalues of the density\nmatrix ρ. QFI originates from metrological studies and\nrefers to the precision of a measurement conditioned on a\nglobal operator ˆOand state ρ. Given a unitary transfor-\nmation of the form ˆU=eiˆOθandρθ=UρU†, the precision\nin parameter θis constrained by the quantum Cramer-\nRao bound ∆θ2=1\nF(ˆO,ρ). For pure states, the uncertainty\ninθis related to the uncertainty in its conjugate operator.\nFor spin systems and operator deﬁned as ˆO=∑i/uni20D7si⋅ˆni,\nQFI is identical to the spin ﬂuctuations about the vector\n/uni20D7Snon the Bloch sphere. For mixed states, F(ˆO,ρ)must\nbe written in terms of an eigendecomposition of the ini-\ntial mixed density matrix. In general, different operators\nwill present different bounds on Eq.3, and there has been\nno analytic insight into determining how this bound can\nbe made tighter in arbitrary quantum systems away from\npurity.\nBeyond metrological optimization, QFI witnesses mul-\ntiparticle entanglement within a state ρand even more\nrecently in the detection of topological quantum phases\n[63, 64]. The Fisher information density f(ˆO,ρ)≡\nF(ˆO,ρ)/slash.leftNgiven Nconstituent particles and ˆObeing a\nsum over local site-operators provides that for f>kthe\nsystem is at least (k+1)-partite entangled. Determining\nthe true multiparticle entanglement of a system requires\noptimization over the set of all possible ˆO.\nThough, QFI depends on the choice of collective opera-\ntorˆO, general relationships exist that connect the devel-\nopment of multipartite and bipartite entanglement. En-\ntropy provides a logarithmic measure of how spread a\nquantum state is throughout the full Hilbert space, while\nmultipartite entanglement examines the development of\noff-diagonal components of the density matrix. For unen-\ntangled initial states, entanglement entropy generically\ngrows proportionally to tunless subjected to localization\nphysics, integrability, or conservation laws, while QFI and\nspin-ﬂuctuations grow like eαtwith some phenomenologi-\ncal exponent α. In quadratic bosonic or fermionic systems\nthis heuristic relation is exactly proportional; SvN(t)∝\nlogF(ˆOmax,ρ(t)), where ˆOmaxis a generic operator that\nwitnesses the maximal ﬂuctuations at each instance in\ntime. This deﬁnition relies on translational invariance\nand a homogeneous collective system [43, 65, 66]. Be-\nyond such systems, for arbitrary Hermitian matrices, the\nequality in Eq.3, is believed to similarly develop like\n∼eSvN;B, but has only been characterized in random ma-\ntrices with Hilbert spaces O(2−102)[67]. Beyond these\nconditions no analyses have been performed to relate bi-\npartite entropy and QFI with regards to general compos-\nite quantum systems and physical, well-characterized op-\nerator dynamics.\nIn this work we consider the simplest realization of\nan interacting composite system. We study an interact-\ning spin-ancilla system to illuminate how observables and4\ninformation in interacting quantum spins behave under\nexchange with a bosonic ancilla. To capture the differ-\nence between ﬂuctuations and true multiparticle entan-\nglement deﬁned in Eq.3 over the reduced spin density ma-\ntrix, we deﬁne the multipartite entanglement loss (MEL)\nas\nMEL(ˆO,ρ)=/uni27E8∆ˆO2/uni27E9−1\n4F(ˆO,ρB). (4)\nThis counts the difference in ﬂuctuating constituent par-\nticles from the internal contribution to the multiparticle\nentanglement, such that for a pure system returns zero\nmissing entanglement. The operator ˆOis speciﬁcally con-\nditioned on the total Hilbert subspace spanning the sys-\ntem and ancilla ˆO∼ˆB⊗ˆA, while in deﬁning QFI, we trace\nover the ancilla degrees of freedom in both ˆOandρ. We\nspeciﬁcally focus on ˆSn⊗ˆIfor collective spin magnetiza-\ntion in the system and identity in the ancilla.\nProvided we are working with an essentially free-\nfermion model (JW transformation of the 1D Ising model)\nand a quadratic fermion-boson coupling, it would be inter-\nesting to observe a similar relationship between ﬂuctua-\ntions and entanglement entropy as mentioned above.\nOur focus in this paper is understanding how the in-\nequality in Eq.3 (MEL) evolves in a speciﬁc model of an\ninteracting spin system coupled to an ancilla described\nbelow. We observe for the ﬁrst time a strong quantitative\nconnection between MEL and SvN∶A:\nMEL(ρ,ˆSmax)∝eSvN,A−1. (5)\nˆSmaxrefers to the maximal MEL optimized over collective\nspin observables. Here we cannot perform such an exten-\nsive search over operators but ﬁnd very strong agreement\nwith simple collective operators in a well-characterized\nmodel. In the pure spin system limit, there is no discrep-\nancy between the spin ﬂuctuations and the correspond-\ning QFI measured over the reduced density matrix of the\nspin sector. Here MEL=0, and the entanglement entropy\nwith the system must similarly be 0. In the opposing limit\nwhere spin ﬂuctuations ∆S2\nn∝L2andF(ˆSn,ρB)=0, this\nrestricts the ancilla entanglement to be at most O(logL).\nWe expect that Eq.5 is only accurate when the environ-\nment experiences an effective semiclassical system, or in\nother words is coupled to collective degrees of freedom to\nthe quantum system of interest. In a completely locally\nthermalizing regime, the entropy of the quantum system\nshould scale like O(L), so observing this semiclassical re-\nsult in a local quantum system is surprising.\nIV. MODEL\nThe transverse-ﬁeld Ising chain (TFIC) is a paradig-\nmatic, exactly solvable example of a 1-D quantum phase\ntransition (QPT) [68] and has garnered signiﬁcant recentinterest in the study of entanglement and scrambling dy-\nnamics [69], dynamical phase transitions [70], and quan-\ntum thermalization [71]. This model has been experimen-\ntally realized using Rydberg atoms when interactions be-\nyond nearest neighbors can be neglected on the relevant\ntimescales [7, 8, 72]. The Hamiltonian for the model is\nHTFIC=−JL\n/summation.disp\ni=1σz\niσz\ni+1+h/summation.disp\ni=1σx\ni. (6)\nwhere periodic boundary conditions have been employed\nsuch that σµ\nL+1=σµ\n1. This model exhibits a ground state\nphase transition at g=h/slash.leftJ=1 that is in the same uni-\nversality class as the classical 2-D Ising model. The crit-\nical point gcseparates two distinct phases, where for\ng<gc, the spins exhibit Z2 symmetry-breaking order\nwith nonzero longitudinal magnetization /uni27E8σz/uni27E9. For h>hc\nthe spins are in a paramagnetic phase with preferential\nalignment along the transverse ﬁeld. The system also ex-\nhibits a dynamical-QPT (DQPT) when the system is pre-\npared in an initial product state /divides.alt0↑↑↑.../uni27E9zand quenched\nacross the ground state critical ﬁeld [70].\nIn our work we employ an ancillary, central qudit\nto probe the TFIC. Depending on the underlying phase\nand transport characteristics, novel real-time dynamics\nand long-time behavior arise in the auxiliary system.\nThe form of the spin-ancilla interaction is a paradig-\nmatic Dicke-Ising construction that combines an essential\nmodel for light-matter interactions and magnetic quan-\ntum matter represented by the following:\nHint=ωca†a+λ√\nLL\n/summation.disp\ni=1(a†+a)σx\ni, (7)\nwhere a(a†)is a bosonic annihilation (creation) operator\nfor a single mode uniformly coupled to each spin with\nstrength λand normalized by√\nLsuch that the effec-\ntive spin-spin interaction induced by the bosonic ancilla\nremains intensive. The second term in Eq.7 represents\nthe rotation of a single spin in exchange for the particle\nnumber within the ancilla. Though here we speciﬁcally\nconsider bosonic ﬁelds with ﬁnite dimension q>L, the\nproblem is identical to a large spin of size S=q/slash.left2 with\nZeeman splitting ωc.\nWe ﬁrst address the general behavior of spin-ancilla ob-\nservables and how they deviate from both the pure Ising\nand the Dicke models. The general relationship between\nobservables is obtained from the exactly solvable ground\nstate in the (J=0)limit where the bosonic occupation ˆN\nis\n/uni27E8ˆN/uni27E9∝/uni27E8ˆS2\nx/uni27E9λ2\nωc, (8)\nwhere here ˆSx=∑iσx\ni. This can be readily observed by\nperforming a unitary transformation on Hamiltonian by5\ntranslating the bosonic creation and annihilation opera-\ntors as\nˆb=ˆa+λ√\nLωcˆSx. (9)\nThe resulting Hamiltonian has no spin-boson coupling\nand can be treated as a classical Ising chain\nˆH=ωcˆb†ˆb+λ2\nLωcˆS2\nx+hˆSx. (10)\nThe relevant timescale translating the exchange of spin\nand bosons is the ancilla splitting ωc, which for all re-\nsults in this work is set to 2 πJ/slash.leftωc=12.6tJ. The sys-\ntem conserves total Sxso when prepared in an eigenstate\nofSx:ψ=/divides.alt0m/uni27E9⊗/divides.alt00/uni27E9, where /divides.alt00/uni27E9represents bosonic vac-\nuum, the spin system will not undergo dynamics follow-\ning a quench, while the bosonic ancilla will ﬂuctuate be-\ntween/divides.alt00/uni27E9and/divides.alt0αm/uni27E9at the characteristic level splitting fre-\nquency. /divides.alt0αm/uni27E9is a coherent state centered at m2λ2\nωc. Once\nIsing interactions ( /divides.alt0J/divides.alt0>0) are included along with the\nnonlocal central ancilla coupling, the model is no-longer\nexactly solvable, but we anticipate in the low energy\nregime that same characteristic behavior captures the\nIsing-boson ground state and low energy quenches. The\nspin-ancilla interaction will shift the underlying ground\nstate and dynamical quantum phase according to the ef-\nfective mean-ﬁeld magnetic interactionλ2\nωc. Greater de-\ntails on the physics of the Ising-Dicke model and magnetic\nphase transition in the absence of a transverse ﬁeld can\nbe found in a recent work by Rohn et al. [73].\nV. ENTANGLEMENT DYNAMICS\nA. Long-time Average Entanglement\nThe spin-ancilla coupling makes the Ising Hamiltonian\nnonintegrable, and in studying its dynamics we employ\nexact diagonalization (ED) and a real-time evaluation of\nthe Schrodinger ODE. We study how a fully polarized,\nnon-equilibrium product state behaves when quenched by\nEqs.6,7. This study sheds light on how the spin-chain\nQPT evolves under a highly non-local coupling, how quan-\ntum ﬂuctuations in many-body systems lead to eventual\nequilibration, and how composite information dynamics\ndevelop as a result of this 2-body (chain-ancilla) construc-\ntion. We study the real-time dynamics of the full spin-\nancilla density matrix and measure spin-ancilla entan-\nglement, spin-spin mutual information, and the QFI con-\ntained in the spin reduced density matrix.\nWe initially prepare the Ising spin-chain and the an-\ncilla bosonic system in an unentangled product state\n/divides.alt0ψ(t=0)/uni27E9=/divides.alt0ψs/uni27E9⊗/divides.alt00/uni27E9with/divides.alt00/uni27E9representing bosonic vac-\nuum and an unentangled polarized state /divides.alt0ψs/uni27E9=/divides.alt0↑↑↑.../uni27E9. A\nquench is then performed at t=0 from the initial wave-\nfunction to (J=−1,h/slash.leftJ,λ)where we then vary handλ.\nFIG. 2. Entanglement metrics: Long-time average entangle-\nment: half-chain MI (a), spin-ancilla von Neumann entangle-\nment entropy SvN,A(b), spin ﬂuctuations ∆s2\n{x,z}(c), and QFI\nalong S{x,z}(d). Results presented as a function of transverse\nmagnetic ﬁeld h/slash.leftJandλfollowing a quench from unentangled\nspin-boson product state. (a) MI grows like the number of excita-\ntions present in the initial quench and saturates at high ﬁeld at\na value that scales with the system size. Entanglement satura-\ntion value decreases with increasing λand proﬁle shifts to lower\nﬁeld has ancilla modiﬁes DQPT. (b) Ancilla SvN,Apicks up the\nsignature of the dynamical phase transition with a peak near\nthe ﬁnite-size resolved critical point and a saturation entropy\nthat grows proportionally to λin the paramagnetic phase. (c)\n∆s2\nzgrows similar to the MI; initially zero in the product state\nand saturates to a constant density ∼3.∆s2\nzhas a ﬁnite-size\nresolved peak about the critical point, modiﬁed with growing λ.\n∆s2\nxremains constant as a function of handλ. Spin ﬂuctuations\nprovide an upper bound (Eq.3) for the respective QFI measures\n(d). Longitudinal and transverse fQmonotonically decreases as\na function of λ. (e) Finite size scaling in the paramagnetic phase\nh=2.0 ofSvN,A(q=20,λ2/slash.leftωc=2.0) and MI in the zero-coupling\nand strong coupling regime λ2/slash.leftωc=2.0. MI shows characteris-\ntic volume-law entanglement scaling even in the presence of the\nexternal environment while SvN,Ashows slow growth with sys-\ntem size and is accurately described by SvN,A∝log(L). Sys-\ntem size L=12,8 for (a-c, d), ﬁnite ancilla dimension q=40,\nJ=−1,ωc=0.5, and tJ∈[0,50].\nIn the non-equilibrium quench scenario ﬁrst ignoring\nthe ancilla λ=0, the half-chain MI transitions from 0\nath=0 toward a volume law entangled state across a\ncritical magnetic ﬁeld (Fig.2a.). Deep into the polarized\nphase, the initial state’s energy begins to lie within the\ncenter of the spectrum, essentially becoming a highly ex-\ncited pure state (Generalized Gibbs Ensemble GGE) with\nextensively scaling entropy. Excitations encoded into the\ninitial non-equilibrium state propagate and distribute en-\ntanglement [74]. The saturation in entanglement occurs6\nFIG. 3. Real-time Entanglement Dynamics: Entanglement metrics following a quench from the polarized state: MI(L/slash.left2),\n∆sz(t)2,fQ(Sz), and SvN;A. The three rows ((a,b,c) correspond to λ2/slash.leftωc={0,0.63,1.13}). (a) For λ=0, MI, spin-ﬂuctuations and\nQFI grow with time t∼L/slash.leftvmax=2.5/slash.leftJ, after which both oscillate about an average value that grows with ﬁeld and saturates above\nthe DQPT critical point. As λincreases, SvN;Agrows in magnitude and similarly oscillates like the characteristic period given by\nthe bosonic level spacing 12 .6tJ. The maximum ancilla entanglement similarly reaches a saturation threshold across the DQPT.\nIncreasing λincreases the initial MI and spin-ﬂuctuation growth rate but decreases saturation value for quenches into the polarized\nphase. The oscillations across all spin density matrix measures (most dramatic across the DQPT in red) are most rapid for times\nt≈τ=12.6tJ, the ancilla period. The feature of greatest interest in the growing difference between ∆sz(t)2and fQ(Sz). In (a) we\nsee that, up to numerical noise in measuring the diagonal operator Szover the spin reduced density matrix, the proﬁles are identical\nas expected. Moving to (b) and (c) the difference grows with λregardless of spin phase. System size L=10,d=30 and hvalues\n(0,0.75,1.5,2.25,3).\nconcurrently with the saturation in excitations as exhib-\nited by the steady-state domain wall count [18]. The\nentanglement saturation value is an interesting focus of\nstudy, where the scaling is indicative of the integrable or\nnonintegrable character of the model and additionally a\nsignature of many-body localized systems [75].\nSpin-ﬂuctuations and QFI (identical in the λ=0 limit)\nseen in Figs.2(b,d) depict the same entanglement growth;\nboth saturating for quenches across h/slash.leftJ>1. Though en-\ntropy and multipartite entanglement both depict exten-\nsively scaling entanglement in the saturated regime, the\nentanglement entropy has a smooth crossover, in agree-\nment with previous results for larger systems, while QFI\ndepicts a second order transition and peak about the\nground state critical point h=J. The ∆ˆS2\nzresults in\nFig.2b. agree with previous theoretical results in the\nparamagnetic regime that show ∆ˆS2\nz(h/slash.leftJ>1,t=∞)=\n3 [76]. Further discrepancies arise due to the difference\nin polarized vs. cat-state initial spin-state.\nWhen additionally quenching the system with nonzero\nλ, identical entanglement behavior is imprinted on the\nancilla entanglement entropy (Fig.2c.), where long-time\nentanglement grows toward the critical ﬁeld and satu-rates across the DQPT. The ancilla shifts the critical ﬁeld\nand at moderate coupling λ2/slash.leftωcdecreases the relative\nentanglement surrounding the critical point, observed in\nFig.2a. MI increases slightly in the ferromagnetic phase\nas the underlying ground state shifts to lower magnetic\nﬁelds. In the saturated phase h/slash.leftJ>1, MI and QFI\n(Fig.2a. and d.) are exclusively removed from the spin\nsubsystem, which leads to a monotonic decrease in the\nsaturation value with coupling. The decrease in spin-\nchain entanglement occurs concurrently with growing\nSvNbetween spins and ancilla. The most interesting ob-\nservation is that this loss is not observed in the spin ﬂuc-\ntuations (Fig.2c.), where the saturation value maintains a\nﬁxed value ∼3. It is intuitive to think that saturated en-\ntangled DoF in the paramagnetic quench share increas-\ning information with the ancilla with growing λ, remov-\ning previously saturated spin-spin entanglement and re-\ndistributing excitations between spin and ancilla. This\nis the same behavior observed in the spin-ﬂuctuations\nand domain-wall density proﬁles, which give credence\nto a possible relationship between excitations and envi-\nronmental entanglement. The most interesting discov-\nery is that MI continues to scale with system size while7\nspin-ancilla entanglement grows at a much slower rate\n∼log(L)regardless of coupling (Fig.2e.).\nThis picture persists even when varying the qudit size\n[see Supplemental for further details]. As the Hilbert\nspace ddecreases, the maximum amount of information\nentropy that can be stored between the spin and ancilla\ndecreases as log (d), so it is expected that a smaller ancilla\nwill have a similarly decreasing impact on entanglement\nwithin the spin-subsystem. This entanglement loss pic-\nture raises important questions: how can we numerically\nevaluate the spin-ancilla entanglement based off of mea-\nsurements on the spin system? How do different phases\nlimit or share stored information? How does this descrip-\ntion hold up in real-time?\nB. Real-time Numerical Results\nIn the non-equilibrium quench, the initial state has\ntrivial entanglement characteristics and reaches a max-\nimum on the order of the maximum quasiparticle veloc-\nityL/slash.leftvmaxwhere L=10 gives t=2.5J/slash.leftmin[1,h]. When\nλ=0 [Fig.3(a)] we see initial growth followed by oscilla-\ntions connected to the integrability of the model and ﬁ-\nnite size effects. As λincreases [Fig.3(b,c)], oscillations\nin MI, spin-ﬂuctuations, and QFI decrease on the order of\nthe spin-ancilla entanglement entropy proﬁle. More inter-\nestingly, we observe a deviation in the QFI proﬁles from\nthe spin ﬂuctuations, as compared to the exaggerated be-\nhavior shown in Fig.1(c). MI and QFI in the paramag-\nnetic quench decay from the λ=0 results with revivals\non the order of the cavity frequency ωc.∆Sz(t)2on the\nother hand maintains a strong average value for all times,\ntaking into account the shift in underlying critical ﬁeld.\nIn contrast to the decaying QFI proﬁle, ancilla entangle-\nment initially grows rapidly and subsequently decays, os-\ncillating at its characteristic timescale for quenches in the\nparamagnetic regime.\nAverage spin ﬂuctuations remain largely unchanged as\na function of λin the real-time evolution as well as at\nlong-times, while QFI shows a clear modiﬁcation in the\nnon-interacting to interacting ancilla limit. The develop-\ning discrepancy between ﬂuctuations and QFI in the pres-\nence of environmental entropy suggest a good ﬁxed point\nto understand how information is distributed between\nspins and the ancilla and how SvN;Aprovides greater in-\nsight in the the inequality in Eq.3.\nC. Quantifying Entanglement Loss\nIn the quench scenario, Ising spins entangle rapidly\nin time as observed by the development of MI and QFI.\nAs coupling with the ancilla increases we see a devia-\ntion from ﬂuctuations and multipartite entanglement like\nthe ancilla entanglement entropy proﬁle. Where ﬂuctua-\ntions remain consistent (Fig.2(b)), spin-spin QFI and an-\nFIG. 4. Multipartite Entanglement Loss: Real-time evolu-\ntion of the transverse and longitudinal (left, right) multipar-\ntite entanglement loss (MEL) (dotted) vs ancilla SvN(solid),\nλ2/slash.leftωc=0.28,0.63,1.13 (a,b,c) from polarized initial state /divides.alt0↑\n/uni27E9⊗N/divides.alt00/uni27E9. Curves offset vertically for clarity. Fisher information\nis reduced compared to the upper bound set by the ﬂuctuations\nalong both spin vectors, leading to the growth in MEL and an-\ncilla entanglement entropy. MEL(Sx)and MEL(Sz)provide\nextremely accurate estimates of the ancilla entanglement en-\ntropy for λ2/slash.leftωc<<Jregardless of underlying phase. With in-\ncreasing coupling, the amount of information gained by the an-\ncilla surpasses the multiparticle entanglement as measured by\nfQ(Sx), saturating MEL(Sx). As more entanglement is en-\ncoded in ˆSz,MEL(Sz)continues to provide a strong estimate\non the ancilla entanglement and is nearly exact in the param-\nagnetic, saturated phase h>J. System size L=10,d=30 and h\nvalues(0,0.75,1.5,2.25,3).\ncilla entanglement undergo highly nontrivial behavior as\na function of the system-environment entropy. We deﬁne\nmultipartite entanglement loss MEL(ρB,ˆO)to capture\nthe difference in observable ﬂuctuations from the true\nmultiparticle spin entanglement in the reduced density\nmatrix over the spin-chain ρB. Here we calculate MEL\nalong the longitudinal and transverse collective spin vec-\ntorSµ=∑iσµ\ni;µ∈[x,z]. These quantities were previously\nexplored using exact solutions to the Ising model and,\nthough may not represent Smaxas we have suggested in\nEq.5, provide two QFI measures that accurately capture\nthe dynamical Ising phases [76].\nFor weak ancilla couplings [Fig.4(a)] we see that both\nMEL(Sx)and MEL(Sz)serve to capture the real-time\nbehavior of the ancilla entanglement as well as accurately\ncapture the amplitude of information gain. As coupling\nincreases and the ancilla begins to sink more entangle-\nment from the Ising spins, SvN;Abegins to saturate the\nentanglement loss observed through MEL(Sx)Fig.4(b,c:\nleft). MEL(Sx)<eSvN;A−1 is most noticeable at half of\nthe ancilla level splittingπ\nωc=6.28tJ. Though MEL(Sx)\nserves as a poor estimate of spin-ancilla entanglement at\nlarge couplings and for h>J,SvN;A,MEL(Sz)proﬁles8\nremain in strong agreement especially in the paramag-\nnetic phase. The transition from MEL(Sx)∼MEL(Sz)∼\neSvN;A−1, to MEL(Sx)<MEL(Sz)∼eSvN;A−1 can be\nattributed to the larger spin ﬂuctuations in Sz. In the\nparamagnetic phase ∆S2\nz=3Land∆S2\nx=L, so as ancilla\nentanglement grows, the loss perceived by MEL(Sx)is\nartiﬁcially low since less entanglement is observed along\nthis direction. As ∆S2\nzis the direction of maximal ﬂuctua-\ntions/QFI, we anticipate that MEL(Sz)remains a strong\nancilla entropy estimate. This is indeed the case where\nin Fig.4(b,c: right) entropy remarkably traces MEL(Sz)\neven beyond λ2/slash.leftωc>J.\nIn all of our results SvN;Ais never parametri-\ncally larger than max [MEL(Sx,t),MEL(Sz,t)]with the\ngreatest discrepancy developing about the critical point.\nSurrounding the critical point we expect the transverse\nand longitudinal QFI to present an inaccurate charac-\nterization of information loss as maximal spin-variation\ngenerally lies in the x−zplane. On the other-hand, in\nthe large-ﬁeld limit, we expect this nonintegrable system\nto reach a relatively stable steady-state as a function of\nﬁeld and ancilla coupling, so MEL(Sz)should continue to\npaint an accurate picture. Deep in paramagnetic regime,\nwhere the system reaches an approximate inﬁnite tem-\nperature state and approaches proximate Sxconservation\n(J<h<λ2/slash.leftωc), QFI along Szwill continue to dominate.\nVI. DISCUSSION\nHere we have extended the scope of how informa-\ntion sharing of entangled degrees of freedom leads to a\nnearly direct connection between spin ﬂuctuations, Fisher\ninformation, and external entanglement entropy. Our\ncharacterization of MEL and the numerically observed\nproportionality with SvN;Aserves as an extension of\nthe exact results observed in quadratic bosonic/fermionic\nsystems [65, 66] and behavior seen in semiclassical,\ninﬁnite/long-range spin systems [43]. These results also\nagree with numerical sampling results over small Her-\nmitian matrices [67], and, here, due to the collective\nspin-boson interaction we connect this collection of works\nby ﬁnding MEL ∝SvN;A∝logL. We ﬁnd that spin\nsubsystem-ﬂuctuation measurements provide a strong es-\ntimate on establishing the number of entangled DoF even\nin composite systems. In Fig.4 we show that entangle-\nment can be stored directly in local spin ﬂuctuations when\nMEL=0 or it can be stored in shared degrees with an en-\nvironment MEL∼eSvN;A−1. This novel characterization\nnot only presents a qualitative framework that captures\nthe long-time equilibration dynamics in our composite\nspin-ancilla system but correlates directly with the real-\ntime entropy ﬂuctuations between spins and ancilla. We\nobserve that when the ancilla acts purely as a bath and\n∆S2\nµ(t,λ)≤∆S2\nµ(t,0), MEL perfectly captures entangle-\nment physics between spins and ancilla. In the highly en-\nergetic regime where the Ising chain is fully excited, the\nancilla can only serve as an effective bath, whereas in thelow-energy quenches for h<<J, the system can be highly\nexcited by ancilla interaction. In the low-ﬁeld quench,\nMEL tends to be parametrically larger than entangle-\nment which suggests a possible upper bound in Eq.5 may\nbe more accurate.\nOur results are in good agreement with recent works\nthat establish a similar picture of spin-ﬂuctuations and\nentanglement in semiclassical models. In the Dicke model\nand its corresponding regular to ergodic phase transi-\ntion it was shown that spin-boson entanglement simi-\nlarly is responsible for the development of additional col-\nlective spin-ﬂuctuations, and in the kicked Ising system\ncollective squeezing occurs in tandem with entanglement\nentropy between bipartitions [43, 45]. Though previ-\nous research presents an intuitive picture, there is no\nestablished/quantitative differentiation between the in-\nternal multipartite spin entanglement and the compos-\nite system-environment entanglement contribution in the\ndevelopment of physical correlations. Our results provide\na deeper heuristic examination into how ﬂuctuations and\nQFI relate in a simple, composite model.\nThis novel characterization of entanglement physics in\na composite spin-boson model provides an intuitive de-\nscription on how DoF are shared with different subsys-\ntems in real-time and helps provide a generalized picture\non information. We see that the proliferation of excita-\ntions that encode collective ﬂuctuation play a critical role\nin the strength of entanglement entropy response when\nprobed by an external, homogeneous ancilla. The col-\nlective interaction relays only semiclassical information\nSvN;A∝logLregarding the dynamics of the local, inter-\nacting quantum spin system and presents a novel trans-\nlation of information. Though this purely numerical in-\nvestigation focuses on highly excited states of the Ising\nmodel and a particular transverse coupling, this physics\nremains consistent in low energy quench scenarios about\nthe Ising ground state as a function of h/slash.leftJ[see Supple-\nmental for greater details].\nVII. CONCLUSION\nFrom our results, experimental work capable of mea-\nsuring global spin-ﬂuctuations and cavity/bosonic correla-\ntions can develop a richer characterization of the full in-\nformation dynamics of multiparticle, composite systems.\nWhere determining QFI in an arbitrary, non-equilibrium,\nmixed system requires full state tomography, our work\nshows that MEL provides a simple extension for pure\nstates by considering external and internal subsystem en-\ntanglement content. Using this method, we recreate ef-\nfective estimates for the amount of entangled information\ncontained in any subsystem of a pure quantum ensemble.\nWe anticipate our results to have an impact on experi-\nmental quantum information studies in cavity-QED and\nNV-center platforms, where the range of interaction con-\nsists of a macroscopic region of the system of interest.\nThis work focuses on adding an environment to the9\nwell-understood Ising model, where quantum thermal-\nization, entanglement, and spin ﬂuctuations have been\nthoroughly characterized. It would be an interesting fu-\nture pursuit to explore less well understood models, or\nfully nonintegrable models where the optimal QFI mea-\nsure is not known, and observing the entanglement con-\nsequences in a similar system-environment construction.\nExploring semiclassical models where analytic forms of\nQFI, MEL, and SvNcan be explored would provide a\nwealth of insight into the physics observed in this local\nquantum system. It is also necessary to investigate how\nthis semiclassical transfer of information breaks down\nwith either inhomogenous bath couplings or a larger/more\ncomplex environment and leads to full thermalization.\nWith regards to thermalization, we have interestingly\nseen that excitations and QFI are intimately related as\ndynamical quasiparticles encode ﬂuctuations, but at the\nsame time they seem to dictate the entanglement re-\nsponse to an external environment. It would be inter-\nesting to develop a more generalized ﬂuctuation dissipa-\ntion relations regarding entanglement susceptibility and\nexcitation exchange/production. As we have seen in our\nresults, MI remains largely preserved in the spin-system\nand only decreases by ∼logLcompared to the noninteract-\ning case. A similar phenomenon was observed in an open\nIsing chain where the system did not completely thermal-\nizeMI(L/slash.left2,t→∞)>0 [50]; so how does the form of the\nsystem-environment interaction preserve certain quasi-\nparticles/operators propagation and encode entanglement\nentropy within the system.\nOur work additionally brings up interesting metrics for\ndetermining how to engineer and transfer quantum infor-\nmation between subsystems; where we initially store or\ngrow multiparticle entanglement from the ground stateor non-equilibrium quench and then transform it into ex-\nternal entanglement entropy. Future work will explore\nthe genuine multiparticle entanglement mediated by an\nancilla and how it is bounded by the size of the ancilla\nHilbert space. A parallel investigation as to how infor-\nmation is scrambled with a weak non-local central vertex\nshows promise in understanding fast scrambling dynam-\nics, where no work has been done on relating operator\ngrowth rate and the size of the ancilla dimension [77]. In\nthe large cavity limit, cavity interactions generate all-to-\nall spin interactions and lead to tunable OTOC physics in\nthe spin subsystem [78], but no work has examined the\npossible ﬁngerprint imprinted on the central mode. This\nwould provide fruitful scrambling protocols and analyses\namenable to cQED and NV center experiments. Finally\nit would be interesting to see if topological signatures in-\nherent in quantum ﬂuctuations or full system entangle-\nment content leave unique signatures in the shared in-\nformation with an external observer. 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Szabo1and Nandini Trivedi1\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\nI. TFIC EXACT SOLUTION\nThe TFIC is exactly solvable following a Jordan-Wigner mapping to non-interacting spins and subsequent Fourier\nand Bogoliubov transformation.\nH=−J(/summation.disp\n/uni27E8i j/uni27E9σz\niσz\nj−h/summation.disp\niσx\ni) (1)\nσx\ni=1−2c†\nici (2)\nσz\ni=−/product.disp\nj<iσx\nj(c†\ni+ci) (3)\nHJW=J(/summation.disp\n/uni27E8i j/uni27E9c†\nicj+cicj+H.C.+h/summation.disp\ni1−2c†\nici) (4)\nThen Fourier transforming the fermionic operators gives us:\nd†\nk=1√\nLL\n/summation.disp\njei(2πjk)/slash.leftLc†\nj,\ndk=1√\nLL\n/summation.disp\nje−i(2πjk)/slash.leftLcj,\nHJW=J/summation.disp\nk([h−cos2πk\nL]d†\nkdk\n−i\n2sin2πk\nL[d−kdk+d†\n−kd†\nk]−h/slash.left2),\nb†\nk=ukd†\nk+ivkd−k,bk=ukdk+ivkd†\n−k,where\nuk=cosθk;vk=sinθk;tanθk=Jsink\nB−Jsink\nHBdG=/summation.disp\nk/epsilon1kb†\nkbk+/summation.disp\nk/epsilon1k\nThis gives us the exact spectrum for the transverse ﬁeld Ising model with ground state energy that can be character-\nized by integrating over the dispersion relation given as\n/epsilon1k=J√\n1+h2−hcosk. (5)\nThe ground state of the model can be determined from the empty fermionic vacuum state /divides.alt00/uni27E9∼/divides.alt0→→→→ .../uni27E9(in terms\nof the spins) and eliminating all momenta from the vacuum state: /divides.alt0Ψ0/uni27E9=Πkbkb−k/divides.alt00/uni27E9.\nThe initial state ( t<0) for a sudden quench experiment will occupy excited modes of the Hamiltonian for t>0.\nFor a sudden quench of the Hamiltonian we can consider that /divides.alt0Ψ0/uni27E9is the ground state of H0. For systems with an\nextensive number of conserved quantities i.e. integrable systems, the inﬁnite time steady state thermalizes according to\na Generalized Gibbs Ensemble [ ?]. This state can be decomposed into the rapidities or conserved quantities (occupation\nof the eigenmodes) of the time-evolution Hamiltonian, written asarXiv:2103.08517v3  [quant-ph]  4 Nov 20212\n0 1 2 3\nk01234k\n0.0\n0.3\n0.7\n1.0\n1.3\n1.7\n2.0\n2.3\n2.7\n3.0\n0 1 2 3\nk0.00.20.40.60.81.0vk=dk\ndk\n0.0\n0.3\n0.7\n1.0\n1.3\n1.7\n2.0\n2.3\n2.7\n3.0\nFIG. S1. TFIM energy spectrum (left) and group velocity for excitations (right) of eigenmode kas a function of transverse magnetic\nﬁeld h/slash.leftJ.\n/divides.alt0Ψ0/uni27E9=/summation.disp\nkc(k)/divides.alt0nk/uni27E9, (6)\nc(k)=/uni27E8Ψ0/divides.alt0nk/divides.alt0Ψ0/uni27E9, (7)\n/divides.alt0Ψ(t)/uni27E9=eiHt/divides.alt0Ψ0/uni27E9, (8)\n/divides.alt0Ψ(t)/uni27E9=/summation.disp\nkei/epsilon1ktc(k)/divides.alt0nk/uni27E9 (9)\nFor all occupied excited modes nkwe can think of an excitation with energy /epsilon1kbeing present within the spin chain.\nWith the excitation present at t=0, this will propagate at a ﬁnite velocity. The velocity of this excitation is given by the\ndispersion relation taken from the spectrum. This is identically what determines the velocity of entanglement growth\nin the integrable Ising model [ ? ?].\nv=d/epsilon1k\ndk(10)\nvmax=max[Jhsink√\n1+h2−2hcosk] (11)\n=Jmin[1,h] (12)\nII. FINITE SIZE AND ADDITIONAL ENTANGLEMENT LOSS RESULTS3\nFIG. S2. MEL dynamics from non-equilibrium quench (λ2/slash.leftJωc=0.125). Real-time evolution of the transverse and longitudinal\n(top, bottom) spin-ﬂuctuations (a,d),Q(b,e), and multipartite entanglement loss (MEL) (dotted) vs ancilla SvN(solid) (c,f) from po-\nlarized initial state. Rapid spin ﬂuctuations are suppressed on the order of ancilla entanglement proﬁle in both x,zspin components.\nFisher information is reduced compared to the upper bound set by the ∆s2\nµ(t)along the two spin vectors. (c,f) Taking the difference\nbetween ﬂuctuations and QFI provides a good estimate on the ancilla entanglement entropy. For weak coupling and numerical noise\nassociated with fQ(Sz),MEL(Sx)provides a good estimate for all spin phases. MEL and SvNcurves are offset vertically for clarity.\nSystem size L=8,d=40 and hvalues(0,0.4,0.8,1.2,1.6,2.0).\nFIG. S3. MEL dynamics from non-equilibrium quench (λ2/slash.leftJωc=2.0). Real-time evolution of the transverse and longitudinal\n(top, bottom) spin-ﬂuctuations (a,d), FQ(b,e), and multipartite entanglement loss (MEL) (dotted) vs ancilla SvN(solid) (c,f) from po-\nlarized initial state. Rapid spin ﬂuctuations are suppressed on the order of ancilla entanglement proﬁle in both x,zspin components.\nFisher information is reduced compared to the upper bound set by the ﬂuctuations along the two spin vectors. (c,f) Taking the dif-\nference between ﬂuctuations and QFI provides the MEL .MEL(Sx)provides a good estimate near J=0 (blue) and is most accurate\nnear the minima in SvN;A. About and across the DQPT, MEL(Sx), saturates and does not capture SvN;A.MEL(Sz)always upper\nbounds SvN;Aforh/slash.leftJ<1 and most accurately captures the entropy at the entropy maxima. MEL(Sz)nearly exactly traces SvN;A\nforh/slash.leftJ>1. MEL and SvNcurves are offset vertically for clarity. System size L=8,d=40 and hvalues(0,0.4,0.8,1.2,1.6,2.0).4\nFIG. S4. MEL dynamics following a quench from the Ising ground state J=1.0,h/slash.leftJ: (a, d, g) transverse spin-ﬂuctuations\n∆S2\nx; (b, e, h) Fisher information FQ; (c, f, i) multipartite entanglement loss (MEL) (dotted) vs ancilla SvN∶A(solid) for L=8,d=\n40,λ/slash.leftωc=0.18,0.5,1.125 (top, middle, bottom row). Spin ﬂuctuations are suppressed on the order of ancilla entanglement proﬁle,\nwhere Fisher information is reduced compared to the upper bound set by the ﬂuctuations. (c,f,i) Taking the difference between\nﬂuctuations and information provides MEL an accurate estimate of the ancilla entanglement entropy in all regimes. This breaks\ndown near the critical point and with increasing λ. MEL and SvNcurves are offset vertically for clarity. System size L=8,d=40\nandhvalues(0,0.4,0.8,1.2,1.6,2.0).\nFIG. S5. Finite size scaling of MEL and entanglement proportionality : We calculate the long time average of MEL and\nSvN;Aand determine the constant of proportionality between the two in regimes where they qualitatively agree: MEL (Sx,λ<<1.0),\nMEL(Sz,h/slash.leftJ≥1.0). Outside of these regimes MEL measures along these vectors do not accurately capture the entanglement proﬁle\nand the general spin-vector of maximal ﬂuctuations is not known. For small systems, the proportionality αgrows with system size\nαµ=1\n2Luntil L=8, where the system in the inﬁnite size limit is 4 −partite entangled. αxandαzare in strong qualitative agreement\nand provide nearly identical relations to SvN;Afor all sizes. We attribute the size dependence for small systems on ﬁnite size effects\nthat are similarly observed in tandem with the Fisher information density fQ.fQgrows with system size to the inﬁnite system size\nlimit fQ∼3 at roughly L=8 as seen in the main text and large system results provided in [ ?].5\nFIG. S6. Long time average features following a quench from the Ising ground state (left) and polarized non-equilibrium state\n(right). The spin-ancilla system is prepared in a product state at t=0 and quenched to regions of the Ising phase diagram J=−1,h/slash.leftJ\nwith an additional ancilla-spin coupling λ. Here λ2/slash.leftωc=0.72,L=12. (a,e) local transverse magnetization /uni27E8σx\ni/uni27E9and (e) nearest\nneighbor correlations /uni27E8σz\niσz\ni+1/uni27E9, (b,f) the 1 /slash.left2 chain mutual information, (c,g) the cavity occupation /uni27E8n/uni27E9, and (d,h) the von Neumann\nentanglement entropy between the qudit and spin-chain. Nearest-neighbor correlations in (e) agree with t-DMRG results presented\nin [?] Results are averaged over the window tJ=[0,50]."    },    {        "title": "2008.07308v2.Terahertz_spin_dynamics_driven_by_an_optical_spin_orbit_torque.pdf",        "content": "Terahertz spin dynamics driven by an optical spin-orbit torque\nRitwik Mondal,1,\u0003Andreas Donges,1and Ulrich Nowak1\n1Fachbereich Physik, Universität Konstanz, DE-78457 Konstanz, Germany\nSpin torques are at the heart of spin manipulations in spintronic devices. Here, we examine the\nexistence of an optical spin-orbit torque, a relativistic spin torque originating from the spin-orbit\ncoupling of an oscillating applied ﬁeld with the spins. We compare the eﬀect of the nonrelativistic\nZeeman torque with the relativistic optical spin-orbit torque for ferromagnetic systems excited by\na circularly polarised laser pulse. The latter torque depends on the helicity of the light and scales\nwith the intensity, while being inversely proportional to the frequency. Our results show that the\noptical spin-orbit torque can provide a torque on the spins, which is quantitatively equivalent to\nthe Zeeman torque. Moreover, temperature dependent calculations show that the eﬀect of optical\nspin-orbit torque decreases with increasing temperature. However, the eﬀect does not vanish in a\nferromagnetic system, even above its Curie temperature.\nI. INTRODUCTION\nInterest in controlling spins by means of circularly po-\nlarised pulses has grown immensely due to its potential ap-\nplications in spin-based memory technologies [1–3]. Apart\nfrom the heat-assisted spin manipulations, the spins can\nalsobe controlled using the inverse Faraday eﬀect (IFE)\n[4–6], the magnetic ﬁeld of the terahertz pulses [7–10] and\nan optical spin-orbit torque (OSOT) that does not impart\nangular momentum into the spin system [11, 12]. To be\nable to explain such eﬀects theoretically, one has to simu-\nlate spin dynamics including several nonrelativistic and rel-\nativistic eﬀects that might appear at ultrashort timescales\n[8–12].\nThe Landau-Lifshitz-Gilbert (LLG) equation of motion,\nconsisting of precession of a spin moment around an eﬀec-\ntive ﬁeld and a transverse relaxation, has been used ex-\ntensively in the past to simulate such spin dynamics [13–\n15]. However, for a spin system excited by ultrashort laser\npulses, a stochastic LLG equation of motion with atomistic\nresolutionisrequiredduetothestrongthermalﬂuctuations\nin order to study the dynamical processes [16–20]. In these\nequations, a stochastic ﬁeld is added to the eﬀective ﬁeld,\nin order to quantify the thermal ﬂuctuations in the spin\nsystem. Nonetheless, at ultrashort timescales, the exact\nform of the LLG equation of motion has to be questioned\nas several other relativistic phenomena can occur. There-\nfore, we seek for an equation of motion that can capture all\nthe possible interactions during ultrashort laser excitations\nof a spin system.\nIn a previous work, starting from the relativistic Dirac\nHamiltonian yet including the magnetic exchange interac-\ntions, a rigorous derivation of the LLG equation of motion\nhas been provided [21]. To treat the action of a laser pulse\nand the corresponding interactions, the Dirac-Kohn-Sham\nequation with external magnetic vector potential was con-\nsidered [22–24]. To this end, a semirelativistic expansion\nof the Dirac-Kohn-Sham Hamiltonian that includes several\nnonrelativistic and relativistic spin-laser coupling terms\nwas derived [25]. Having these coupling terms, an extended\nequation of motion that includes not only the spin preces-\nsion and damping, but also other relativistic torque terms\n\u0003ritwik.mondal@uni-konstanz.dewas obtained [26]. One of these torque terms is the ﬁeld-\nderivative torque which appears due to the time-dependent\nﬁeld excitation, e.g., in case of THz pulse excitation [27].\nAnother new torque term is the OSOT, which stems from\nthe spin-orbit interaction of the applied ﬁeld with the elec-\ntron spins i.e., it imparts spin-angular momentum of the\napplied ﬁeld to the spins. The new equation of motion for\nthe reduced magnetization vector mi(t)including OSOT\ncast in the form of an LLG equation is\n@mi\n@t=\u0000\rmi\u0002\u0000\nBe\u000b\ni+BOSOT\u0001\n+mi\u0002\u0012\nD\u0001@mi\n@t\u0013\n: (1)\nWe deﬁne the gyromagnetic ratio as \randD, representing\nthe damping parameter, is in general a tensor. For sim-\npliﬁcation, we consider only a scalar damping parameter \u000b\nthat can be expressed as \u000b=1\n3Tr(D)[21]. The eﬀective\nﬁeld is represented as Be\u000b\ni, which is the derivative of the\ntotal magnetic energy without the relativistic light-spin in-\nteraction with respect to mias will be speciﬁed later on\nin detail. The additional ﬁeld BOSOTdescribes the opto-\nmagnetic ﬁeld induced by the laser pulse due to the OSOT\nphenomenon.\nInthisarticle, weinvestigatetheeﬀectoftheOSOTterm\nwithin atomistic spin dynamics simulations. First, we re-\nvisit the derivation of the OSOT from a general spin-orbit\ncoupling Hamiltonian that has been derived within a rel-\nativistic formalism [21]. We ﬁnd that the OSOT depends\non the helicity, frequency and intensity of the light pulse.\nIf the intensity is high and frequency is low, we expect the\nOSOT terms to show the most signiﬁcant eﬀects. We sim-\nulate the spin dynamics for a spin model, representative for\nbcc Fe, with the OSOT and ﬁnd that the OSOT can pro-\nvide signiﬁcant contributions at the THz regime. Studying\nthe temperature dependence, we ﬁnd that the OSOT ef-\nfects is robust against thermal ﬂuctuations, i.e., we observe\nno signiﬁcant reduction of the strength of the OSOT even\nup to the critical temperature.\nII. OPTICAL SPIN-ORBIT TORQUE\nThe generalized spin-orbit coupling (SOC) Hamiltonian,\nas derived within the fully relativistic Dirac framework, canarXiv:2008.07308v2  [cond-mat.mes-hall]  15 Feb 20212\nFigure 1. (Color Online) ZT and OSOT for a single spin (big\nred arrow) excited by an elliptically polarised laser pulse. The\nﬁelds and torques are represented by yellow and blue arrows,\nrespectively. Due to the presence of elliptical polarisation, the\nZF and ZT are drawn as blurry.\nbe written as [21, 23, 26]\nHSOC=\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot];\n(2)\nwhere the electric ﬁelds are represented as Etot=Eint+\nEextandEext=\u0000@A\n@t\u0000r\bwith(A;\b)asmagneticvector\nand scalar potentials. The physical constants have their\nusual meanings and \u001bdenotes the electron’s spin through\nPauli spin matrices and pis the electron momentum.\nNote that the SOC can occur in several ways, as de-\nscribed in the following: (i) the angular momentum of an\nelectron couples to the spin of the electron — this can be\nexpressed as\u001b\u0001(Eint\u0002p), (ii) the ﬁrst-order magnetic vec-\ntor potential of the electromagnetic (EM) ﬁeld couples to\nthe spins — this can be expressed as \u001b\u0001(Eint\u0002A), (iii)\nthe spin angular momentum of the EM ﬁeld couples to the\nspin — this can be expressed as \u001b\u0001(Eext\u0002A).\nIn a spherically symmetric potential, the ﬁrst type SOC\ncan be written as traditional l\u0001scoupling, where lands\nrepresent the orbital and spin angular momentum respec-\ntively. It provides explanations to several relativistic eﬀects\nin magnetism e.g., magnetic Gilbert damping [21, 28, 29]\nand many others [30, 31]. Such a SOC exists even without\nan external ﬁeld. In contrary, the latter two types of SOC\ndepend explicitly on the external ﬁeld and can be written\nas\nH0\nSOC=e2~\n4m2c2\u001b\u0001[(Eint+Eext)\u0002A]:(3)\nThe total internal ﬁeld depends on the intrinsic ﬁeld E0\nint\nthat even exist without the applied ﬁeld and the applied\nﬁeld itself. Within the linear response theory we write\nEint=E0\nint+\u0010Eext, where\u0010deﬁnes the coupling strength\nof the applied EM ﬁeld relative to the intrinsic one. We\nthus consider the optical spin-orbit coupling as\nHOSOC =e2~(1 +\u0010)\n4m2c2\u001b\u0001(Eext\u0002A): (4)\nUsing the deﬁnition of Zeeman coupling of \u0000\u0016s\u001b\u0001BOSOT,\nthe optical spin-orbit couplingcan be shown to induce a ﬁeld (see Appendix A for de-\ntails)\nBOSOT =e2~(1 +\u0010)\n4m2\u0016sB2\n0\n!sin\u0011^ex; (5)\nwhereB0(=E0=c),\u0011and!are the envelope of the oscillat-\ningZeemanﬁeld, helicityandangularfrequencyoftheellip-\ntically polarised light, respectively. \u0016sdeﬁnes the magnetic\nmoment. Note, that due to the OSOT, the induced ﬁeld\npoints along the direction of the energy ﬂux of the prop-\nagating wave. Additionally, note that the ﬁeld BOSOTis\nlargest for circularly polarised light ( \u0011=\u0006\u0019=2). However,\nif the coupling strength \u0010is not the same for right and left\ncircularly polarised light, their combined eﬀect could lead\nto nonzeroBOSOTfor linearly polarised light. The param-\neter\u0010depends on the electron density and absorption of\nthe light that can be diﬀerent for right and left circularly\npolarised light, leading to a magnetic circular dichroism\n(MCD). In the following, we simulate the Zeeman eﬀect\nfrom the electromagnetic THz ﬁeld and the optical spin-\norbit coupling eﬀects simultaneously, and make a compar-\nison between these two eﬀects in a ferromagnetic system.\nIII. ATOMISTIC SPIN SIMULATIONS\nInordertocomputethespindynamics, itisconvinientto\ntransformtheimplicitformofourequationofmotiontothe\nexplicit Landau-Lifshitz (LL) form. For a scalar damping\nparameters, \u000b, the LLG equation (1) can be recast as\n@mi(t)\n@t=\u0000\r\n(1 +\u000b2)mi\u0002\u0000\nBe\u000b\ni+BOSOT\u0001\n\u0000\r\u000b\n(1 +\u000b2)mi\u0002\u0002\nmi\u0002\u0000\nBe\u000b\ni+BOSOT\u0001\u0003\n:(6)\nThe eﬀective ﬁeld, Be\u000b\ni, is the derivative of total en-\nergy with respect to the magnetic moment, Be\u000b\ni=\n\u00001\n\u0016is@H=@mi. The LLG equation, Eq. (6), hereby consists\nof the so called ﬁeld-like (see also Fig. 1) and the weaker\ndamping-like torque which is proportional to the Gilbert\ndamping coeﬃcient \u000b= 0:01and describes the coupling of\nthe spins to a heat bath.\nIn the following we consider a spin model for bcc Fe. The\ntotal Hamiltonian of the system (without the relativistic\nspin-light coupling term) Hcan be expressed as\nH=\u0000X\ni<jJijmi\u0001mj\u0000X\nidz(mz\ni)2\n\u0000\u0016sB(t)\u0001X\nimi: (7)\nThe ﬁrst term describes the traditional Heisenberg ex-\nchange energy, considering the exchange constants Jijup\nto the third nearest neighbour. The second term rep-\nresents the uniaxial anisotropy with energy constant dz\nand the last term is the Zeeman coupling with a time-\ndependent ﬁeld B(t) =Rh\nB0exp\u0010\n\u0000t2\n2\u001c2\u0000i!t\u0011i\n. The el-\nlipticity of the applied pulse is taken into account through\nB0=B0p\n2\u0000\n^y+ ei\u0011^z\u0001\n, which considers the major and mi-\nnor axes of the ellipse and \u001cdeﬁnes the pulse duration. In3\nthe following we use the abbreviations Zeeman ﬁeld (ZF),\nZeeman torque (ZT), and optical spin-orbit ﬁeld (OSOF).\nA striking diﬀerence between ZT and OSOT is that the\nZT does not depend on the frequency of the pulse and is\nproportional to the amplitude of the applied ﬁeld pulse.\nHowever, the OSOT depends inversely on the frequency\nand is proportional to the square of the ﬁeld amplitude,\ni.e. the intensity. To quantify the OSOT eﬀects, one can\nestimateitsamplitudebycomputingthecharacteristicﬁeld\nconstant of the OSOF\n~B(!;\u0010) =4m2\ne\u0016s!\ne2~(1 +\u0010): (8)\nConsidering a central frequency of f= 1 THz, we ﬁnd that\n~B(2\u0019f;0) = 157 T . Hence an electromagnetic ﬁeld ampli-\ntudeofabout 10 TcouldinduceaOSOFofaround1T,even\nfor the low limit of the coupling strength, \u0010!0. Needless\nto mention that the OSOT eﬀects might exceed the ZT if \u0010\nis higher. However, the results shown here were computed\nfor the limit \u0010!0. Such ultra-intense THz ﬁelds are cur-\nrently only achievable with linear polarisation, with recent\nexperiments pointing towards achieving circular polarized\nlaser pulses in the terahertz frequency regime [32, 33].\nIV. APPLICATION TO FERROMAGNETS\nIn our following simulations, we consider bcc Fe as a fer-\nromagnetic sample. For the zero-temperature simulations\nit is suﬃcient to solve the LLG equation for a single spin,\ndue to the homogeneity of the THz pulse excitation. The\nchoice of exchange parameters is only relevant at elevated\ntemperatures. We shine an electromagnetic pulse having\n1 pspulse duration along the ^x-direction with yandzﬁeld\ncomponents as shown in the Fig. 1. We will mainly discuss\nthe limitdz\u00190, since the uniaxial anisotropy does not\nhave a signiﬁcant impact on the ultrafast spin dynamics\n(see Appendix B for details).\nA. Zero temperature simulations\nIn particular we compare the eﬀect of ZT and OSOT at\nzero temperature. As the optical pulse is applied along the\n^x-direction (k=jkj^x), we calculate the change in mag-\nnetization for yandzcomponents in Fig. 2(a). For the\ncase of only ZT the induced OSOT remains obviously zero.\nFor a maximum of 10 Tapplied ZF, the change in magne-\ntization is about 50% for the y-component and 5% for the\nz-component. The reason for the triggered magnetisation\ndynamics is the following: the ZF from the optical pulse\nhas two components ByandBz. TheBycomponent exerts\na torque along\u0000^xon the equilibrium spin direction:\n\u0001m/m0^z\u0002By^y=\u0000m0By^x; (9)\nwithequilibriummagnetisation m0. Ontheotherhand, the\nBzcomponent of the EM ﬁeld does not exert any torque\non the Fe spins at ﬁrst as the spins are initially aligned\nalong ^z. However, as soon as the above mentioned torque\ninduced some magnetisation \u0001mx, theBzcomponent ofthe EM pulse, one quarter period later, exerts a torque\nalong ^ydirection:\n\u0001m/\u0001mx^x\u0002Bz^z=m0ByBz^y:(10)\nTherefore, there exists a superposition between two torques\nfor the ZF. Note that the change in \u0001mymagnetization is\nantisymmetric in the helicity of light, which suggests that\nthe eﬀect is similar to IFE [34]. Further note that the\nab initio calculations of the IFE were calculated using a\nnonrelativistic theory [35, 36]. However, present theory is\nbased on a relativistic interaction Hamiltonian. According\nto Eq. (5), the induced ﬁeld diverges in the limit !!0,\nwhich can be explained by the previous theories of IFE [37–\n39], and is in accordance with the ab initio calculations of\nIFE [35, 36, 40]. We would also like to mention that these\nab initio calculations showed the IFE being asymmetric in\nhelicity for ferromagnets which includes the absorption of\nlight [36].\nThe OSOF, on the other hand, has only one component\nBxalong ^xdirection, seeFig.2(b). Thus, itexertsatorque\nalong ^y-direction\n\u0001m/m0^z\u0002Bx^x/m0B2\n0^y: (11)\nand changes the magnetization by about 50 %as shown in\nFig. 2(b). According to our theory in Eq. (5), the right\nand left circular polarization will induce an OSOF along\nthe+^xand\u0000^xdirections respectively. Therefore, theright\ncircular pulse exerts a torque along ^y(viz. ^z\u0002^x=^y) and\nsimilarly, the left circular pulse exerts a torque along \u0000^y.\nThe change in magnetization is also antisymmetric in the\nhelicity of the light pulse, similar to ZT eﬀects in \u0001my.\nNote that we have assumed that the material dependent\nparameter\u0010is zero, which dictates that the antisymmetric\nbehavior is not universal. In fact, the parameter \u0010could be\ndiﬀerent for right and left circular pulse, meaning that the\nantisymmetry of the plots above would be broken [36].\nThe eﬀective contributions of the combined ZT and\nOSOT have been computed in Fig. 2(c). We note that\nthe magnetization change in myis opposite in helicity for\nZT and OSOT (e.g., see ﬁfth row plots in Figs. 2(a) and\n2(b)). Therefore, the ﬁnal eﬀective contribution becomes\nonly about 5% change in \u0001my. For thez-component of\nmagnetization, the eﬀective \u0001mzis negligibly small, even\nthough individual changes are about 5% due to ZT and\nOSOT.\nTo quantitatively understand the ZT and OSOT eﬀects,\nwe compute the ﬁeld dependent contributions in terms\nof the maximum change of each magnetization compo-\nnents \u0001mias a function of the applied ﬁeld B0for left\ncircular polarised light, shown in Fig. 3. From a sim-\nple scaling argument we would expect the spin excitation\n\u0001m?= \u0001mx^x+ \u0001my^yscaling with the magnitude of\nBEMandBOSOF, respectively. For the EM ﬁeld, this is\nsimply the amplitude BEM/B0, whereas for the OSOF,\nthat isBOSOF/B2\n0according to Eq. (5).\nAs already described above, Eq. (9), the spin excitation\n\u0001mxfor the EM ﬁeld is induced by the ﬁeld-like torque\nfrom itsBycomponent, thus, Fig. 3 (a) shows a linear scal-\ning with the EM ﬁeld amplitude B0. The \u0001mycomponent\nin Fig. 3 (b), however, is induced in a second order process,4\n0.5\n0.00.5BOSOF,x [T](a) Effect of only ZT\nLeft\nRight(b) Effect of only OSOT\nLeft\nRight(c) Effect of ZT + OSOT\nLeft\nRight\n5\n05BZF,y [T]\n5\n05BZF,z [T]\n0.2\n0.00.2mx [s]\n0.2\n0.00.2my [s]\n4\n 2\n 0 2 4 6\nTime [ps]0.981.00mz [s]\n4\n 2\n 0 2 4 6\nTime [ps]4\n 2\n 0 2 4 6\nTime [ps]\nFigure 2. (Color Online) The dynamical eﬀects of ZT and OSOT for Fe at an maximum applied ﬁeld of 10 T. The calculations\ninclude (a) onlyZT, (b) onlyOSOT and (c) both the ZT and OSOT. The ﬁrst row shows the applied ZF which has yandz\ncomponents, the second row represents the induced OSOF which has only an x-component. The other three rows show the change\nin magnetization along x,yandzcomponents respectively.\ndescribed in Eq. (10) and thus scales quadratically with\nthe EM ﬁeld amplitude B0, i.e., \u0001my/B2\n0. A deviation\nof this scaling law is observed at lower ﬁeld amplitudes of\nB0.100 mT, where the damping-like torque of the EM\nﬁeld with \u0001my/\u000bB0is taking over. This eﬀect can also\nbe seen by comparing the torque amplitudes, which diﬀer\nby a factor of \u0001my=\u0001mx\u0019\u000bin this regime. The \u0001mz\ncomponent, i.e., the deviation of the equilibrium compo-\nnent then follows from conservation of magnetization am-\nplitude: \u0001mz=m0\u0000p\nm0\u0000\u0001m2\n?. At low excitation the\ndominant contribution here is the \u0001mxcomponent and one\ncan simplify \u0001mz\u0019\u0000\u0001m2\nx=2m0/\u0000B2\n0.\nIn contrast to the EM ﬁeld, there is only one ﬁeld-like\ntorque acting along the +^ydirection due to the OSOF in\n^xdirection. This torque can be seen in Fig. 3 (b), and\nshows the quadratic scaling \u0001my/\u0000Bx/\u0000B2\n0implied\nby Eq. (11). The \u0001mxexcitation in Fig. 3 (a) on the other\nhand is due to the damping-like OSOT and thus following\nthe same power law, but a factor of \u000bsmaller compared\nto the ﬁeld-like excitation in Fig. 3 (b). Finally, the \u0001mzexcitation of the OSOT follows again from conservation of\nmagnetization amplitude, leading to \u0001mz\u0019\u0000\u0001m2\ny=2m0\nand implying \u0001mz/\u0000B4\n0as shown in Fig. 3 (c).\nThese scaling observations are once more summarized in\nTab. I where we display the scaling exponents obtained\nby ﬁtting our simulation data. An interesting observation\nhere is that although the characteristic ﬁeld of the OSOT\nis on the order of 100 Tin this frequency regime, due to the\ndiﬀerent torque symmetry compared to the ZT, the OSOT\nis by no means negligible. For the \u0001myexcitation we ﬁnd\nthat the strength of ZT and OSOT are comparable, though\nopposite in sign, at ﬁelds on the order of 102mT—a value\nthat is in much closer reach experimentally. Altogehter,\nthe OSOT eﬀects can thus provide an equivalent torque\ncompared to the Zeeman eﬀect. Therefore, the OSOT ef-\nfects cannot be neglected when a circularly polarised light\nacts on a magnetic system even at the weak coupling limit\n\u0010!0.\nUp to now we did not take into account the role of a ﬁ-\nnite anisotropy. In order to investigate this, we performed5\n102\n101\n100101\nB0[T]106\n105\n104\n103\n102\n101\nmi, max [s]\nmx\nZT\nOSOT\n102\n101\n100101\nB0[T]\ndamping\ntorque2ndorder\nfield torque\nmy\nZT\nOSOT\n102\n101\n100101\nB0[T]\nmz\nZT\nOSOT\nΔmy<0Δmy>0(a) Effect ofonlyZT (b) Effect ofonlyOSOT (c) Effect of ZT + OSOT\nFigure 3. (Color Online) Maximum of the magnetization change as a function of applied Zeeman ﬁeld for the application of circular\npolarised THz pulses (absolute values). Lines are ﬁts to the data according to a single (double) power law, see Tab. I for coeﬃcients.\nTable I. Fit parameters to a scaling function fN(B0) =PN\nn=1ay;n(B0=T)\fi;nwhereai;nis given in units of \u0016s. For\nthe ZF ﬁt of \u0001mya double power law N= 2has been used,\nwhereas for the remaining ﬁts a monomial N= 1was suﬃcient.\nZF OSOF\nn a i;n \fi;n ai;n \fi;n\n\u0001mx1 1:98\u000210\u000021:00 9:90\u000210\u000052:00\n\u0001my1 8:80\u000210\u000040:96 1:98\u000210\u000032:00\n2 1:74\u000210\u000032:09 - -\n\u0001mz1 2:03\u000210\u000042:03 1:96\u000210\u000064:00\nthe same calculation depicted in Fig. 2 with the anisotropy\ndz= 7:66µeVfor Fe included. We found that the main ef-\nfect of the anisotropy ﬁeld is the precession of the induced\nmagnetization \u0001m?around the z-axis over time (see Ap-\npendix B for details). On the other hand, no direct eﬀect of\nthe anisotropy can be observed on the ultrafast time scales,\ni.e., on the time scale of the pulse duration.\nB. Finite temperature simulations\nFor the ﬁnite temperature simulations, the exchange pa-\nrameters of Pajda et al.[41] are used, where the ﬁrst two\nnearest neighbors are strongly ferromagnetically, however,\nthe third nearest neighbor is weakly antiferromagnetically\ncoupled. For these simulations, we also use the uniaxial\nanisotropy of dz= 7:66µeValong thez-axis, in order to\nalign the magnetization. Calculating the temperature de-\npendence of the OSOT for ferromagnetic Fe, we use a sim-\nulation grid of 483spins and add a stochastic ﬁeld to the\neﬀective ﬁeld in Eq. (6), in order to treat the thermal ﬂuc-\ntuations [17]. The calculated Curie temperature of this sys-\ntem isTC= 1368 K and thus slightly higher than the true\nvalue of Fe. However, since the Curie temperature is the\nonly temperature scale in our simulations, we can basically\ntreat it as a free scaling parameter.\nWe compare in Fig. 4 the OSOT eﬀects at T= 0and\nT= 0:73\u0002TC(0K and 1000K). We ﬁnd in Fig. 5 that\nthough the OSOT eﬀect in \u0001miappears to decrease withincreasing temperature, this reduction is only related to\nthe thermal reduction of magnetic order meq(T)at ﬁnite\ntemperature. In other words the rotation angle of the nor-\nmalized magnetization is not sensitive to the temperature.\n0.5\n0.00.5Bx\nOSOF (T)Left\nRight\n2.5\n 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5\nTime (ps)0.2\n0.1\n0.00.10.2my (s)\nT=0\nT=0.73×TC\nFigure 4. (Color Online) OSOT eﬀects at ﬁnite temperature.\nTop panel: the OSOT-induced ﬁeld. Bottom panel: the cor-\nresponding spin dynamics for the change of myatT= 0and\nT= 0:73\u0002TC(0K and 1000K).\nTo illustrate this further, we have systematically calcu-\nlated the temperature dependence of the OSOT by cal-\nculating the diﬀerence between maximum spin excitation\nfor right and left circular pulses in Fig. 5. For each tem-\nperature, we performed ten simulations for each circular\npulses, and took the average to determine max [\u0001mR;y]\u0000\nmax [\u0001mL;y]as a function of temperature, which ensures\nthatthethermalﬂuctuationsareminimised. Farawayfrom\nTCthe spin excitation amplitude is proportional to the\nequilibrium magnetization curve. Only in the close vicinity\nof the critical temperature, we ﬁnd an increase of the net\nspin excitation, relative to the equilibrium magnetisation.\nThis is simply due to the large amplitude of both ZF and\nOSOF which induce a transient magnetic order.6\nFigure 5. (Color Online) Temperature dependence of OSOT,\nby the diﬀerence between the maximum changes of mydue to\nright and left polarised light pulses at maximum Zeeman ﬁeld\namplitude of 10 T. For comparison, the right axis shows the\nequilibrium magnetization.\nTherefore, OSOT eﬀects should be observed for bcc Fe\neven at elevated temperatures i.e., at realistic conditions\nfor ultra-intense spin excitations.\nV. CONCLUSIONS\nTo conclude, we incorporated a new torque into the LLG\nequation that should appear in ultrafast spin dynamics,\nnamely OSOT, and investigated this eﬀect via computer\nsimulations of an atomistic spin model, representative for\nbcc Fe. The OSOT originates from the spin-orbit coupling\nof the electron spins to an external EM ﬁeld. The strength\nof this OSOT, unlike the ﬁrst-order ZT, depends on the\nintensity of the ultrafast light pulse, as well as on the fre-\nquency. Inaddition, theOSOTdependsontheellipticityof\nthepulseanditprovidesmaximumtorqueforcircularlypo-\nlarised light pulses. Throughout the simulations presented\nhere, we considered the weak coupling limit \u0010!0. How-\never, the coupling depends on the electronic conﬁgurations\nof the system and could potentially further increase the\nstrength of the OSOT. Although the OSOT is a higher or-\nder contribution in the external ﬁeld, we found that the ZT\nandtheOSOTprovidequantitativelyequivalenttorqueson\nthe spins for circularly polarised laser pulses in the mag-\nnetization component perpendicular to the k-vector and\nequilibrium magnetisation m0. The eﬀect of the OSOT\nresembles an already known eﬀect, namely the IFE and\nit can be considered as a relativistic contributions to the\nIFE. The temperature dependence study of OSOT shows\nthat the OSOT eﬀect is present at elevated temperatures,\neven up to the Curie temperature.\nACKNOWLEDGMENTS\nWe thank László Szunyogh for valuable discussions and\nacknowledge ﬁnancial support from the Alexander vonHumboldt-Stiftung, Zukunftskolleg at Universität Kon-\nstanz via grant No. P82963319 and the Deutsche\nForschungsgemeinschaft via NO 290/5-1.\nAppendix A: Derivation of optical spin-orbit torque\nFollowing Eq. (4), the induced ﬁeld can be written as\nBOSOT =\u0000e2~(1 +\u0010)\n4m2c2\u0016s(Eext\u0002A):(A1)\nWeusethetimedependentﬁeldas Eext=R\u0000\nE0ei(k\u0001r\u0000!t)\u0001\nand the amplitude as the elliptically polarised light E0=\nE0p\n2\u0000\n^y+ ei\u0011^z\u0001\n, with\u0011as the ellipticity of the light. There-\nfore,theelectricﬁeldcanbewrittenas(whenonlythetime-\ndependent part is taken) Eext=@A\n@t)A=\u0000R\nEextdt\nthat can be calculated as follows\nA=\u0000R\u0014\nE0Z\ne\u0000i!tdt\u0015\n=\u0000R\u0014\niE0e\u0000i!t\n!\u0015\n:(A2)\nTherefore, the induced ﬁeld can be taken from Eq. (A1)\nas\nBOSOT =e2~(1 +\u0010)\n4m2c2\u0016s!R[i(E0\u0002E?\n0)]\n=e2~(1 +\u0010)\n4m2c2\u0016s!E2\n0sin\u0011^x\n=e2~(1 +\u0010)\n4m2\u0016s!B2\n0sin\u0011^x (A3)\nIn the last step of the calculation, we used the relation\nE0=cB0. In our simulations, we apply time-dependent\nZeeman ﬁelds along yandz-directions and the correspond-\ning induced optical spin-orbit ﬁeld acts along x-direction.\nAppendix B: Eﬀect of anisotropy\nHere, we compute the inﬂuence of uniaxial magnetic\nanisotropy on the spin dynamics induced by the ZT and\nOSOT. Fig. 6 shows the magnetization dynamics taking\ninto account the uniaxial anisotropy for bcc Fe. The main\neﬀect of anisotropy can be noticed by comparing with the\nzero-anisotropy simulations in Fig. 2. A small increase in\n\u0001mxis observed in the case of only ZF or OSOF after the\npulse has passed. This is due to the slow precession of the\ninduced \u0001mycomponentwhichstartstoprecessaroundthe\nanisotropy ﬁeld along ^zaxis. In case of the superposition\nof ZF and OSOF the excitation of \u0001myis mostly compen-\nsated and therefore no \u0001mxemerges either. Additionally,\nwe mention that the anisotropy energies do not aﬀect the\n\u0001myand\u0001mzon the ultrafast time scale and the net ex-\ncitation remains the same irrespective of the anisotropy —\nat least for the typically weak magnetic anisotropies of the\n3d ferromagnets.7\n0.5\n0.00.5BOSOF,x [T](a) Effect of only ZT\nLeft\nRight(b) Effect of only OSOT\nLeft\nRight(c) Effect of ZT + OSOT\nLeft\nRight\n5\n05BZF,y [T]\n5\n05BZF,z [T]\n0.2\n0.00.2mx [s]\n0.2\n0.00.2my [s]\n4\n 2\n 0 2 4 6\nTime [ps]0.981.00mz [s]\n4\n 2\n 0 2 4 6\nTime [ps]4\n 2\n 0 2 4 6\nTime [ps]\nFigure 6. (Color Online) The dynamical eﬀects of ZT and OSOT have been calculated for Fe including the uniaxial anisotropy at an\napplied ﬁeld of 10 T. The calculations have been performed accounting (a) onlyZeeman eﬀect, (b) onlyOSOT eﬀect and (c) both\nthe Zeeman and OSOT eﬀects. In all the plots, the ﬁrst row represents the induced spin-orbit ﬁeld which has only x-component, and\nthe rows two and three show the applied ZF which has yandzcomponents. The other three rows show the change in magnetization\nalongx,yandzcomponents respectively. The orange and blue colors represent the action of right and left circular pulses.\n[1] A. V. Kimel, A. Kirilyuk, and Th. Rasing, Laser Photon.\nRev.1, 275 (2007).\n[2] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n[3] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev,\nA. M. Balbashov, and Th. Rasing, Nature 435, 655 (2005).\n[4] R. John, M. Berritta, D. Hinzke, C. Müller, T. San-\ntos, H. Ulrichs, P. Nieves, J. Walowski, R. Mondal,\nO. Chubykalo-Fesenko, J. McCord, P. M. Oppeneer,\nU. Nowak, and M. Münzenberg, Sci. Rep. 7, 4114 (2017).\n[5] S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uh-\nlíř, L. Pang, M. Hehn, S. 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B 64, 174402 (2001)."    },    {        "title": "1701.05471v1.Spin_separation_and_exchange_for_quantum_dots_in_the_Overhauser_field.pdf",        "content": "arXiv:1701.05471v1  [cond-mat.mes-hall]  19 Jan 2017Spin separation and exchange for quantum dots in the Overhau ser ﬁeld\nJ. Leśnicki and B. Szafran\nAGH University of Science and Technology,\nFaculty of Physics and Applied Computer Science,\nal. Mickiewicza 30, 30-059 Kraków, Poland\nWe describe the spin and charge dynamics of the system of two e lectrons conﬁned within a\ndouble quantum dot deﬁned in a quantum wire. The spin dynamic s is driven by the electron\nmotion in presence of the spin-orbit interaction and the ran domly varying local Overhauser ﬁeld\ndue to the nuclear spins. The Schroedinger equation is solve d with the time-dependent conﬁguration\ninteraction method that allows for an exact description of t he system dynamics. The procedures\nof the spin separation, exchange and read-out by the spin to c harge conversion all induced by the\ndetuning variation are simulated. The rates of the potentia l variation that are necessary for the\nspin separation and spin to charge conversion in the context of the Landau-Zener transitions are\ndetermined. The average over random conﬁgurations of the hy perﬁne ﬁeld produce spin exchange\nresults which qualitatively agree with the experimental da ta.\nI. INTRODUCTION\nConstruction of a universal quantum gate for informa-\ntion processing on spins of electrons conﬁned in quantum\ndots1requires implementation of a controllable coupling\nbetween the spins conﬁned in adjacent quantum dots.\nThe proposed procedure2employs time evolution of the\nsystem with switching on the exchange energy4–6J, de-\nﬁned as the diﬀerence of the singlet and triplet energy\nlevels, for a short time. A nonzero exchange energy re-\nquires interdot tunnel coupling, and the control can be\nachieved by modulation of the tunnel coupling between\nthe dots with either a tunable interdot barrier2or tunable\npotential inside one or both quantum dots3.\nThe exchange energy2in the absence of spin-orbit in-\nteraction and hyperﬁne interaction is isotropic7, i.e. de-\npends only on relative orientation of the spins and con-\nserves the total spin in the dynamics of the few-electron\nsystems. Variation of the spin polarization of the electron\nsystem is possible in presence of the spin-orbit coupling\nwhich translates the electron motion in space to rotations\nof its spin as it precedes in the eﬀective magnetic ﬁeld\nthat accompanies the spin-orbit coupling8–14. The spin-\norbit coupling allows also for initialization of the state\nof the spin qubits, for separation of the spins of moving\nelectrons15–17in particular.\nIn the present paper we report on a simulation of the\nspin separation and spin exchange in a GaAs double\nquantum dot two-electron system. We use a quasi one-\ndimensional model with the assumption of a strong lat-\neral conﬁnement and the time-dependent conﬁguration-\ninteraction approach that allows for an exact account of\nthe two-electron spin and charge dynamics. For separa-\ntion of the spins we use the texture of the internal mag-\nnetic ﬁeld arising due to a non-zero magnetic moments\nof atomic nuclei18–20. The ﬁeld is expressed as a local\nclassical Overhauser ﬁeld obtained by summation of ran-\ndomly oriented nuclear magnetic moments over the span\nof the lateral electron wave function21.\nThe coupling of the carrier spins to the changing nu-clear spin ﬁeld is considered the main source of dephasing\nand spin relaxation in III-V materials18. Here, the ﬂuc-\ntuations of the nuclear ﬁeld are used as a resource for sep-\naration of the spins, which is performed via an adiabatic\nevolution of the system kept in the ground state when\nthe tunnel coupling between the dots is quenched. The\nexchange interaction is then switched on for the electrons\nto exchange their spins, as in Ref. 3. Next, the procedure\nof the read out of the spin exchange result is implemented\nby projecting the spin states on the charge conﬁgurations\nof the double dot. The spin exchange probability aver-\naged over a number of ﬂuctuations of the Overhauser\nﬁeld reproduce the fringe pattern of the experimental re-\nsults of Ref. 3as a function of the potential diﬀerence\nin quantum dots (which is related to detuning ε) and\nthe spin exchange time. We discuss the rate of the po-\ntential modulation for preparation of the initial states,\nthe eﬀects of the potential asymmetry on the exchange\ninteraction, and the reproducibility of the spin manip-\nulation procedures in presence of the Rashba spin-orbit\ninteraction and the Overhauser ﬁeld.\nII. THEORY.\nWe consider a quasi one-dimensional double quantum\ndot deﬁned within a GaAs quantum wire. The single-\nelectron Hamiltonian for the considered system reads\nˆH3D(1)(t) =/planckover2pi12/vectork2\n2m∗+V(/vector r)+ˆHB+ˆHSO, (1)\nwith/vectork=−i∇, the electron eﬀective mass in GaAs\nm∗=0.067m0, andV(/vector r)stands for the conﬁnement\npotential. The third term ( /vectorBHF),\nˆHB=gµb\n2/vector σ·(/vectorB+/vectorBHF(/vector r)), (2)\naccounts for the spin Zeeman eﬀect that includes the ex-\nternal magnetic ﬁeld /vectorBand the internal Overhauser ﬁeld\ndue to the hyperﬁne interaction18with the nuclear spins,2\nwhereg=−0.44is the GaAs electron Landé factor, µb\nthe Bohr magneton, and /vector σthe vector of Pauli matrices.\nThe last component of the Hamiltonian ( 1) is responsible\nfor the Rashba-type spin-orbit coupling.\nˆHSO=α(σzkx−σxkz). (3)\nWe apply the coupling constant α=0.44 meV nm21,22.\nWe assume that the system is strongly conﬁned along\nthe axis of the wire, so that the electrons occupy the\nground state of the lateral quantization only and the\ncharge dynamics involves time evolution exclusively in\nthe axial direction. For the lateral single-electron wave\nfunctions we adapt the Gaussian form,\nΨ(x,y) =1√πlexp(−x2+y2\n2l2), (4)\nwithl= 10nm.\n0255075100V (meV)\nL D Ra)\n-20-1001020-20-1001020-20-1001020\n-100 -75 -50 -25 0 25 50 75 100BHF(mT)\nz (nm)x\ny\nzb)\nFigure 1. (a) Schematic picture of fragment of a quantum\nwire hosting a double quantum dot, and its representation as\na value of potential V[z]along thez-axis. (b) A sample of the\neﬀective HF magnetic (Overhauser) ﬁeld generated at random\nwith the approach of Ref. 21. Each vector component ( x,y,\nz) was plotted separately.\nThe Hamiltonian ( 1) neglects the orbital eﬀects of the\nexternal ﬁeld which is justiﬁed by the assumption of a\nstrong lateral conﬁnement. Based on this assumption we\nconsider the electron wave functions frozen in the form\ngiven by Eq. 4which allows us to introduce a quasi-one-\ndimensional version of the energy operator ( 1) obtainedby integrating the Hamiltonian /angb∇acketleftΨ|ˆH3D(1)|Ψ/angb∇acket∇ightoverxand\nydimensions, which yields:\nˆH(1)=−/planckover2pi12\n2m∗∂2\n∂z2+V(z)+gµb\n2/vector σ·(/vectorB+/vectorBHF(z))+iασx∂\n∂z,\n(5)\nwhereV(z)is the conﬁnement potential along the axis\nof the wire [Fig. 1(a)], with two quantum dots of length\nLandR, separated by a barrier of length D. Two bar-\nrier regions ﬁll the remainder of the computational box\nof length 213.6 nm. We took L=R=61.02 nm and\nD=30.51 nm for all the presented results. The po-\ntential is taken 0 inside the right quantum dot, 15 meV\nwithin the central barrier, and 100 meV in the outer bar-\nriers [Fig. 1(a)]. The potential in the left quantum dot is\nset equal to ∆Vthat is varied in the simulation.\nIn most of the calculations the external magnetic ﬁeld\nis taken |/vectorB|=100 mT as in the experiment 3. For this\nvalue, the nuclear spins separate from the electron spins\nin the sense that the electron-nuclear spin ﬂips18,24,25are\nforbidden by the energy conservation. We approximate\nthe hyperﬁne ﬁeld by a local magnetic ﬁeld of a random\norientation which is considered constant during the time\nevolution of the electron spin. The latter assumption is\njustiﬁed by a long ﬂuctuation time of the nuclear ﬁeld\nwhich is of the order of 10 to 100 µs18. The ﬂuctuation\nof the hyperﬁne ﬁeld in the experimental conditions is ac-\ncounted for by averaging the results of the spin dynamics\n– which is also used in the experimental data processing3.\nThe procedure for derivation of the ﬁeld /vectorBHF(z)is\nadapted from Ref. 21with a classical vector of the ef-\nfective magnetic ﬁeld of length 5T generated at random\norientation at each ion of the crystal followed by the aver-\naging of all the local nuclear vectors with the probability\ndensity corresponding to the Gaussian lateral wave func-\ntions ( 4). A sample of the generated ﬁelds is plotted in\nFig.1(b). The values in Fig. 1(b)are given for each grid\npoint in the ﬁnite diﬀerence approach that we employ\nfor determination of the single-electron eigenstates. The\neﬀective HF ﬁeld as seen by the electron spin is further\nreduced to a few milliteslas19,20by averaging over the\nprobability density along the axis of the quantum dot.\nThe integration of the two-electron stationary Hamil-\ntonian over the lateral degrees of freedom produces the\noperator\nˆH(0)\n(2)=2/summationdisplay\ni=1ˆH(1)(σi,zi)+/radicalbig\nπ/2e2\n4πε0εlerfcx/parenleftbigg|z1−z2|√\n2l/parenrightbigg\n,\n(6)\nwhere the last term is the electron-electron interac-\ntion potential that results from the integration of the\nCoulomb interaction with the lateral wave function26,\nε= 12.9is the dielectric constant, and erfcxis the scaled\ncomplementary error function.\nThe two-electron Hamiltonian ( 6) is diagonalized by\nthe conﬁguration interaction method with the basis of\nSlater determinants χmof antisymmetrized products of3\nthe single-electron eigenfunctions of operator ( 5),\nψn(σ1,σ2,z1,z2) =M/summationdisplay\nm=1vnmχm(σ1,σ2,z1,z2).(7)\nWe used at least 120 Slater determinants for the basis\n(7) to obtain the eigenstates ψnand the energies of the\ntwo-electron Hamiltonian.\nFor simulation of the system dynamics we separate the\nexternal potential variation from the time-independent\nHamiltonian,\nˆH(2)(t) =ˆH(0)\n(2)+ˆW(2)(t). (8)\nThe simulated potential variation amounts in changing\nthe∆V(additional left quantum dot potential)\nˆW(2)(t) = ∆V(t)·(l(z1)+l(z2)), (9)\nwherel(z)is equal 1 in the left quantum dot and 0 else-\nwhere.\nWe solve the time-dependent Schrödinger equation\ni/planckover2pi1∂\n∂tψ(t) =ˆH(0)\n(2)ψ(t)+ˆW(2)(t)ψ(t), (10)\nin the basis of ˆH(0)\n(2)eigenstates ψn(7):\nψ(t) =N/summationdisplay\nn=1an(t)ψn(σ1,σ2,z1,z2)exp(−iEnt//planckover2pi1).(11)\nThis form of the wave function when plugged into the\nSchrödinger equation produces a set of diﬀerential equa-\ntions for the akcoeﬃcients,\n˙ak(t) =−i\n/planckover2pi1N/summationdisplay\nn=1an(t)/angb∇acketleftψk|ˆW(2)(t)|ψn/angb∇acket∇ightei(Ek−En)t//planckover2pi1,(12)\nthat we solve using the Crank-Nicolson scheme, for which\nthe subsequent steps of the wave function are given by\nsolution of an algebraic linear system of equations,\n/bracketleftbigg\nI−1\n2W(t+∆t)∆t/bracketrightbigg\n/vector a(t+∆t) =/bracketleftbigg\nI+1\n2W(t)∆t/bracketrightbigg\n/vector a(t),\n(13)\nwithIthat stands for the identity matrix, and W(t)that\nstands for a matrix with elements\nWk,n(t) =−i\n/planckover2pi1∆V(t)/angb∇acketleftψk|l(z1)+l(z2)|ψn/angb∇acket∇ightei(Ek−En)t//planckover2pi1.\n(14)\nIII. RESULTS\nA. Two-electron eigenstates\nThe lowest two-electron energy levels are plotted in\nFig.2(a)as functions of the diﬀerence of potentials in8.228.248.268.288.308.328.348.368.388.40E (meV)a)\nST+T0T-S′\n012\n4.64.654.74.754.84.854.94.955qR (e)\n∆V (meV)I. preparation II. manipulation(1,1)S(0,2)S\nIII. readout(0,2)S′\n(1,1)S′b)\n10-510-410-310-20.11\n 0  1  2  3  4  5  6J (meV)\n∆V (meV)c)\nFigure 2. (a) Energies Eand (b) electron charge qRstored by\nthe right quantum dot, as functions of the potential diﬀeren ce\n∆V. The sequence of ∆Vchanges is presented in (b). The\nsystem preparation amounts to an adiabatic charge transfer\nfrom (0,2)S state to the state with separated electrons (1,1 )S.\nThe manipulation of the state is carried at ∆Vbelow the\navoided crossing. For the readout of the spin state after the\nmanipulation stage the system is ramped back to the voltages\nfor which (0,2)S is the ground state. An axial magnetic ﬁeld\nofBz= 100 mT is applied here and below, unless stated\notherwise. (c) The exchange energy Jdeﬁned as the energy\ndiﬀerence between T0and the lowest-energy singlet S. The\ncenter of the avoided crossing is marked by the vertical dott ed\nline.\nleft and right quantum dot. The charge localized in the\nright dot for the corresponding energy levels is plotted in\nFig.2(b). The splitting of the triplet energy levels in Fig.\n2(a)is due to the axial magnetic ﬁeld set to Bz= 100\nmT. The spectrum in Fig. 2(a)contains the triplet states\nwith a single electron per quantum dot [charge conﬁgu-\nration denoted by (1,1)] as well as the singlet states with\nseparated electrons (1,1)S, and both electrons localized\nin the right quantum dot (0,2)S (S′in the following de-\nnotes the higher-energy singlet state). Figure 2(a)shows4\nan avoided crossing of the singlets near ∆V= 4.79meV.\nThe (0,2)S energy level does not react to the potential\nvariation in the left quantum dot with ∆V, hence its\nweak dependence on the potential variation outside the\navoided crossing.\n 3.94 3.95 3.96 3.97 3.98\n 0  100  200  300  400  500E (meV)\nB (mT)S\nT+T0T-(1,1) charge configuration, ∆V = 0\n 3.9632 3.9633 3.9634 3.9635\n 50 52 54 56 58 60 62ST+\nFigure 3. Energies of the lowest four states of the system as\na function of external magnetic ﬁeld B. Inset shows enlarged\nanticrossing of states S and T +. Curves are colored according\nto symmetry (singlet/triplet) and not to the actual order of\nstates (which changes due to crossings).\nThe magnetic ﬁeld dependence of the energy levels\nwith separated electrons – obtained for symmetric con-\nﬁning potential (i.e. for ∆V= 0) – is depicted in\nFig.3. The narrow avoided crossing between the S and\nT+energy levels near 56 mT is induced by the spin mix-\ning factors of the HF ﬁeld and the SO coupling.\nIn the absence of the HF ﬁeld and the SO coupling the\nwave functions of the two-electron eigenstates are sepa-\nrable into the spatial and spin components, in particular,\nfor the states with zero total spin projection:\nΨS=ψS(z1,z2)1√\n2(χ↑(σ1)χ↓(σ2)−χ↑(σ2)χ↓(σ1)),\n(15)\nand\nΨT0=ψT0(z1,z2)1√\n2(χ↑(σ1)χ↓(σ2)+χ↑(σ2)χ↓(σ1)),\n(16)\nwhereχare the spin eigenstates. For the states ( 15) and\n(16) the spin-up and spin-down densities are equal in each\npoint in space, so that the dots store zero average spin.\nThe spatial wave functions for states ( 15) and ( 16) can\nin the ﬁrst approximation be expressed by the single-dot\nφlandφrground-state orbitals, localized in the left and\nright dots, respectively\nψS(z1,z2) =1√\n2(φl(z1)φr(z2)+φr(z1)φl(z2)),(17)\nψT0(z1,z2) =1√\n2(φl(z1)φr(z2)−φr(z1)φl(z2)).(18)In the weak interdot tunneling regime the S and T 0states\nare nearly degenerate and can be mixed by either the\nspatial variation of the eﬀective Landé gfactor28–30, the\nspin-orbit coupling27, or the HF ﬁeld3. For the maximal\nmixing case one obtains the spin separation over the dots\nΨ↑↓=1√\n2(ΨS+ΨT0) = (19)\n1√\n2(φl(z1)χ↑(σ1)φr(z2)χ↓(σ2)−φl(z2)χ↑(σ2)φr(z1)χ↓(σ1))\nwith the left (right) dot storing the spin-up (spin-down)\ndensity and a state with interchanged spins\nΨ↓↑=1√\n2(ΨS−ΨT0) = (20)\n1√\n2(φr(z1)χ↑(σ1)φl(z2)χ↓(σ2)−φr(z2)χ↑(σ2)φl(z1)χ↓(σ1)).\nThe spin-orbit coupling alone can separate the spins\nover the dots in the external ﬁeld but only provided that\nthe double dot system is strongly asymmetric, with one\ndot larger by a factor of three than the other27. We ﬁnd\nthat the spin separation by the HF ﬁeld occurs also for\nquantum dots of the same size. The average spin in the\nright dot calculated for the second excited state (T 0) is\ndisplayed in Fig. 4. The average was taken over ﬁve\nrandom HF ﬁeld distributions. Naturally, for each ran-\ndom distribution the average spin in the left and right\nquantum dots is diﬀerent. Note, that the experiment3\nalso applies averaging the results of the spin evolution\nover many runs – for which the HF ﬁeld varies. Figure\n4shows that for a high energy barrier the spin in the\nright dot is close to |/angb∇acketleftSz/angb∇acket∇ight|=/planckover2pi1\n2, i.e. the spin separation\nis a typical result. We ﬁnd that the spin conﬁguration\nin the second excited state is then either ↑↓or↓↑, de-\npending on the speciﬁc HF ﬁeld distribution, the sign\nof the spin was hence neglected, instead, the values were\ntaken as negative (orange) for the cases where the second\nstate participating in the spin separation is the ground\nsinglet S. In nanowire double quantum dots a substan-\ntial variation of the gfactors in both the dots has been\nfound28–30. The variation should ﬁx the orientation of\nthe spins in the Hamiltonian eigenstates in the absence\nof the exchange interaction. The spin separation in the\nHamiltonian eigenstates is very rarely obtained for the\n∆Vvalues which correspond to the avoided crossing be-\ntween the (1,1) and (0,2) singlets, near ∆V= 5meV. In\nthis energy range the exchange energy is strong and pre-\nvails over the HF ﬁeld ﬂuctuations. Figure 4shows that\nthe closer we are to this value, the larger the interdot\nbarrierVbneeds to be in order to induce the spin separa-\ntion. The spin separation in the weak coupling regime is\ncrucial for the charge and spin dynamics to be discussed\nbelow.5\n 0 1 2 3 4 5 6 7\n∆V (meV) 5 10 15 20Vb (meV)\n-0.5 0 0.5\nL = 61.02 nm, D = 30.51 nm\nFigure 4. Averaged absolute values of spin sRin the right\nquantum dot in the second excited state, for diﬀerent values of\ndot potential diﬀerence ∆Vand barrier potential Vb. Values\nare taken with negative sign (orange) if the separation invo lves\nS and T 0states, and with positive sign (red) when T 0and\nS′are involved.\nB. The spin separation and exchange sequence\nThe spin separation and exchange procedure that is\nsimulated below is adapted from the experiment of Ref. 3\nand depicted in Fig. 5. The procedure starts with a\nstrong potential diﬀerence ∆Vwith two electron ground\nstate singlet localized in the right quantum dot S(0,2). A\nslight change of ∆Vis applied to pass across the singlets\navoided crossing of Fig. 2with the evolution time that is\nfast on the time scale of the hyperﬁne ﬁeld spin ﬂipping\nbut slow on the time scale deﬁned by the exchange inter-\naction, in order to change the charge occupation of the\ndots but keep the singlet spin state. Next, the symmetry\nof the conﬁnement potential is restored slowly on the hy-\nperﬁne interaction time scale, which – as we show below –\nin presence of the HF ﬁeld leads to the appearance of the\nstate with deﬁnite spin orientation in each of the dots.\nNext, large asymmetry of the potential is reintroduced for\nduration of tE. The asymmetry of the conﬁnement po-\ntential enhances the exchange interaction31and produces\nthe spin ﬂips between the dots as a result of time evolu-\ntion of the superposition of (1,1)S and (1,1)T 0states. Af-\ntertEthe exchange energy is ﬁrst rapidly quenched and\nthen the potential is adiabatically changed towards the\ninitial state. The right dot is occupied by two electrons\nwith a maximal probability provided that an even num-\nber of spin-ﬂips was performed during the spin exchange\ntimetE. In the subsections to follow we ﬁrst explain\nthe electron structure of the eigenstates, next we move\nto the description of the system initialization, readout\nby the spin to charge conversion, and the interdot spin\nexchange.C. Time evolution: Spin-dependent charge\ndynamics.\nThe sequence that is simulated (see Fig. 2(b)and Fig.\n5) starts as in the experiment3by the two-electron singlet\nwith both electrons in the right dot (0,2)S. In this subsec-\ntion we deal with the charge separation that is achieved\nby the small drop of ∆Vfrom∆V= 5meV to∆V= 4.7\nmeV that is visible at the beginning of the sequence in\nFig.5(a). The drop takes the system across the avoided\ncrossing of the singlets in Fig. 2(a). The initial states\nare taken as eigenstates of the stationary Hamiltonian\nfor∆V= 5meV. The ﬁnal state of the time evolution\nis plotted in Fig. 6(a-d) as a function of the switching\ntime for the ﬁrst four eigenstates of the Hamiltonian for\n∆V= 5meV, and in terms of the eigenstates of the\nHamiltonian for ∆V= 4.7meV (lower panels). Addi-\ntionally, the resultant value of charge in the right dot\n(upper panels) is provided to emphasize states having\n(0,2) occupation. The rate of the changes can be com-\npared with two time scales: the one given by the energy\nsplitting between the singlets, which at the center of the\navoided crossing [see Fig. 2(c)] is∆S≃0.02meV that\ncorresponds to τE≃/planckover2pi1\n2∆S= 16.5ps, and the one given by\nthe Zeeman splitting in the HF nuclear ﬁeld of the singlet\nand T +state,∆EZ=gµBB′\nHF, whereB′\nHFis given by\naveraging the nuclear magnetic ﬁeld of Fig. 1(b)with the\nwave function along the zcoordinate, typically B′\nHF= 1\nmT, for which τHF≃/planckover2pi1\n2∆Ez= 12.9ns.\nFor the (0,2)S state as the initial state [Fig. 6(a)]\nthe ﬁnal one is (0,2)S′when the switching is fast on\ntheτSscale. The transition has then the Landau-Zener\ncharacter32–35The charge occupation of the dots is left\nunchanged for the nonadiabatic abrupt switching. To be\nmore precise, the abrupt potential change leaves a small\nadmixture of (1,1)S state to (0,2)S′- which produces os-\ncillations of the charge localized in the left dot in the lim-\nits that are marked in the upper panel of Fig. 6(a)with\nthe gray area. For an adiabatic switching time compa-\nrable withτSthe ﬁnal state is the spin separated singlet\n(1,1)S, while for a very slow switching time – comparable\nwithτHFthe evolution is adiabatic on the HF coupling\ntime scale and the destination state is the T +ground\nstate [see Fig. 6(e)]. Summarizing, Fig. 6(a)indicates\na sequence of transitions across the two avoided cross-\nings that involves both the (1,1) and (0,2) singlets as\nwell as the T +triplet. The T +– S avoided crossing is\nmuch tighter than the (1,1)S and (0,2)S one, and thus it\nrequires a much slower potential variation for the elec-\ntron to pass from (0,2)S to the T +ground state. Con-\nversely, for the initial state [Fig. 6(b)] set at T +– an\nexcited state at ∆V= 5meV – a very slow switching\ntime≃τHFis required to keep the electron in the excited\nstate [(0,2)S for lower ∆V], otherwise the time evolution\nends at T +triplet. T 0state does not enter any avoided\ncrossing. For the initial state set at T 0one stays in the\nT0independent of the switching time [Fig. 6(c)]. Finally,6\n012345∆V (meV)> 1 µs tE > 1 µs(0,2)S\n(1,1)S\n↑↓[S + T0 exp(-iJt/h-)] / √2\n? ↑↓\n↓↑(1,1)S\n(1,1)T0(0,2)S\n(1,1)T0a)\n10-510-30.1J (meV)> 1 µs tE > 1 µsb)\nFigure 5. (a) Variation of the potential asymmetry in time wi th the targeted evolution of the two-electron state explain ed\nschematically. (b) The exchange energy deﬁned as the energy diﬀerence between the triplet T 0state and the lowest energy\nsinglet for the potential asymmetry in (a).\nfor (1,1)S′in the initial state the ﬁnal one is (1,1)S for a\nfast switching (much shorter than τS), (0,2)S′for a longer\nswitching time (between τSandτHF), and T –for an ex-\ntremely slow ( ≃τHF) switching. Therefore, both singlets\nin the initial state evolve to spin-polarized triplets in th e\nlimit of slow potential variation.\nThe time evolution presented in Fig. 6with∆Vvaried\nfrom 5 meV to 4.7 meV corresponds to the initial state\npreparation in the experimental sequence3. The reverse\npotential variation is used3in the spin state detection\nby the spin-charge conversion. In the experiment3the\ncharge detection of the right quantum dot is used for\ndetermination of the result of the spin dynamics. The\nresults of the simulation for the potential variation from\n∆V= 4.7meV to∆V= 5meV – the small and abrupt\nrise of the potential in the left dot at the end of the\nsequence of Fig. 5(a), are given in Fig. 7. The initial\nstate is set as one of the eigenstates of the Hamiltonian\nfor∆V= 4.7meV with the ﬁnal states projected onto\nthe Hamiltonian eigenstates for ∆V= 5meV. Figure\n7shows that the (1,1)S singlet evolves to (0,2)S only\nprovided that the switching time is longer than ≃0.1ns\nbut not longer than ≃200ns. For a slower switching the\nsystem evolves to T+. A slower switching time for T +in\nthe initial state produces (0,2)S in the ﬁnal state [Fig.\n7(a)]. However, in order to produce a detectable increase\nof the charge in the right dot above a single electron\ncharge, the switching time needs to exceed 10 ns. For\nT–in the initial state a slow switching produces the spin\nsinglet but with electrons separated over the dots [Fig.\n7(d)].\nFirst two panels of Figure 8shows the time evolution\nfor the read-out sequence with the HF ﬁeld switched oﬀ\nand the orientation of the external ﬁeld changed to par-\nallel to the x[Fig.8(a)] andydirections [Fig. 8(b)]. For\nthexdirection of the external ﬁeld the spin of the two-\nelectron state remains unchanged. In this case the eﬀec-\ntive SO magnetic ﬁeld and the external ﬁeld are aligned,\nso that the electron motion only changes the Zeeman\nsplitting energy and no precession of the spin is present.\nFor the external ﬁeld oriented parallel to the yaxis the\nprecession reappears since the external and the eﬀective\nﬁelds are no longer aligned. In presence of the HF in-teraction - the intrinsic anisotropy of the SO interaction\nis masked by the nuclear ﬁeld, and the results remain\nquantitatively similar for varied external ﬁeld orientati on.\nHowever, the switching times diﬀer within a certain range\nfrom one HF ﬁeld conﬁguration another. The results for\nthree random conﬁgurations of the HF ﬁeld are given in\nFig.8(c,d) with (d) or without (c) the SO interaction.\nThe transition between the singlets – which are spin con-\nserving – ignore the details of the HF ﬁeld, however the\ntransition to the spin polarized triplet does depend on\nthe random HF ﬁeld. Without the SO interaction the\nS-T+switching times diﬀer by two orders of magnitude.\nThe presence of the SO coupling reduces this variation\nrange signiﬁcantly - to a single order of the magnitude\nonly.\nD. Time evolution: Spin-separation\nAccording to the preceding subsection the single-dot\n(0,2)S singlet can be transformed into the state with sep-\narated carriers (1,1)S provided that the switching time\nacross the avoided crossing is of the order of 1 to 10 ns.\nFor∆V= 4.7meV – considered above the spins are gen-\nerally not polarized within the separate quantum dots\n[see Fig. 4]. The spin separation in the system can then\nbe induced by an adiabatic variation of the potential from\n∆V=4.7 to 0.7 meV – see the part of the sequence with\nthe slow potential diﬀerence drop in Fig. 5(a). The re-\nsults for the system evolution are depicted in Fig. 9for a\ntypical random HF ﬁeld. Figure 9shows the projections\nof the ﬁnal states in the basis of the destination Hamil-\ntonian eigenstates. For the switching time that exceeds\n100 ns the ﬁnal states are the Hamiltonian eigenstates\nwith separate spins, and there is one to one correspon-\ndence between the S, and T 0states to the ↑↓and↓↑ones.\nConversely, for the potential variation in the opposite di-\nrection one obtains either the S or T 0state, depending\non the spin distribution ↑↓or↓↑in the left and right\nquantum dots respectively. This fact is next used in the\ndetection of the spin exchange with the spin and charge\nconversion induced by the potential variation. Obviously,\nthe correspondence between S, T 0and↑↓,↓↑states can7\n 1 1.5 2qR (e)c)\nψ(t = 0) = T0\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)T0 1 1.5 2qR (e)d)\nψ(t = 0) = (1,1)S′\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)(1,1)S (0,2)S′ T- 1 1.5 2qR (e)a)\nψ(t = 0) = (0,2)S\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)(1,1)S T+ (0,2)S′ 1 1.5 2qR (e)b)\nψ(t = 0) = T+\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)(1,1)S T+E\n∆Ve)\nS S′T-T+T0\ntswitch\nFigure 6. Final absolute values of charge qRin the right quantum dot (upper panels) and squared moduli of projections of\nﬁnal stateψ(+∞)onto the ﬁnal hamiltonian basis X∈ {S, T+, T0, T–, S′} (lower panels), as a function of switching time\ntswitch, if the system was initially in the (a) (0,2)S (b) T +(c) T 0and (d) (1,1)S′state. Potential diﬀerence ∆Vwas switched\nfrom 5.0 meV to 4.7 meV (state preparation). The gray areas in the upper panels of (a) and (d) indicate the range within\nwhich the charge in the right dot changes in the ﬁnal state, wh ich is a superposition of two ﬁnal Hamiltonian eigenstates w hen\nthe switching time is short. (e) The schematics of the spectr um and the switching direction.\nbe opposite with equal probability for a random HF ﬁeld\ndistribution, however one or the other is typical for the\nHF ﬁeld generated at random (i.e. spins tend to separate\nat low∆V).\nE. Time evolution: Spin exchange\nAbove we described how the system initialized in the\n(0,2)S ground state is taken across the avoided crossing\nto the (1,1)S state and next into the separated spin state\n↑↓or↓↑, depending on the state of the HF ﬁeld. Here, we\nconsider the rapid rise of the diﬀerence of potentials (the\ncenter of Fig. 5) for which a nonzero exchange energy\nJ=ET0−ESappears. Then, the solution of the time\ndependent Schrödinger equation reads\nΨ(t) =1√\n2exp(−iESt\n/planckover2pi1)/parenleftbigg\nΨS+ΨT0exp(−iJt\n/planckover2pi1)/parenrightbigg\n(21)and the spins in both the dots ﬂip with the period of\nTf=2π/planckover2pi1\nJ. The wave function of this form switches the\nspin orientations within the dot, as it varies from Ψ↑↓(at\nt= 0) toΨ↓↑att=T/2. The spin ﬂips are stopped\nwhen the system is taken down to a small value of ∆V\nagain. For spin exchange times which are odd multiples\nofTf, a slow rising of the potential takes the system\nto (1,1)T 0eigenstates. For spin exchange times that are\neven multiples of Tfthe potential variation returns the\nsystem to (1,1)S and next to (0,2)S.\nFigure 10shows the projection of the wave function\nonΨ↑↓– taken as the initial state for the central high\n∆Vpoint in the time sequence as a function of time.\nThe gray lines indicate three sample evolutions for some\nﬁxed random HF ﬁeld orientations and the green one is\nan average over 100 such runs. We can see that the aver-\nage has a decreasing amplitude of the oscillations which\ncorresponds to the inhomogeneous broadening due to the8\n 1 1.5 2qR (e)c)\nψ(t = 0) = T0\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)T0 1 1.5 2qR (e)d)\nψ(t = 0) = T-\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)T- (1,1)S′ 1 1.5 2qR (e)a)\nψ(t = 0) = T+\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)(0,2)S T+ 1 1.5 2qR (e)b)\nψ(t = 0) = (1,1)S\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)(0,2)S T+ (1,1)S′\nFigure 7. Final absolute values of charge qRin the right quantum dot (upper panels) and squared moduli of projections of ﬁnal\nstateψ(+∞)onto the ﬁnal hamiltonian basis X∈ {S, T+, T0, T–, S′} (lower panels), as a function of switching time tswitch,\nif the system was initially in the (a) T +(b) (1,1)S (c) T 0and (d) T –state. Potential diﬀerence ∆Vwas switched back from\n4.7 meV to 5.0 meV (state readout).\nrandom ﬁeld. In Fig. 11we show the projection of the\nwave function on the Ψ↑↓state as a function of the po-\ntential diﬀerence ∆Vduring the spin exchange and the\nspin exchange time tE. The inset shows the result ob-\ntained by Eq. ( 21) for the exchange energy as calculated\nin the absence of the HF ﬁeld. The simulated pattern of\nthe fringes agrees with the analytical one. The result of\nthe simulation contains the eﬀect of the inhomogeneous\npotential – the visibility of the oscillations deteriorate s\nwith the exchange time. The oscillations are closer to the\nideal value for a large potential variation ∆Vin which\nthe interdot tunnel coupling and the exchange energy is\nlarger. The spin exchange is not only faster but occurs\nwith the larger ﬁdelity for larger ∆V, i.e. closer to the\n(1,1) – (0,2) avoided crossing.\nIV. SUMMARY AND CONCLUSIONS\nWe presented the results of the simulation of the spin\nseparation, exchange and spin-to-charge conversion for\na model of a two-electron system conﬁned in a double\nquantum dot deﬁned within GaAs quantum wire and atexture of the HF nuclear magnetic ﬁeld with the SO in-\nteraction using the conﬁguration interaction method and\nan eﬀective Overhauser ﬁeld distribution.\nFor the potential diﬀerence (detuning) sweeps through\nthe (0,2)S – (1,1)S avoided crossing – that is used for\npreparation of the initial state as well as for the spin to\ncharge conversion applied for the readout, the HF ﬁeld\nand the SO interaction play similar roles, and one can\nbe replaced by the other. The transition times in HF\nﬁeld diﬀer from one random distribution to the other,\nwhile the SO interaction introduces an anisotropy of the\nevolution in the external magnetic ﬁeld. A simultaneous\npresence of the HF ﬁeld and the SO interaction stabilize\nthe variation range of the transition times and reduce the\nanisotropy as a function of the external magnetic ﬁeld\norientation.\nThe spin-separation at the preparation stage is\nachieved due to the HF ﬁeld and an adiabatic evolution\nat the scale of the nuclear Zeeman eﬀect. The rate of the\nspin exchange induced by a pulse of potential ∆Vdif-\nfers strongly from one random nuclear spin distribution\nto another, but the averaged spin evolution closely follow\nthe time scale set by the exchange energy in the absence\nof the HF ﬁeld.\n1D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).2G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B9\n 0 0.5 1\n 0.001  0.01  0.1  1  10  100  1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)→SO\nB || xa)\n(0,2)S(1,1)S′\n 0 0.5 1\n 0.001  0.01  0.1  1  10  100  1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)→SO\nB || yb)\n(0,2)ST+ (1,1)S′\n 0 0.5 1\n 0.001  0.01  0.1  1  10  100  1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)→HF\nB || zc)\n(0,2)S\nT+(1,1)S′\n 0 0.5 1\n 0.001  0.01  0.1  1  10  100  1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)→HF + SO\nB || zd)\n(0,2)ST+ (1,1)S′\nFigure 8. Same as Fig. 7for varied conditions of the simula-\ntion. The HF ﬁeld is switched oﬀ (a,b) and the magnetic ﬁeld\noriented parallel to the x(a) andy(b) directions. The SO\ninteraction is switched oﬀ in (c). In (d) both interactions a re\npresent. Three random conﬁgurations of the HF ﬁeld were\nconsidered in (c,d).\n 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)b)\n↑↓↓↑ ψ(t = 0) = T0 0 0.5 1\n0.001 0.01 0.1 1 10 100 1000| 〈 X | ψ(+∞) 〉 |2\ntswitch (ns)a)\n↑↓\n↓↑ψ(t = 0) = (1,1)S\nFigure 9. Squared moduli of projections of ﬁnal state ψ(+∞)\nonto the ﬁnal hamiltonian states with separated spins X∈\n{↑↓,↓↑}, as a function of switching time tswitch, if the sys-\ntem was initially in the (a) (1,1)S or (b) T 0state. Potential\ndiﬀerence ∆Vwas switched from 4.7 meV to 0.7 meV.0.00.20.40.60.81.0\n 0  20  40  60  80  100| 〈 ↑↓ | ψ(+∞) 〉 |2\ntE (ns)sample oscillations\n100 samples avg.\nFigure 10. Probabilities PS=|/an}bracketle{tS|ψ/an}bracketri}ht|2of ﬁnding the system\nin state S as a function of exchange time tE, for which larger\npotential diﬀerence was reintroduced (to ∆V= 4.5meV).\nGray plots are sample Rabi oscillations, while the green is a n\naverage over 100 random distributions of the HF ﬁeld.\n| 〈 S | ψ(∆V, tE) 〉 |2\n 0  20  40  60  80  100\ntE (ns)4.354.404.454.504.554.604.65∆V (meV) 0  0.5  1\n0 40 804.44.54.6\nFigure 11. 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Our approach is\nbased on a quantum description of the electron spin in combination with the coherent semiclassical\ndynamics of nuclear spins. We consider single and double Landau-Zener passages across the S-T+\nanticrossings. For linear sweeps, the electron dynamics is expressed in terms of parabolic cylinder\nfunctions. The dynamical nuclear polarization is described by two complex conjugate functions\n\u0003\u0006related to the integrals of the products of the singlet and triplet amplitudes ~ c\u0003\nS~cT+along the\nsweep. The real part Pof \u0003\u0006is related to the S-T+spin-transition probability, accumulates in the\nvicinity of the anticrossing, and for long linear passages coincides with the Landau-Zener probability\nPLZ= 1\u0000e\u00002\u0019\r, where\ris the Landau-Zener parameter. The imaginary part Qof \u0003+is speci\fc\nfor the nuclear spin dynamics, accumulates during the whole sweep, and for \r&1 is typically\nan order of magnitude larger than P.PandQalso show critically di\u000berent dependences on the\nshape and the duration of the sweep. Qhas a profound e\u000bect on the nuclear spin dynamics, by\n(i) causing intensive shake-up processes among the nuclear spins and (ii) producing a high nuclear\nspin generation rate when the hyper\fne and spin-orbit interactions are comparable in magnitude.\nEven in the absence of spin-orbit coupling, when the change in the the total angular momentum\nof nuclear spins is less than ~per single Landau-Zener passage, the change in the global nuclear\ncon\fguration might be considerably larger due to the nuclear spin shake-ups. We \fnd analytical\nexpressions for the back-action of the nuclear reservoir represented via the change in the Overhauser\n\felds the electron subsystem experiences.\nPACS numbers: 73.63.Kv,72.25.Pn,76.70.Fz\nI. INTRODUCTION\nElectron spin states in semiconductor quantum dots\nare investigated for their potential use as quantum bits\nin quantum computing architectures.1{3To this end, con-\ntrol of the spin states and their couplings to the environ-\nment is essential. In GaAs and InAs semiconductors, a\nmajor source of electron spin decoherence is the coupling\nto the surrounding nuclear spins.1,4{8Since the quan-\ntum dots are large compared to the interatomic spacing,\neach electron interacts with typically one million nuclei.\nAchieving control over this many-body interaction is a\nkey for manipulating semiconductor quantum bits.\nIn two electron double quantum dots, the singlet Sand\ntripletT0states de\fne the elementary qubit. The cou-\npling between these states is governed by the gradient\nin the longitudinal magnetic Zeeman splitting between\nthe two dots. Controlling this coupling enables singlet-\ntriplet qubit manipulations. Beyond the two-state S-T0\nqubit operation, the gradient in the transverse magnetic\nZeeman splitting between the two dots de\fnes the cou-\npling of the singlet Sto the triplet T+andT\u0000states.\nFinally, the longitudinal magnetic Zeeman splitting de-\ntermines the relative energies of the triplet states. This\nZeeman splitting arises from the external \feld Band the\nnuclear spin background via the Overhauser \feld, and by\nchanging the nuclear spin polarization the basic electron\nparameters can be tuned.Polarization of nuclear spins can be created and de-\nstroyed by \rip-\rop processes by pumping the elec-\ntronic states via time-dependent gate voltages. This\nhas recently been investigated in many interesting\nexperimental9{12and theoretical papers in double quan-\ntum dots in the regime of Pauli blockade.13{17Experi-\nmentally, it has been demonstrated that an Overhauser\n\feld gradient of several hundred milli Tesla can be gener-\nated and sustained.9The dephasing time of the electron-\nspin qubits has been extended to more than 200 \u0016s.11\nBecause the dynamical interaction of an electron spin\nwith a nuclear spin reservoir is enormously complicated,\ndi\u000berent theoretical e\u000borts were focused on the various\naspects of it. The two aspects most closely related to\nour paper are the theoretical modeling of the connec-\ntion between the generation of dynamical nuclear spin\npolarization at short and long time scales13,15,18and the\nin\ruence of the spin-orbit interaction on the build-up of\nthe nuclear polarization.16,17\nThe aim of this paper is to study in detail the elec-\ntron and nuclear spin dynamics as the system passes\nacross aS-T+anticrossing. In GaAs and InAs quan-\ntum dots in an external magnetic \feld, T+is the lowest\nenergy component of the electron triplet state because\nof the negative electron g-factor,g < 0. During a S-\nT+(or aT+-S) passage, electrons trade their spin with\nthe nuclear reservoir, and multiple passages are used in\ncreating a di\u000berence (\"gradient\") of the e\u000bective nucleararXiv:1104.4591v1  [cond-mat.mes-hall]  23 Apr 20112\n(Overhauser) \felds between two parts of the double dot\nthat are used for qubit rotations. The study of a sin-\ngle passage (or two passages during a single cycle) pro-\nvides a \frm basis for investigating events on longer time\nscales. Also, the progress in experimental techniques cur-\nrently allows, instead of averaging data over thousands\nof sweeps, to perform single-shot measurements,19and\nmost recently such measurements have been achieved for\ndouble quantum dots.12Also, the double dot dynamics\nduring a single sweep manifests itself explicitly in beam\nsplitter experiments.20We expect the approach devel-\noped in our paper to become a useful tool in discussing\nsuch types of experiments and, more widely, to facilitate\nbetter understanding and utilization of the nuclear spin\nenvironment in solid state based quantum computing.\nSpeci\fcally, we take into account the spatial distribu-\ntion of the hyper\fne coupling between the electron and\nnuclear spins and compute the change in the topography\nof the nuclear spin polarization and the related changes\nin the gradient and average Overhauser \felds governing\nthe dynamics of the electron spin. These \felds, that the\nelectrons experience in the singlet and triplet states, de-\npend on the spatial variation of the electron-nuclear cou-\npling and we take this dependence into account. We em-\nploy the Zener approach21and \fnd analytically explicit\nexpressions for the electron and nuclear spin dynamics\nduring a single linear sweep and during cycles consisting\nof two linear sweeps.\nLet us give an overview of the main results. We express\nthe whole electron and nuclear spin dynamics in terms of\ntwo complex conjugate functions \u0003\u0006(Ti;Tf) depending\non the initial and \fnite times ( Ti;Tf) and the shape of\nthe path between them. These \u0003\u0006functions are integrals\nof the products of the singlet and triplet amplitudes dur-\ning theS-T+passage. The real part P= Ref\u0003\u0006gis\nthe transition probability between the singlet Sand the\ntripletT+states. The imaginary part Q= Imf\u0003+gin-\ncludes basic information about the nuclear spin dynam-\nics including the nuclear shake-ups. The Landau-Zener\nprobability, PLZ= 1\u0000e\u00002\u0019\r, where\ris the Landau-\nZener parameter, is the asymptotic value of P(Ti;Tf) for\na single sweep when Ti!\u00001 andTf!1 . Usually,\nall results are expressed in terms of PLZ. Our approach\nprovides a more detailed information about the nuclear\nspin dynamics away from the S\u0000T+anticrossing.\nOscillations of the transition probability P(Ti;Tf) as\na function of its arguments reveal typical interference\npatterns. These oscillations are highly anharmonic for\nsmall Landau-Zener transition probabilities PLZ\u001c1\nand might persist for a long time with a large ampli-\ntude for intermediate Landau-Zener transition probabili-\ntiesPLZ\u00180:5. However, it is not typically the transition\nprobability Pthat determines the nuclear spin dynam-\nics. Instead, the other S-T+quantity,Qis non less im-\nportant. While Pis constrained to be in the interval\n0\u0014P\u00141, there are no such constraints on Qand it\nis typically larger than P. We \fnd that Qcontrols the\nshake-up processes among the nuclear spins. In the ab-sence of spin-orbit coupling, at most ~of the angular\nmomentum can be transferred to the nuclear spin bath.\nGiven that there are around a million nuclear spins in the\nquantum dots, of which around a thousand are aligned\ninitially, a change in one out of a thousand nuclear spins\nwould have only a minor e\u000bect. However, the nuclear\nspins are allowed to interchange their spins during the\nS-T+passage without violating the conservation of the\nangular momentum. Although the interchange does not\nchange the total nuclear spin angular momentum, the re-\ndistribution of the nuclear spins can lead to considerable\nchanges in the various gradient and average Overhauser\n\felds that the electrons experience. This is because the\nOverhauser \felds depend on weighted average values of\nthe nuclear spin distribution with respect to the electron-\nnuclear couplings and not just the total nuclear spin. We\n\fnd that such shake-ups are very sensitive to the initial\nnuclear spin distribution and that they are often much\nlarger than the average nuclear spin production because\nQis typically ten times larger than P.\nFurthermore, when the spin-orbit coupling competes\nwith the hyper\fne interaction and Qis considerably\nlarger than P, then theQ-enhanced spin generation dom-\ninates for a generic direction of the nuclear spin polariza-\ntion and can become considerably larger than P. How-\never, after averaging over the direction of the transverse\nnuclear spin polarization, Qcancels and the results of\nRefs. [16,17] are recovered.\nAnother \fnding is that even geometrically symmetric\ndouble quantum dots acquire asymmetric behavior be-\ncause of the spatial inhomogeneity of the hyper\fne cou-\npling. The sign of the asymmetry depends on B, and its\nmagnitude is largest close to the (0,2) or (2,0) con\fgura-\ntion. The consequences of this B-controlled asymmetry\nfor building nuclear \feld gradients are similar to that\nenvisioned in Ref. 15 for geometrically asymmetric dots.\nThis paper is organized in the following way. In Sec. II,\nwe describe the model of a double quantum dot that fol-\nlows the lines of Refs. [15,22,23]. We introduce the basic\nnotations related to the electron-nuclear hyper\fne inter-\naction and the nuclear dynamics induced by it in Secs. III\nand IV, respectively. In Sec. V, a linear Landau-Zener\nsweep is treated analytically and the time-dependence of\nthe e\u000bective magnetic \felds acting on the nuclei is dis-\ncussed in detail. Because Sec. V is rather technical, a\nreader interested in experimental applications can skip\nto Sec. VI, where numerical data for the linear in time\nLandau-Zener sweeps and cycles are discussed. In Sec.\nVII, the back action of the nuclear spin dynamics on the\nOverhauser \felds in the electron spin Hamiltonian is es-\ntimated. Appendix A outlines the notations for electron\nspin operators. Appendix B discusses the spatial depen-\ndence of the hyper\fne interaction. We demonstrate that\neven for two symmetric quantum dots, the hyper\fne cou-\npling acquires asymmetries controlled by the overlap in-\ntegral and the external magnetic \feld. Appendix C in-\ncludes two new identities for parabolic cylinder functions.\nWe conclude and summarize our results in Sec. VIII.3\nII. MODEL\nWe consider two electrons in a double quantum dot.\nWhen the electron spin is conserved, the classi\fcation of\nthe electron states as a singlet state Sand three triplet\n(T\u0017,\u0017= 0;\u00061) states is exact. Spin-orbit interaction\nand the interaction with the nuclear spins mixes these\nstates. We use the singlet and triplet stationary states\nas our basis. They are\n\tS(1;2) = S(1;2)\u001fS(1;2); (1a)\n\tT\u0017(1;2) = T(1;2)\u001fT\u0017(1;2); (1b)\nwhere 1 and 2 denote the 1st and 2nd electron. The\nspin wave functions obey the symmetries \u001fS(1;2) =\n\u0000\u001fS(2;1) as well as \u001fT\u0017(1;2) =\u001fT\u0017(2;1) and are speci-\n\fed in Appendix B. The orbital wave functions  S(1;2)\nand T(1;2) obey the symmetries  S(1;2) = S(2;1)\nand T(1;2) =\u0000 T(2;1);and we consider only the low-\nest energy orbital states so there are no additional quan-\ntum numbers labeling the orbital wave functions.\nThe electrons interact with each other, external gate\npotentials, an external magnetic \feld, and with the nu-\nclear spins predominantly via the hyper\fne interaction.\nThe latter interaction, as well as spin-orbit coupling, in-\nduce transitions between the singlet and triplet states\nthat we compute. The nuclei interact with the exter-\nnal magnetic \feld, the electrons through the hyper\fne\ninteraction, and with each other via the magnetic dipole-\ndipole interaction. The latter interaction a\u000bects the nu-\nclear spin dynamics on long time scales of around milli\nseconds, and we disregard it in what follows. However, we\ntake into account (in a semiclassical Born-Oppenheimer\napproach and in the leading order in the large elec-\ntron Zeeman splitting) an indirect RKKY-like interac-\ntion between nuclear spins originating from the hyper\fne\nelectron-nuclear coupling (see Sec. V D). Near the ST+\nanticrossing it manifests itself at the scale of about 10 \u0016s.\nOf central importance is the hyper\fne electron-nuclear\ninteraction\n^Hhf=AX\nj2X\n`=1\u000e(Rj\u0000r`)(^Ij\u0001^ s(`)); (2)\nwhereAis the electron-nuclear interaction strength, `\nnumerates electrons and jnuclei, ^s(`) =1\n2^\u001b(`) are the\nelectron spin operators in terms of the vector of Pauli ma-\ntrices ^\u001b(`) for each electron `, and ^Ijare the nuclear spin\noperators. The electron and nuclear spin operators are\ndimensionless in our notations. Carets denote quantum\nmechanical operators and bold variables are vectors.\nIn the 4\u00024 singlet and triplet space ( S;T +;T0, and\nT\u0000), the Hamiltonian that describes the electrons and\ntheir interaction with the nuclear spins can be written as\n^H=\u0012\u000fS ^ vT\nn\n^ v\u0003\nn\u000fT\u0000^\u0011\u0001^S\u0013\n; (3)\nwhere the total electron spin ^S=^s(1) + ^s(2). Addition-\nally, the spin-orbit interaction induces terms in Eq. (3)that we discuss below. The nuclear spins are also af-\nfected by the external magnetic \feld through the nuclear\nZeeman e\u000bect that we take into account below in the de-\nscription of their dynamics. However, we disregard the\ne\u000bect of the nuclear Zeeman energy on the equilibrium\nspin populations because of the high temperature of the\nnuclear spin bath. The \u000fSand\u000fTterms in the diagonal\nmatrix elements of Eq. (3) describe the singlet and triplet\nenergies in the absence of the nuclear and external mag-\nnetic \felds. They depend on the electrostatic gate po-\ntentials and the interactions between the electrons. The\no\u000b-diagonal operator components ^ vT\nn= (^v+\nn;\u0000^vz\nn;\u0000^v\u0000\nn);\nare nuclear spin dependent (a superscript Tdenotes the\ntranspose of a vector and the subscript ndenotes that\nthis coupling is due to the nuclear spins)\n^v\u000b\nn=AX\nj\u001aj^I\u000b\nj; (4)\nwith\u000b= (+,\u0000,z),^I\u0006\nj=\u0010\n^Ix\nj\u0006i^Iy\nj\u0011\n=p\n2 are the trans-\nverse nuclear spin components, and the singlet-triplet\nelectron-nuclear coupling coe\u000ecients\n\u001aj=\u001a(Rj) =Z\ndr \u0003\nS(r;Rj) T(r;Rj) (5)\ndependent on the positions Rjof nucleij. Roughly, \u001aj\nvaries from positive in one quantum dot to negative in\nthe other. Therefore, ^ v\u0006\nnand ^vz\nnrepresent di\u000berences\nin the e\u000bective nuclear magnetic \felds in the two dots\nin the directions transverse and parallel to the external\nmagnetic \feld, respectively. The e\u000bective splitting of the\ntriplet states due to the external magnetic \feld Band\nthe nuclei is\u0000^\u0011\u0001^S, where\n^\u0011=\u0011Zez+^\u0011n=\u0011Zez\u0000AX\nj\u0010j^Ij; (6)\n\u0011Zis the electron Zeeman splitting in the \feld Bk^z,\n^Sis the spin-1 operator for the electrons (as de\fned in\nAppendix A), and the position dependent coupling con-\nstants of the triplet states to the nuclei are\n\u0010j=Z\ndr \u0003\nT(r;Rj) T(r;Rj): (7)\nThis completes the description of the Hamiltonian that\ngoverns the coupling between the electron and nuclear\nspin dynamics.\nTheST+anticrossings arising due to ^ v\u000b\nnand also\ntheST0level splittings were investigated by the beam-\nsplitting technique20and Rabi-oscillations9,10,24, respec-\ntively.\nIII. ELECTRON AND NUCLEAR SPIN\nDYNAMICS\nThe Hamiltonian of Eq. (3) de\fnes a many-body prob-\nlem of the coupled electron-nuclear dynamics. Our in-\nterest is in the dynamical nuclear polarization that is4\nachieved by changing the gate voltages in such a way\nthat the electronic subsystem makes a transition from\nthe singletSto the lowest energy triplet T+state or vice\nversa. The many-body interaction can be simpli\fed by\nemploying the Born-Oppenheimer approach.15The elec-\ntrons are fast as compared to the nuclei. The electrons\nalso interact with a large number of nuclei, around one\nmillion. These two features imply that the electron dy-\nnamics is una\u000bected by the dynamics of a single nucleus\nand electrons see only a quasi-static con\fguration of all\nnuclei during a single ST+crossing. This motivates an\nansatz where the wave function is separable into elec-\ntronic and nuclei parts.15\nThe electron dynamics can be solved from the Hamil-\ntonian of Eq. (3) with the assumption that the nuclear\nspin operators can be approximated by their expectation\nvalues before the transition, ^ vn!vn. The detuning\nenergy\u000fis de\fned as the di\u000berence between the triplet\nenergy\u000fT0and the singlet energy \u000fS,\u000f=\u000fT0\u0000\u000fS, and\nis controlled by the variations in the gate voltages. We\nrestrict ourselves to the limit of a rather large external\nmagnetic \feld so that the splitting between the triplet\nstates is larger than the magnitude of the o\u000b-diagonal\nmatrix elements that mix the singlet and triplet states.\nWhen the separation between the energy levels is much\nlarger than the matrix elements that mix the singlet and\ntriplet states, the singlet and triplet states are well sepa-\nrated. The singlet-triplet matrix elements produce anti-\ncrossings between the singlet and triplet levels when their\nenergies are tuned to be close to resonance. Our focus is\non situations where the system is tuned close to the S-T+\ntransition as shown in Fig. 1 There, the energies of the\nST+T0T-\nS\nT+0.81.01.21.4Detuning Ha.u.L\n-1.0-0.50.51.01.5Energy Ha.u.L\nFIG. 1: Schematics of the singlet and triplet energy levels as\na function of the detuning energy \u000f=\u000fT0\u0000\u000fSclose to the\nS-T+anticrossing. The Zeeman splitting \u0011Z= 1 is chosen as\nthe energy unit, o\u000b-diagonal matrix elements are v?=jv\u0006j=\n0:07.\ntriplet states T0andT\u0000are of the order the electron Zee-\nman splitting \u0011zaway from the energies of singlet Sand\ntripletT+states, which is a large energy as compared to\ntheS-T+anticrossing width. In this case, the electron\ndynamics can be approximated by the 2 \u00022 dynamics for\nthe singlet Sand triplet T+amplitudes of the electron\nwave function. The reduction of the original 4 \u00024 elec-\ntron dynamics problem to a 2 \u00022 problem also facilitates\n\fnding an exact solution for the electron dynamics forlinear sweeps and allows to reveal the role of the long\ntime \\tails\" of the singlet and triplet amplitudes crucial\nfor the nuclear spin dynamics. In the 2 \u00022 basis, the elec-\ntron dynamics is described by the singlet cSand triplet\ncT+amplitudes that obey a Schr odinger equation\nH(ST+)\u0012\ncS\ncT+\u0013\n=i~@t\u0012\ncS\ncT+\u0013\n(8)\nwith the Hamiltonian\nH(ST+)=\u0012\n\u000fSv+\nv\u0000\u000fT+\u0013\n; (9)\nwhere\u000fT+=\u000fT\u0000\u0011z, and following Refs. [16,17] we have\nincluded the spin-orbit matrix elements v\u0006\nsothat couple S\nandT+states into the total o\u000b-diagonal matrix elements\nv\u0006=v\u0006\nn+v\u0006\nso: (10)\nWhile the coupling between SandTlevels in GaAs\ndouble quantum dots is usually attributed to the hy-\nper\fne interaction, spin-orbit coupling is inevitably\npresent while di\u000ecult to evaluate quantitively for spe-\nci\fc devices.25It manifests itself in spin relaxation,26,27\nlevel anticrossings in InAs single and double dots,28,29\nand in the EDSR30,31both in GaAs32,33and InAs34dou-\nble dots. It is important to emphasize the existence of\ndi\u000berent mechanisms that couple the electron spin to the\norbital degrees of freedom. They include the traditional\n(Thomas) spin-orbit interaction that couples the electron\nspin to the electron momentum and the Zeeman interac-\ntion in a inhomogeneous magnetic \feld B(r) that couples\nthe electron spin to the elecron coordinate.35In Ref. 32,\nthe \frst mechanism dominated while in Refs. 36 and 33\ndi\u000berent versions of the second one were important. We\nshow in what follows that spin-orbit coupling also has a\nprofound e\u000bect on the nuclear spin polarization produc-\ntion rate.\nBy carrying out a unitary transformation of the orig-\ninal 4\u00024 Hamiltonian, it can be shown that the cor-\nrections to the reduced 2 \u00022 Hamiltonian of Eq. (9) are\nquadratic in the small ratio between v\u0006and the Zeeman\nsplitting\u0011Zprovided the gate-voltage induced S-T+tran-\nsition is slow so that ~(_\u000fS\u0000_\u000fT)=\u00112\nZ\u001c1. We assume that\nthis criterion is satis\fed.\nIn turn, the dynamics of nuclear spins is driven by\nthe e\u000bective magnetic \felds \u0001jarising from the electron\ndynamics\n~d^Ij\ndt=\u0001j\u0002^Ij; (11)\nwhere the components of the \felds \u0001jacting on the nu-\nclei are the transverse \u0001\u0006\nj=\u0000\n\u0001x\nj\u0006i\u0001y\nj\u0001\n=p\n2 and longi-\ntudinal \u0001z\nj\felds:\n\u0001+\nj=A\u001ajcSc\u0003\nT+; (12a)\n\u0001\u0000\nj=A\u001ajc\u0003\nScT+; (12b)\n\u0001z\nj=A\u0010jjcT+j2\u0000\u0011j(nZ); (12c)5\nand\u0011j(nZ)is the nuclear Zeeman splitting for the nu-\ncleusj. Because the dynamics of electron amplitudes\n(cS(t);cT+(t)) depends not only on the potentials on the\ngates but also on the nuclear spins through the matrix\nelementsv\u0006, \felds \u0001jcan be considered as dynamical\nRKKY \felds.\nIn the next section we show how the changes in the\nelectronic states as they pass across the S-T+anticrossing\nchange the spatially dependent nuclear polarization.\nIV. DYNAMICAL NUCLEAR POLARIZATION\nWe consider a situation where the changes in the gate\nvoltages can induce a singlet Sto tripletT+transition or\nvice versa, so that the total electron angular momentum\nmay be increased or reduced by 1. In the absence of\nspin-orbit coupling, this implies that the change in the\nz-projection of the total nuclear spin equals the change\nin the elecron spin (but with the opposite sign). There\nis no conservation law for the spatial distribution of the\nnuclear spin. We are interested in how this change of\nangular momentum is distributed among the nuclei. As\nalready mentioned above, the typical time scale for nuclei\ndynamics is long as compared to the time scale for the\nelectron dynamics, in particular, with the singlet-triplet\ntransition time. Let us denote the initial time of the\nsweep asTiand the \fnal time as Tf. We assume that the\nduration of the Landau-Zener sweep, Tf\u0000Ti, is short as\ncompared to the typical nuclear spin precession time and\ntake the nuclear dynamics into account as a perturbation.\nAlso, since the total change of the angular momentum is\nof the order 1, the typical change in the individual nuclear\nspins is much less than 1. With these assumptions, the\nchange of a nuclear spin \u0001 ^Ij=^Ij(Tf)\u0000^Ij(Ti) during a\nLandau-Zener transition is\n\u0001^Ij=\u0000j(Tf;Ti)\u0002^Ij(Ti); (13)\nwhere the total e\u000bect of the electrons on the nuclei is the\nintegrated e\u000bect of the magnetic splitting in Eqs. (12a),\n(12b), and (12c) :\n\u0000j(Tf;Ti) =ZTf\nTidt\n~\u0001j(t): (14)\nIn order to \fnd explicit expressions for the dependence\nof the electron states on the e\u000bective \feld induced by the\ntransverse nuclear spin polarization v\u0006\nn, it is convenient\nto make a transformation of the singlet and triplet am-\nplitudes\ncs= ~cs,cT+= ~cT+v\u0000=v?: (15)\nThen the Hamiltonian becomes real, and\n\u0012\n\u000fsv?\nv?\u000fT+\u0013\u0012\n~cS\n~cT+\u0013\n=i~@t\u0012\n~cS\n~cT+\u0013\n; (16)wherev?=jv\u0006j. Eq. (16) depends, in addition to the\nexternal magnetic \feld, on the absolute value of the com-\nbined e\u000bect of the nuclear spin induced transverse e\u000bec-\ntive \feld and spin-orbit interaction, but does not depend\non its direction.\nIn this basis, we can express the total e\u000bect of the ( x;y)\ncomponents of the e\u000bective \feld of Eq. (14) in terms of\n\u0000\u0006\nj=\u0006iA\u001aj\u0003\u0006v\u0006=(2v2\n?); (17)\nwhere the dimensionless functions \u0003\u0006(Ti;Tf) are de\fned\nas\n\u0003\u0000=i2v?ZTf\nTidt\n~~c\u0003\nS(t)~cT+(t); (18)\nand \u0003+= (\u0003\u0000)\u0003. This expression can be transformed\nby using the equation ~ cT+=v\u00001\n?(i~@t\u0000\u000fs(t))~csfollowing\nfrom Eq. (16),\n\u0003\u0000=\u00002ZTf\nTidt~c\u0003\nS(t)@~cs(t)\n@t\u0000i2ZTf\nTidt\n~\u000fs(t)jcs(t)j2;\n(19)\nso that\nRef\u0003\u0006g=P=jcs(Ti)j2\u0000jcs(Tf)j2; (20)\nis the transition probability P(Ti;Tf) from the singlet S\nto the triplet T+state. There is no such a simple relation\nbetween the imaginary parts of \u0003\u0006and the transition\nprobability, and this fact is important for the following\ndiscussion of the e\u000bect of the Landau-Zener sweeps on\nnuclei. However, we observe that when the Hamiltonian\nin the Schr odinger equation (16) is stationary, i.e., when\nthe gate voltages are \fxed and \u000fSand\u000fT+are constant\nin time, and the system is in an eigenstate of the Hamil-\ntonian of Eq. (16), the \feld ~ c\u0003\nS~cT+is real implying a non-\nvanishing imaginary contribution to \u0003\u0006. The imaginary\npart of \u0003\u0006thus includes contributions that can be un-\nderstood in terms of RKKY-like static nuclear spin-spin\ninteraction mediated by the electronic state, but this in-\nteraction also depends on the spin-orbit coupling. We\nwill diagonalize the stationary Hamiltonian of Eq. (16)\nin Sec. V D and relate the imaginary part of \u0003\u0006to the\nstatic electronic properties and show how this in\ruences\nthe dynamical nuclear dynamical properties. The imag-\ninary part of \u0003\u0006is central for the understanding of the\ndynamical nuclear polarization and we de\fne\nQ= Imf\u0003+g=\u0000Imf\u0003\u0000g: (21)\nWe also express the total e\u000bect of the \feld along zas\n\u0000z\nj=A\u0010j\u0003z=(2v?)\u0000\u0011j(nZ)(Tf\u0000Ti)=~; (22)\nwhere\n\u0003z= 2v?ZTf\nTidt\n~\f\f~cT+(t)\f\f2: (23)6\nUsing Eqs. (12a), (12b), and (12c), as well as express-\ning \u0000\u0006\njand \u0000z\njof Eqs. (17) and (23) in terms of \u0003\u0006\nand \u0003z, we arrive at the spin production during a single\nS!T+transition both in the transverse\n\u0001^I\u0006\nj=A\n2v?\u0014v\u0006\nv?\u0003\u0006\u001aj^Iz\nj\u0006i\u0003z\u0010j^I\u0006\nj\u0015\n\u0007i\u0011j(nZ)\n~(Tf\u0000Ti)^I\u0006\nj (24)\nand the longitudinal components\n\u0001^Iz\nj=\u0000A\n2v2\n?h\n\u0003\u0000v\u0000\u001aj^I+\nj+ \u0003+v+\u001aj^I\u0000\nji\n: (25)\nNext, substituting operators ^ v\u0006\nnin Eq. (4) by their semi-\nclassical values v\u0006\nnand using Eq. (10), we \fnd the change\nin thez-component of the total nuclear spin, \u0001 Iz=P\nj\u0001Iz\nj,\n\u0001Iz=\u0000P+1\n2v2\n?\u0002\n\u0003\u0000v\u0000v+\nso+ \u0003+v+v\u0000\nso\u0003\n; (26)\nor\n\u0001Iz=\u0000P\n2v2\n?(v\u0000v+\nn+v+v\u0000\nn)\u0000iQ\n2v2\n?(v\u0000v+\nn\u0000v+v\u0000\nn):(27)\nNote that the change in the z-component of the total\nnuclear spin is computed under the constraint that the\ntransverse nuclear \felds are v\u0006\nnbefore the sweep.\nRemarkably, \u0001 Izof Eq. (26) only depends on the basic\nparameters of the Hamiltonian H(ST+)of Eq. (9) and the\nshape of the sweep and does not depend on the detailed\ntopography of nuclear spins. Therefore, the result is very\ngeneral and convenient to use. In this respect, transfer\nof the longitudinal component of the angular momentum\ndi\u000bers from the transfer of its transverse component that,\naccording to Eq. (24), depends on the speci\fc spin con-\n\fguration.\nIn the absence of spin-orbit interaction, v\u0006\nso= 0, the\ntotal change in the electron spin equals the transition\nprobability P, as expected for a (partial) transition be-\ntween the singlet Sand tripletT+states. Conservation of\nthezcomponent of the angular momentum then dictates\nthat the change in the zcomponent of the total nuclear\nspin equals\u0000P. Spin-orbit coupling breaks the conser-\nvation law for the angular momentum transfer from the\nelectronic to the nuclear spin system since angular mo-\nmentum can be transferred to or from the lattice as well.\nSuch processes manifest themselves in the second term\nin Eq. (26). It depends on the relative phase between\nthe spin-orbit and hyper\fne interaction matrix elements.\nFurthermore, this term depends not only on the transi-\ntion probability P, but also on the imaginary parts of\n\u0003\u0006.Qacquires contributions not only from the part of\nthe sweep near the anticrossing point but also from its\nlong tails. As a result, the magnitude of Qcan be much\nlarger than Pfor certain classes of sweeps. This genericfeature suggests that Qcan be made large, and the spin-\norbit coupling can strongly in\ruence nuclear dynamics\neven when it is weaker than the hyper\fne coupling.\nOur results con\frm the prediction of Ref. 17 that the\nspin-orbit coupling in\ruences the nuclear spin generation\nrate profoundly. The quantity computed in Ref. 17 is the\ntotal change of the nuclear spin \u0001 Izaveraged over the\nphase of the transverse nuclear \feld v\u0006\nn. This averaging\nannihilates the second term of Eq. (27) while the \frst\nterm coincides with Eq. (9) in Ref. 17.37SinceQcan be\nconsiderably larger than P, we expect enhancement of\nspin production rate in experiments performed at a \fxed\n(while generic) values of v\u0006\nn.\nV. LINEAR SWEEPS - LANDAU-ZENER\nELECTRON TRANSITIONS\nWhen the changes in the gate voltages are such that\nthe di\u000berence in the energy betwen the singlet Sand the\ntripletT+varies linearly in time, Eq. (16) reduces to the\nstandard Landau-Zener problem. Because the Landau\napproach based on analytical continuation allows \fnding\nonly the transition probabilities,38we employ in the fol-\nlowing the Zener approach21allowing \fnding explicit ex-\npressions for the time dependence of electron wave func-\ntions that drives the coherent nuclear spin dynamics. We\nconsider a transition from the singlet Sstate to the triplet\nT+state, but because of the symmetries of the Hamilto-\nnian the solution can also be used to \fnd the wave func-\ntions that describe the transition from the triplet T+to\nthe singlet Sstate. We derive this relation in Sec. V C.\nDe\fningt= 0 as the time when the energies \u000fsand\u000fT+\nof the singlet Sand triplet T+are equal, we introduce\n\u000fs=\f2t=2~; \u000fT+=\u0000\f2t=2~; (28)\nwhere\fis a positive number with dimension of energy.\nThis representation implies that the singlet state has the\nlowest energy at early (negative) times and the triplet\nstate has the lowest energy for large \fnal (positive) times.\nA natural time-scale is ~=\fso that Eq. (16) with \u001c=\nt\f=~reads\n\u0012\n\u001c=2p\rp\r\u0000\u001c=2\u0013\u0012\n~cS\n~cT+\u0013\n=i@\u001c\u0012\n~cS\n~cT+\u0013\n; (29)\nwhere\n\r= (v?=\f)2(30)\nis the Landau-Zener parameter. When \ris small, the\ntransition probability from the singlet Sto the triplet\nstateT+is small. In the opposite limit, when \ris large,\nthe transition probability is close to 1. As above, we\ndenote the initial time from where the sweep starts as Ti\nand the \fnal time where it ends as Tf. In dimensionless\nunits, we have \u001ci=Ti\f=~and\u001cf=Tf\f=~.\nIn order to determine the change in the nuclear spin\npolarization, we need to compute not only the transi-\ntion probability P, but also the singlet Sand triplet T+7\namplitudes, ~ cSand ~cT+. Because the nuclear dynamics is\ncontrolled by the electron dynamics via the e\u000bective \felds\nof Eqs. (12a), (12b), and (12c), explicit expressions for\nthe amplitudes (~ cS(\u001c);~cT+(\u001c)) should be found not only\nnear the anticrossing point \u001c= 0, but along the whole\nsweep,\u001ci\u0014\u001c\u0014\u001cf. Therefore, it is necessary to em-\nploy Zener's derivation of the Landau-Zener transition\nprobability21and complement it with a detailed infor-\nmation about the asymptotic behavior of the amplitudes\nand e\u000bective magnetic \felds.\nEliminating ~ csfrom Eq. (29) by substituting\n~cs=1p\r\u0010\u001c\n2+i@\u001c\u0011\n~cT+; (31)\ninto its \frst row, we \fnd\n@2\n\u001c~cT++\u0012\n\r\u0000i\n2+1\n4\u001c2\u0013\n~cT+= 0: (32)\nThen, by changing the variable \u001cto\nz=ei3\u0019=4\u001c; (33)\nEq. (32) transforms to\n@2\nz~cT+(z) +\u0012\nn+1\n2\u00001\n4z2\u0013\n~cT+(z) = 0; (34)\nwheren=i\r. This is the Weber equation39,40\nwhose solutions are the parabolic cylinder (Weber) func-\ntionsDn(z),Dn(\u0000z),D\u00001\u0000n(\u0000iz) andD\u00001\u0000n(iz) of\nwhich only two are linearly independent. When ex-\npressed as functions of the real argument \u001c, they cor-\nrespond to Di\r(ei3\u0019=4\u001c),Di\r(\u0000ei3\u0019=4\u001c),D\u00001\u0000i\r(ei\u0019=4\u001c)andD\u00001\u0000i\r(\u0000ei\u0019=4\u001c), respectively. In a similar way, we\n\fnd the di\u000berential equation that the singlet amplitude\nobeys. Eliminating ~ cT+by substituting\n~cT+=1p\r\u0010\n\u0000\u001c\n2+i@\u001c\u0011\n~cS; (35)\ninto the second row of Eq. (29) and taking its complex\nconjugate, we \fnd\n@2\n\u001c~c\u0003\nS+\u0012\n\r\u0000i\n2+1\n4\u001c2\u0013\n~c\u0003\nS= 0: (36)\nHence ~c\u0003\nSsatis\fes the same di\u000berential equation (32) as\n~cT+; its solutions are the Weber functions listed above. In\nSec. V A we discuss the asymptotic behavior of the singlet\nSand triplet T+amplitudes that is critical for imposing\nthe initial conditions and \fnding long time scale nuclear\nspin dynamics.\nA. Asymptotic Expansions\nFor the following, the asymptotic behavior of the so-\nlutions in both limits, \u001c!\u00061 , is required. However,\nbecause the solutions appear in pairs, with opposite signs\nof\u001c, it is su\u000ecient to \fnd their \u001c >0 asymptotics. We\nnote that the indeces of all above D-functions are imagi-\nnary or complex [ i\ror (\u00001\u0000i\r)] while the asymptotics of\nRefs. 39,40 are valid only for Dn(z) functions with inte-\nger indeces.41In what follows, we employ the asymptotic\nexpressions from Mathematica 8 which are valid for arbi-\ntrary complex indices. For large positive times \u001c!1 ,\nthey are\nDi\r(ei3\u0019=4\u001c)\u0019e\u00003\u0019\r=4ei\u001c2=4\u001ci\r+ei\u0019=4p\n2\u0019\n\u0000(\u0000i\r)e\u0000\u0019\r=4e\u0000i\u001c2=4\u001c\u00001\u0000i\r+O(\u001c\u00002); (37a)\nDi\r(\u0000ei3\u0019=4\u001c)\u0019e\u0019\r=4ei\u001c2=4\u001ci\r+O(\u001c\u00002); (37b)\nD\u00001\u0000i\r(ei\u0019=4\u001c)\u0019e\u0000i\u0019=4e\u0019\r=4e\u0000i\u001c2=4\u001c\u00001\u0000i\r+O(\u001c\u00003); (37c)\nD\u00001\u0000i\r(\u0000ei\u0019=4\u001c)\u0019p\n2\u0019\n\u0000(1 +i\r)e\u0000\u0019\r=4ei\u001c2=4\u001ci\r+ei3\u0019=4e\u00003\u0019\r=4e\u0000i\u001c2=4\u001c\u00001\u0000i\r+O(\u001c\u00002): (37d)\n[asympt1-4] One can see that as \u001c!1 the function\nD\u00001\u0000i\r(ei\u0019\r= 4\u001c) vanishes as \u001c\u00001while the absolute val-\nues of the three other D-functions saturate. We note\nthat all asymptotic expressions for the D-functions in-\nclude two oscillatory factors. The Fresnel-type factors\nexp(\u0006i\u001c2=4) originate from the accumulation of the adi-\nabatic Schr odinger phases during a linear sweep, and the\nfactors\u001c\u0006i\rdepending on \rre\rect the non-adiabaticity.It follows from Eq. (37c) that for a sweep starting from\nthe singlet Sstate at large negative initial time \u001ci, the\nfunctionD\u00001\u0000i\r(\u0000ei\u0019\r= 4\u001c) should be chosen as one of\nthe basis functions for the triplet T+state because it\nvanishes when \u001c!\u00001 . We choose Di\r(ei3\u0019\r=4\u001c) as the8\nsecond basis function. Then\n~cT+(\u001c) =ap\re\u0000i3\u0019=8D\u00001\u0000i\r(\u0000ei\u0019=4\u001c)\n\u0000bp\re\u0000i3\u0019=8Di\r\u0010\nei3\u0019=4\u001c\u0011\n; (38)\nwhereaandbare coe\u000ecients that depend on the initial\ntime\u001ci. The overall phase factor as well as the factorsp\r\nand\u00001=p\rhave been chosen as a matter of convenience\nin the following transformation. One can check that b/\n\u001c\u00002\niforj\u001cij\u001d1.\nEq. (36) implies that ~ c\u0003\nS, the complex conjugate of the\nsingletSamplitude, can be expressed in terms of the\nsame Weber functions as the triplet amplitude ~ cT+. An\nexplicit connection between them can be found by em-\nploying Eq. (31), and the expression for the singlet com-\nponent ~cscan be further simpli\fed by using the standard\nrecurrence relations for D-functions.39,40As applied to\ntheD-functions of Eq. (38), they read\n\u0010\u001c\n2+i@\u001c\u0011\nDi\r(ei3\u0019=4\u001c) =\u0000\rei3\u0019=4D\u00001+i\r(ei3\u0019=4\u001c)\n(39)\nand\n\u0010\u001c\n2+i@\u001c\u0011\nD\u00001\u0000i\r(\u0000ei\u0019=4\u001c) =ei3\u0019=4D\u0000i\r(\u0000ei\u0019=4\u001c)\n(40)\nTheD-functions of the right hand side of Eqs. (39) and\n(40) di\u000ber from the D-functions of Eq. (38), but are\nrelated to them by complex conjugation\nD\u00001+i\r(ei3\u0019=4\u001c) =h\nD\u00001\u0000i\r(\u0000ei\u0019=4\u001c)i\u0003\n;(41)\nD\u0000i\r(\u0000ei\u0019=4\u001c) =h\nDi\r(ei3\u0019=4\u001c)i\u0003\n: (42)\nTherefore, the general solution for the singlet ampli-\ntudes is\n~cS(\u001c) =ah\ne\u0000i3\u0019=8Di\r(ei3\u0019=4\u001c)i\u0003\n+bh\ne\u0000i3\u0019=8D\u00001\u0000i\r(\u0000ei\u0019=4\u001c)i\u0003\n:(43)\nAs a consequence, the function \u0003\u0000of Eq. (18) depend-\ning on the product ~ c\u0003\nS(t)~cT+(t) and describing the re-\nsponse of nuclear spins to a Landau-Zener pulse can be\nexpressed in terms of two functions D\u00001\u0000i\r(\u0000ei\u0019\r= 4\u001c)\nandDi\r(ei3\u0019\r=4\u001c). In Sec. V B, we consider the Landau-\nZener scenario when the initial electron state is prepared\nat\u001ci!\u00001 and the sweep runs to \u001cf!1 , as well as\nthe asymptotic behavior of e\u000bective \felds ~ c\u0003\nS~cT+at large\nbut \fnite timesj\u001cj\u001d1.B. In\fnite Limits and Asymptotics\nWhen the system is in the singlet state at early times,\nj~cS(\u001c!\u00001 )j= 1 and ~cT+(\u001c!\u00001 ) = 0, then b= 0\nandjaj2e\u0019\r=2= 1, as follow from Eq. (37b), and\n~cS(\u001c) =ei'e\u0000\u0019\r=4h\ne\u0000i3\u0019=8Di\r(ei3\u0019=4\u001c)i\u0003\n; (44a)\n~cT+(\u001c) =ei'e\u0000\u0019\r=4p\rh\ne\u0000i3\u0019=8D\u00001\u0000i\r(\u0000ei\u0019=4\u001c)i\n;\n(44b)\nwhere'is an arbitrary phase. For a \fnite but large initial\ntime\u0000\u001ci(\u001ci>0), this description remains satisfactory\nwith the accuracy to the terms of the order \u001c\u00002\niin the\nsinglet amplitude of Eq. (44a) and of the order \u001c\u00001\niin\nthe triplet amplitude of Eq. (44b).\nFor completeness, let us also consider the situation\nwhen the system is in the triplet state T+at early times\n\u001c!\u00001 . Then it follows from Eqs. (37b) and (37c) that\na= 0 ande\u0019\r=2jbj2=\r= 1, so that\n~cS(\u001c) =ei'0e\u0000\u0019\r=4p\rh\ne\u0000i3\u0019=8D\u00001\u0000i\r(\u0000ei\u0019=4\u001c)i\u0003\n;\n(45a)\n~cT+(\u001c) =\u0000ei'0e\u0000\u0019\r=4h\ne\u0000i3\u0019=8Di\r(ei3\u0019=4\u001c)i\n;(45b)\nwhere\u001e0is an arbitrary phase.\nWe can now \fnd the transition probability for the S!\nT+transition of Eq. (20). It is\nPLZ=j~cS(\u001c!\u00001 )j2\u0000j~cS(\u001c!1 )j2= 1\u0000e\u00002\u0019\r:\n(46)\nwhich is the celebrated Landau-Zener result. The trans-\nverse components of the e\u000bective \feld acting on the nu-\nclear spins are controlled by the product\n~c\u0003\nS~cT+=p\re\u0000\u0019\r=2e\u0000i3\u0019=4\u0002 (47)\nDi\r(ei3\u0019=4\u001c)D\u00001\u0000i\r(\u0000ei\u0019=4\u001c):\nIts asymptotic behavoir following from Eqs. (37b) and\n(37d) is\n~c\u0003\nS~cT+\u0019p\r\n\u001c+O(\u001c\u00002) (48)\nfor the early times \u001c!\u00001 and\n~c\u0003\nS~cT+\u0019\u0000p\r\n\u001c\u0002\n1\u00002e\u00002\u0019\r\u0003\n+p\re\u0000i3\u0019=4p\n2\u0019\n\u0000(1 +i\r)e\u00003\u0019\r=2ei\u001c2=2\u001c2i\r+O(\u001c\u00002) (49)9\nfor the late times \u001c! 1 . The absolute value of\nthe second term of Eq. (49) is e\u0000\u0019\rp\n1\u0000e\u00002\u0019\ras\ncan be checked by using the identity j\u0000(1 +i\r)j2=\n\u0019\r=sinh(\u0019\r). This result is easy to understand since it\nequalsj~cSjj~cT+jin the asymptotic regime \u001c!1 , where\nj~cT+j2= 1\u0000e\u00002\u0019\randj~cSj2=e\u00002\u0019\r. The second term of\nEq. (49) exhibits very fast Fresnel-like oscillations ei\u001c2=2\nwhen\u001c!1 and does not contribute signi\fcantly to\nthe integral \u0003\u0000of Eq. (18) describing the total e\u000bec-\ntive \feld applied to the nuclei as a result of the sweep.\nThis factor originates from the accumulation of the phase\nexp\bR\u0002\n\u000fS(t)\u0000\u000fT+(t)\u0003\ndt=~\t\nalong the sweep.\nThe origin of the coe\u000ecients in the 1 =\u001cterms in Eqs.\n(48) and (49) can also be made quite transparent. By\nusing the time-dependent Schr odinger equation (29), we\n\fnd\n(i@\u001c+\u001c)\u0000\n~c\u0003\nS~cT+\u0001\n=p\rh\nj~cS(\u001c)j2\u0000\f\f~cT+(\u001c)\f\f2i\n\u0001(50)\nKnowing that for early times, \u001c!\u00001 , the amplitudes\napproachj~cSj2= 1 and\f\f~cT+\f\f2= 0, we recover Eq. (48).\nFor late times,j~cSj2\u0000\f\f~cT+\f\f2!\u00001+2 exp(\u00002\u0019\r), which\nexplains the 1 =\u001cterm in Eq. (49). Furthermore, we note\nthat in the leading order the operator ( i@\u001c+\u001c) annihilates\nthe second term of Eq. (49).\nThe integrals of Eqs. (48) and (49) diverge logarith-\nmically when the integration limits approach \u00061. This\nmeans that while PLZof Eq. (46) and the total spin\ntransfer \u0001Izof Eq. (26) (for vso\u0006= 0) are controlled by\nthe vicinity of the anticrossing point, the e\u000bective \felds\n\u0001jand shake up processes in the nuclear subsystem pro-\nduced by them are controlled by the global shape of the\npulse. The same is true for \u0001 Izwhenvso\u00066= 0. We note\nthat while the presence of logarithmic terms is a general\nproperty of linear sweeps, they contribute to \u0001 Izonly in\nthe presence of spin-orbit coupling.\nC. Reverse sweep from the triplet T+to the singlet\nS.\nLet us relate the reverse sweep, starting in a triplet\nstateT+and sweeping to a singlet state S, to theS!T+\nsweep elaborated above. Since now the rates of the\nchange of the singlet Sand triplet T+energies have\nthe signs opposite to the signs in Eq. (28), the dynam-\nical equations for the amplitudes (~ cS;~cT+) di\u000ber from\nEq. (16) by the interchange ~ cS$~cT+. Furthermore,\nfor aT+!Stransition, the system was initially in the\ntripletT+state, hence, the singlet Samplitude vanishes\nat the early time. Therefore, the initial conditions are\nalso ~cS$~cT+interchanged as compared to the S!T+\nsweep. This implies that their product transforms as\n~c\u0003\nS~cT+!\u0000\n~c\u0003\nS~cT+\u0001\u0003, and \u0003\u0006! \u0000 (\u0003\u0006)\u0003according to\nEq. (18g. In other words the transition probability\nP= Ref\u0003\u0006gchanges sign, but the imaginary parts\nQ= Imf\u0003+gremain unchanged. The change of thesign of Ref\u0003\u0006gis obvious because of the S$T+in-\nterchange, so that the longitudinal component of the an-\ngular momentum transfer changes sign. However, the\ne\u000bective \feld Im f\u0001\u0006gdoes not change, and this indi-\ncates that the imaginary components of \u0003\u0006should add\nduring aS!T+!Scycle.\nIn conclusion of this section, for linear sweeps the di-\nmensionless function \u0003\u0000(Ti;Tf) that re\rects the e\u000bect of\na single Landau-Zener sweep on nuclei diverges logarith-\nmically when Ti!\u00001 andTf!1 . In Sec. VI, we\ndiscuss in more detail the dependence of \u0003\u0006(Ti;Tf) on\nthe limits ( Ti;Tf) and the Landau-Zener parameter \r.\nD. Adiabatic Regime\nSome more insight on the long- \u001ctails of the products\n~c\u0003\nS~cT+comes from the stationary solution of Eq. (16).23\nFor a large detuning \u000e=\u000fT+\u0000\u000fSfrom theS\u0000T+an-\nticrossing, whenj\u001cj\u001d1, the stationary solution of Eq.\n(16) provides an adiabatic approximation to the singlet\nand triplet amplitudes. Note that we still assume the du-\nration of the sweep is short as compared to the nuclear\nLarmor precession time.\nThen the eigenenergies of the electronic states of the\nHamiltonian of Eq. (16) are\n\u000f\u0006=1\n2\u0000\n\u000fs+\u000fT+\u0001\n\u0006q\nv2\n?+ (\u000e=2)2; (51)\nand at the lower branch of the energy spectrum the prod-\nuct of the amplitudes equals\n~c\u0003\nS~cT+=\u0000v?=2p\nv2\n?+ (\u000e=2)2: (52)\nHere the oscillatory \u001c-dependent phase factors cancel be-\ntause ~cSand ~cT+belong to the same eigenvalue. It im-\nmediately allows calculating the transverse components\n\u0001+\nj=A%j~cS~c\u0003\nT+v\u0006=v?and \u0001\u0000\nj=A%j~c\u0003\nS~cT+v\u0000=v?of\u0001j\nand the e\u000bective \felds from Eqs. (12a) and (12b). The\ntransverse components \u0001\u0006\njvanish asv?=\u000ewhenj\u000ej=v?\n!1 . Similarly, the longitudinal component found from\nEq. (12c) equals\n\u0001z\nj=\u0000A\u0010j\n22\n41\u0000\u000e=2q\nv2\n?+ (\u000e=2)23\n5\u0000\u0011j(nZ): (53)\nFar from the intersection, when \u000e=v?!\u00001 and the\neigenstate is almost a pure triplet, \u0001z\nj!\u0000A\u0010j\u0000\u0011j(nZ).\nIn the opposite limit, when \u000e=v?!1 and the eigenstate\nis almost a pure singlet, \u0001z\nj!\u0000\u0011j(nZ). The point \u000e= 0\nhas been identi\fed as \\spin funnel\" in Ref. 42.\nIn the adiabatic limit, the \felds \u0001jacquire the usual\nmeaning of RKKY \felds with a nuclear dynamic time\nscale oft\u0018~=\u0001j. Near the level anticrossing point \u000e= 0,\n\u0001j\u0018An0=Nwheren0is the concentration of nuclei and10\nNis the number of nuclei in the dot. With An0\u001910\u00004\neV andN\u0019106,t\u001910\u0016s.\nFor a slow linear sweep between \u001ci=\u0000\u001cfand\u001cf, with\n\u000e!\f\u001c, one \fnds from Eqs. (18) and (52) the quantity\n\u0003\u0006\n(a)which, according to Eq. (21), result in\nQ(a)= 4\rln0\n@q\n\u001c2\nf+ 4\r+\u001c2\nf\n2p\r1\nA; (54)\nand from Eq. (20) we \fnd P(a)= 0. The results for P(a)\nandQ(a)hold with logarithmic accuracy; the subscript\n(a) indicates that they were derived in the adiabatic ap-\nproximation. In the same way, one can check that ~ c\u0003\nS~cT+\nof Eq. (52) is in agreement with the 1 =\u001cterms of Eqs. (48)\nand (49).\nApplying Eq. (52) to a nonlinear dependence \u000e=\u000e(\u001c),\none easily concludes that \u0003\u0006converges if \u000e(\u001c) is super-\nlinear and diverges by some power law if it is sublinear.\nEquation (52) implies important consequences for the\nnuclear spin dynamics under the condition of time-\nindependent detuning. Indeed, it follows from Eqs. (10),\n(11) { (12b), and (52) that the rate of change of the total\nnuclear spin is\n~@Iz\n@t=\u0000i\n2v+\nsov\u0000\nn\u0000v\u0000\nsov+\nnp\nv2\n?+ (\u000e=2)2: (55)\nTherefore, time-independent detuning results in produc-\ning a magnetization Izthat increases linearly in time as\nlong as the parameters of the electronic Hamiltonian re-\nmain unchanged. This generation of spin magnetization\nby time-independent electrical bias is possible because\nthe time-inversion symmetry is violated by a strong ex-\nternal \feld Bproducing Zeeman splitting of the electron\ntriplet state, and the simultaneous presence of hyper\fne\nand spin-orbit interactions. The magnitude of the ef-\nfect reaches its maximum at \u000e= 0, when the system is\nbrought to the center of the ST+anticrossing. The time\nscales of the parameter change can be estimated similarly\nto Sec. VII D. Under the usual conditions, the shortest\nof them corresponds to the precession of v\u0006\nnin the ex-\nternal \feld. These conclusions seem to agree with the\nobservations of Ref. 43.\nVI.S!T+SWEEPS AND ROUND CYCLES\nComplex functions \u0003\u0006(Ti;Tf) of Eq. (18) describe\nthe e\u000bect of a sweep on the nuclear spins. As seen\nfrom Eqs. (20) and (26), the probability of the electron\nS!T+transitionPis completely controlled by the\nreal part of \u0003\u0006,P= Ref\u0003\u0006g, while the angular mo-\nmentum transfered to the nuclear system \u0001 Izdepends\nboth on the real and imaginary parts of \u0003\u0006. Imaginary\nparts of \u0003\u0006are always present but manifest themselves\nin the nuclear spin accumulation only when there are two\ncompeting mechanisms of the electron spin transfer, hy-\nper\fne and spin-orbit.In this section, we \frst present data on the dependence\nof \u0003\u0006on the integration limits and the Landau-Zener pa-\nrameter\robtained by numerical integration of Eq. (29),\nand then develop an analytical approach for describing\nthe oscillatory dependence of the transition probability\nPon the cycle length.\nA. Linear sweeps\nWe begin with linear S!T+sweeps of Sec. V. For\nsuch sweeps, we denote the initial time \u0000\u001ci(\u001ci>0) and\nthe \fnal time \u001cf(\u001cf>0) so that the duration of the\nsweep is\u001ci+\u001cf. To reduce the number of parameters,\nwe assume \u001ci=\u001cf. Transition probabilities P(\u001cf) are\nplotted in Fig. 2(a) as a function of the sweep half-time\n\u001cffor two values of \r. While for large \u001cfboth curves sat-\nurate to the Landau-Zener probabilities PLZof Eq. (46),\noscillations of P(\u001cf) are very pronounced. They decay\nat a rather long time scale, and their shape cannot be\ndescribed by a single characteristic time. We attribute\nthe oscillations to the interference pattern between two\nspectrum branches and estimate their period \u001coscfrom\nthe Schr odinger exponent exp( \u0000iv?t=~) in the anticross-\ning point, what results in \u001cosc\u0019\r\u00001=2. The rate of their\ndecay is controlled by the passage time ~v?=\f2across the\navoided crossing that results in a decay time \u001cdec\u0019\r1=2.\nFinally, we arrive at a rough estimate of the transient\nregime\u001ctr\u0018maxf\r1=2;\r\u00001=2g. Actually, this only is\na lower bound on \u001ctr. The saturation takes a longer\ntime and the di\u000berence in the shapes of the \r= 1 and\n\r= 0:1 curves deserves more comments. The \r= 0:1\ncurve strongly resembles plots of Fresnel integrals, and\nwe attribute the oscillatons to the ei\u001c2=2factors in the\nasymptotics of Eq. (49). With increasing \r, the pat-\nterns of oscillations are getting less regular due to the\nsecond oscillatory factor \u001c2i\rin the asymptotics of ~ c\u0003\ns~cT+.\nThe switching of regimes happens at 2 \u0019\r\u00191 as is seen\nfrom the expression e\u00002\u0019\rfor the Landau-Zener transi-\ntion probability.\nIn agreement with the asymptotics found in Sec. V B,\nthe imaginary parts of \u0003\u0006displayed in Fig. 2(b,c) exhibit\na behavior quite di\u000berent from the behavior of their real\npartsP. They increase nearly logarithmically with \u001cf,\nwith weak oscillations superimposed on this monotonic\ngrowth. Their magnitudes increase with \r, and for\r\u00191\nand\u001cf\u001910 they are by one order of magnitude larger\nthanP. Therefore, even with a moderate spin-orbit cou-\npling, the imaginary parts of \u0003\u0006are expected to con-\ntribute essentially to the spin transfer \u0001 Izof Eq. (26).\nThis contribution should not only change the magnitude\nof \u0001Izbut also smoothen its \u001cf-dependence.\nIn Fig. 2(b), we also plot Q= Imf\u0003+gfor\r= 1 by\nusing the approximate adiabatic expression of Eq. (54)\nto compare it to the exact numerical result. Apart from\nsome details of the behavior for early and late times,\nwhich are expected, we see that the dominant contribu-\ntion toQcan be explained in terms of the adiabatic \feld11\n246810tfH=tiL0.20.40.60.81.0P=Re@L±D\n(a)\n246810tfH=tiL2468Q=Im@L+D\n(b)\n246810tfH=tiL0.10.20.30.40.5Q=Im@L+D\n(c)\nFIG. 2: (a) Transition probability P= Ref\u0003\u0006(\u001cf)gfor a\nlinear sweep starting in the Sstate at the initial time \u0000\u001ci\nand ending at the \fnal time \u001cf=\u001ciplotted as a function\nof the half-sweep time \u001cffor two values of the Landau-Zener\nparameter\r. Full (blue) line \r= 1, dashed (red) line \r=\n0:1. The anticrossing point is passed in the middle of the\nsweep at time \u001c= 0. The full (blue) lines in (b) and (c) are\nQ= Imf\u0003+(\u001cf)gfor\r= 1 and\r= 0:1, respectively. In (b)\nand (c), the dashed (green) lines are the adiabatic solutions\nof Eq. (54).\nof Eq. (54). Fig. 2(c) provides a similar comparison,\nbut for a faster sweep with \r= 0:1. Even in this situa-\ntion, the adiabatic approximation is a reasonable starting\npoint for describing the basic shape of Qof Eq. 21.\nThe above analysis of linear sweeps, together with\nthe arguments of Sec. V D, allow to make some conclu-sions about the generic (non-linear) S-T+sweeps as well.\nImagine the sweeps with the rate unchanged near the\nanticrossing but increasing away from it. As long as the\nspeed-up happens at times \u001c >\u001ctr(this inequality should\nbe ful\flled strong enough), the probability P= Ref\u0003\u0006g\nchanges only modestly, while the long time tails of the\nproducts ~c\u0003\nS(\u001c)~cT+(\u001c) contributing to Q= Imf\u0003+gare\ncut-o\u000b. Thus, increasing the sweep rate away from the\nanticrossing reduces Qand might have a profound e\u000bect\non \u0001Iz. However, its speci\fc magnitude depends on the\nvalues of a number of parameters such as v\u0006\nn;v\u0006\nso;\u001ctr, and\nthe speed-up time.\nB. Cyclic linear sweeps\nRound sweeps are of the most practical interest for\nexperiment, and their detailed shapes are nontrivial be-\ncause of the oscillating tails of Re f\u0003\u0006gof Fig. 2(a).\nTherefore, we provide below the data on \u0003\u0006for two dif-\nferent round sweeps starting in the singlet states Sat\n\u001ci<0.\nFig. 3 presents data for a round sweep of the total\nduration of 4 \u001cfthat includes the sweep of Fig. 2 from\n\u001ci=\u0000\u001cfto\u001cfand the backward sweep that begins im-\nmediately after the end of the forward sweep. According\nto Eq. (20), P= Ref\u0003\u0006gdisplays the probability of\nS!T+transition. Remarkably, Fig. 3(a) shows that for\n\r= 1 the decay of Pis rather long and includes deep\nand irregular oscillations. For \r= 0:1,P(\u001cf) shows a\nwide maximum at \u001cf\u00192, and the following oscillations\nwithout any visible decay up to \u001cf= 10. In this case,\na double dot in the linear sweep regime resembles a res-\nonator of a length decreasing as \u001c\u00001\nf. We expect that\n\frst peaks can be resolved experimentally, e.g., in beam\nsplitter experiments20while higher peaks should merge\ninto a background with P\u00190:5. Using \frst sharp peaks\nfor ultrafast spin operation is highly tempting.\nAs distinct from P= Ref\u0003\u0006g,Q= Imf\u0003+gof\nFig. 3(b) is a nearly monotonic function of \u001cffor\r= 1\n(with irregular oscillations superimposed), and is about\n10 for\u001cf= 10. Therefore, it can heavily contribute to\n\u0001Iz. However, Imf\u0003\u0006gis small and strongly oscillates\nat\r= 0:1.\nTo demonstrate the e\u000bect of the tunneling process near\nthe anticrossing point, in Fig. 4 are plotted the data for\na cycle that begins in the Sstate at\u0000\u001ci, reaches the\nanticrossing at \u001c= 0, and then runs immediately back\nwith the same speed until \u001cfwith\u001ci=\u001cf. Compari-\nson of Figs. 3(a) and 4(a) for \r= 1 shows quite similar\npatterns of the oscillations of P(\u001cf) that are more regu-\nlar in Fig. 4(a). However, the patterns for \r= 0:1 are\nrather di\u000berent demonstrating essential decrease in the\nspin transfer. The magnitudes of Q= Imf\u0003+gare small\nin both cases, but their \u001cfdependences are rather di\u000ber-\nent.12\n246810tfH=tiL0.20.40.60.81.0P=Re@L±D\n246810tfH=tiL51015Q=Im@L+D\nFIG. 3: (a) Transition probability of a S!T+transition\nP= Ref\u0003\u0006gand (b) the imaginary part Q= Imf\u0003+(\u001cf)g\nfor a round sweep plotted versus \u001cf(one fourth of the sweep\ntime). The \frst part of the sweep is the same as the sweep of\nFig. 2, and the second part sweeps in the opposite direction\nwith the same speed immediately after reaching the turning\npoint. Full (blue) lines \r= 1, dashed (red) lines \r= 0:1.\nC. Analytical theory of the probability oscillations\nWe can explain the oscillations in the transition prob-\nability as a function of the total duration of the cycle\nemploying the analytical results in Sec. V. The forward\nsweep from\u0000\u001cito the turning point \u001cmgives rise to the\nsinglet and triplet amplitudes of Eq. (44). Assuming\n\u001cm\u001d1 and\u001ci\u001d1, and employing (37a) and (37d), the\nsinglet and triplet amplitudes at the turning point \u001cmare\n~c(fS)\nS\u0019e\u0000\u0019\rei(\u001c2\ni\u0000\u001c2\nm)=4\u001ci\r\ni\u001c\u0000i\r\nm; (56)\n~c(fS)\nT+\u0019e\u0000i3\u0019=4e\u0000\u0019\r=2p2\u0019\r\n\u0000(1 +i\r)ei(\u001c2\ni+\u001c2\nm)=4\u001ci\r\ni\u001ci\r\nm:(57)\nHere the superscripts indicate that we started in the sin-\ngletSstate and carried out a forward linear sweep. The\nphase of the early time singlet state is arbitrary and is\nomitted because it only modi\fes the overall phase of the\nwave function and does not in\ruence the \fnal result for\nthe probability. The amplitudes of Eqs. (56) and (57) are\nderived under the assumption that ~ cS= 1 and ~cT+= 0\nat time\u001c=\u0000\u001ci.\nNext, we consider the backward sweep and include\nthe contributions from two channels passing through the\nT+andSstates at the turning point. As discussed in\n246810tfH=-tiL0.20.40.60.81.0Re@L±D\n246810tfH=-tiL2468-Im@L-DH=Im@L+DLFIG. 4: Real part P(a) and imaginary part Q(b) of the\nfunction \u0003+(\u001cf) for a round sweep plotted vshalf-sweep time\n\u001cf. First part of the sweep, starting in the Sstate at\u001ci<\n\u001c < 0, stops in the anticrossing point at \u001c= 0 and runs\nimmediately in the opposite direction until \u001cf=j\u001cij. Full\n(blue) lines \r= 1, dashed (red) lines \r= 0:1.\nSec. V C, the dynamical equations for the amplitudes\nfor the backward sweep (~ c(b)\nS;~c(b)\nT+) di\u000ber from Eq. (16)\nby the interchange ~ cS ! ~cT+. In order to make\ncontact with our results in Sec. V, we change the\ntime\u001c!\u001c\u00002\u001cmfor the backward sweep. Using the\ninterchange ~ cS ! ~cT+, it follows from Eq. (44b) that\nfor the triplet T+channel the ratio of the \fnal and initial\namplitudes along the backward sweep is ~ c(bT+)\nS=~c(fS)\nT+=\np\re\u0000i3\u0019=8D\u00001\u0000i\r\u0000\n\u0000ei\u0019=4\u001cf\u0001\n=\u0002\ne\u0000i3\u0019=8Di\r\u0000\n\u0000ei3\u0019=4\u001cm\u0001\u0003\u0003,\nT+in the superscript of ~ c(bT+)\nS indicates the channel. In\nthe limit\u001cf\u001d1, Eqs. (37d) and (37c) imply that this\nratio equals\n~c(bT+)\nS=~c(fS)\nT+\u0019e\u0000i3\u0019=4e\u0000\u0019\r=2p2\u0019\r\n\u0000(1 +i\r)ei(\u001c2\nf+\u001c2\nm)=4\u001ci\r\nf\u001ci\r\nm:\n(58)\nSimilarly, by using the interchange ~ cS ! ~cT+, it follows\nfrom Eq. (45b) that for the singlet Schannel the ratio\nof the \fnal and initial amplitudes along the backward\nsweep ~c(bS)\nS=~c(fS)\nS=Di\r\u0000\ne3i\u0019=4\u001cf\u0001\n=Di\r\u0000\n\u0000e3i\u0019=4\u001cm\u0001\n. In\nthe limit\u001cf\u001d1;Eqs. (37d) and (37c) imply that the\nratio of the singlet amplitudes after the backward sweep\nequals\n~c(bS)\nS=~c(fS)\nS\u0019e\u0000\u0019\rei(\u001c2\nf\u0000\u001c2\nm)=4\u001ci\r\nf\u001c\u0000i\r\nm: (59)\nThe singlet amplitude at the \fnal time \u001cfafter the cycle13\nof duration ( \u001ci+\u001cm)+(\u001cm+\u001cf) is a sum of the contribu-\ntions coming from both channels, ~ c(tot)\nS= ~c(bT+)\nS + ~c(bS)\nS.\nFinally,\n~c(tot)\nS\u0019ei(\u001c2\nf+\u001c2\ni\u00002\u001c2\nm)=4\u0012\u001cf\u001ci\n\u001c2m\u0013i\r\n\u0002h\n(1\u0000PLZ) +PLZei#(\u001cm)i\n; (60)\nwhere the Landau-Zener transition probability PLZfor a\nsingle-passage is de\fned by Eq. (46) and the phase #(\u001cm)\nat the turning point \u001cmis de\fned as\nei#(\u001cm)=ei\u001c2\nm\u001c4i\r\nme\u0000i\u0019=2\u0000(\u0000i\r)=\u0000(i\r): (61)\nThe dependence of the transition probability P= 1\u0000\nj~cS(\u001cf)j2on the position \u001cmof the turning point is\nP(\u001cm) = 4PLZ(1\u0000PLZ) sin2#(\u001cm)=2; (62)\nwhere#(\u001cm) is the St uckelberg phase.44{46It is acquired\nbetween the two passages and includes both the adiabatic\nand non-adiabatic ( \r-dependent) parts. From Eq. (62)\nwe can make several observations that are consistent\nwith the numerical data of Fig. 3. First, when \u001cf\u001d1\nand\u001ci\u001d1,Pdoes not depend on the initial and \fnal\ntimes. The transition probability only depends on the\nLandau-Zener probability PLZof Eq. (46) and the turn-\ning point\u001cm. This means that the oscillations of P(\u001cm)\nare a robust feature of a coherent double passage across\na Landau-Zener anticrossing. The transition probability\noscillates around the average value\nPav= 2PLZ(1\u0000PLZ): (63)\nFor fast sweeps PLZ\u001c1 so thatPoscillates between\n0 and 4PLZ. For slow sweeps PLZis close to 1 and the\nprobability oscillates between 0 and 4(1 \u0000PLZ). The\nmaximum in the oscillation amplitudes is achieved at\nPLZ= 1=2. WhenPLZ= 1\u0000e\u00002\u0019\r= 1=2 (\r\u00190:11),\nthe transition probability Poscillates between 0 and 1.\nThe amplitudes of the oscillations are smaller for all other\nvalues of\r. This is exacly the behavoir we see in the nu-\nmerical plots. One more remarkable feature of Fig 3(a),\nthat all oscillations pass through P= 0, is also re\rected\nby Eq. (62).\nOscillatory patterns of \r= 0:1 curves in Figs. 2(a) and\n3(a) show strikingly di\u000berent behavior. In Fig. 2(a), the\namplitude of oscillations decreases with \u001cf, andPgradu-\nally approaches its Landau-Zener limit PLZ. On the con-\ntrary, in Fig. 3(a) the oscillations, after some transitional\nperiod, acquire a stationary amplitude. Eqs. (62) and\n(63) clarify the origin of this behavior typical of double\npassages across the anticrossing.44{46Indeed,Pavof Eq.\n(63) is a Landau-Zener probability P(2)\nLZfor a double pas-\nsage across the anticrossing that can be derived directly\nby the above two-channel procedure with quantum am-\nplitudes substituted by probabilities, see Ref. 38. There-\nfore, suppression of these long-time scale oscillations andapproaching the double-passage Landau-Zener limit P(2)\nLZ\nare only achieved when the decoherence is taken into ac-\ncount, and can allow measuring decoherence times.\nIn conclusion, prolonged oscillations of the electronic\namplitudes (~ cS;~cT+) are a generic property of the co-\nherent electron dynamics during the single- and double-\npassages across the S-T+anticrossing. Their amplitudes\nand durations are controlled by the Landau-Zener param-\neter\rand by dephasing on longer time scales, and the\npatterns are rather di\u000berent for the single- and double-\npassages.\nVII. BACK ACTION OF NUCLEAR SPIN\nDYNAMICS ON OVERHAUSER FIELDS\nThe Hamiltonian ^Hof Eq. (3) describing the electron\nstates depends on the Overhauser \felds created by the\nspatially dependent nuclear spin con\fguration. The elec-\ntrons experience the nuclear \felds v\u000b\nnand\u0011nof Eqs. (4)\nand (6), where the \frst represents the components of the\ne\u000bective di\u000berence magnetic \feld in the dots, and the sec-\nond represents the induced average magnetic \feld. When\ngoing through the S-T+transition, the electrons will ex-\nperience a change of these nuclear Overhauser \felds. It\nis a unique property of Eq. (26) for the change in the\ntotal longitudinal nuclear spin \u0001 Izthat it expresses a\nglobal property of a double dot in terms of the parame-\nters of the electronic Hamiltonian and does not depend\nof a speci\fc con\fguration of nuclear spins. For di\u000ber-\nent elements in the Hamiltonian ^H, we calculate their\nmean-square values as well as their variances.\nThe expression for the change of the total zcompo-\nnent of the nuclear spin of Eq. (27) makes the role of\nQexplicit due to the mediation of spin-orbit coupling.\nWithv\u0006\nso= 0, the total spin transfer is protected by\nthe momentum conservation law and Qmanifests itself\nthrough shake-up processes in the nuclear spin reservoir\nrespecting the conservation of the total angular momen-\ntum. The electron dynamics induces changes in the nu-\nclear spin con\fguration that in turn induce changes in\nthe in the diagonal and o\u000b-diagonal elements of the elec-\ntron Hamiltonian (3). In what follows, we compute these\nchanges.\nA. Changes in Overhauser \felds\nElectrons experience an e\u000bective Zeeman splitting in\nthe Overhauser \feld of ^\u0011jof Eq. (6). The associated\nchange in the z-component of ^\u0011j, \u0001^\u0011z\nn=\u0000AP\nj\u0010j\u0001^Iz\nj,\nis\n\u0001^\u0011z\nn=A2\n2v2\n?X\nj\u001aj\u0010j(\u0003\u0000v\u0000^I+\nj+ \u0003+v+^I\u0000\nj); (64)\nIn the multicycle regime, the \feld of Eq. (64) has been\nmeasured by Petta et al.42and by Foletti et al.9by the14\nshift in the position of the ST+anticrossing. In contrast\nto \u0001Iz, the change \u0001^ \u0011z\nnin the longitudinal \feld depends\non the detailed nuclear spin con\fguration and on the spa-\ntially dependent electron-nuclear couplings \u001ajof Eq. (5)\nand\u0010jof Eq. (7).\nThe singlet-triplet terms ^ v\u0006\nnand ^vz\nnin the Hamiltonian\n^Hof Eq. (3) are sums over all nuclear spins. ST0level\nsplittings characterized by ^ vz\nnwere measured in Ref. 24\nand a number of follow-up papers, and ST+splittings\ndescribed by ^ v\u0006\nnin Ref. 20. The changes in these terms\nduring a cycle are \u0001^ v\u000b\nn=AP\nj\u001aj^I\u000b\nj. By using Eq. (24),\nwe \fnd changes in the components \u000b=\u0006that couple S\ntoT\u0006\n\u0001^v\u0006\nn=A2\n2v?2\n4v\u0006\nv?\u0003\u0006X\nj\u001a2\nj^Iz\nj\u0006i\u0003zX\nj\u001aj\u0010j^I\u0006\nj3\n5\n\u0007iAX\nj\u001aj\u0011j(nZ)\n~(Tf\u0000Ti)^I\u0006\nj; (65)\nand, by using Eq. (25), in the component \u000b=zcoupling\nStoT0\n\u0001^vz\nn=\u0000A2\n2v2\n?2\n4\u0003\u0000v\u0000X\nj\u001a2\nj^I+\nj+ \u0003+v+X\nj\u001a2\nj^I\u0000\nj3\n5:\n(66)\nWe note that while ^ vz\nnonly produces a longitudinal Over-\nhauser \feld mixing SandT0, \u0001^vz\nnincludes operators ^I\u0006\nj\nand therefore mixes SandT+belonging to our 2 \u00022\nsubspace.\nIn the next sub sections, mean values and variances of\nthese operatores are computed.\nB. Constraints and mean values\nWhile nuclear spins are distributed in the bath ran-\ndomly, the magnetization \ructuations v\u0006\nncontrolling\nelectron dynamics during the cycle impose on their values\nthe constraints\nAX\nj\u001ajI\u000b\nj=v\u000b\nn; (67)\nadding also a constraint related to vz\nn. To simplify cal-\nculations, we consider below the nuclear spins Ijas ran-\ndom Gaussian variables that are normalized, in the ab-\nsent of constraints, as hI\u0015\njI\u00150\nj0i=1\n3Ij(Ij+ 1)\u000ejj0\u000e\u0015\u00150, with\n\u0015= (x;y;z ). Then the mean values of I\u0015\njare\nhI\u0015\nji=R\ndI\u0015\njI\u0015\njP(I\u0015\nj)Q\nj06=jR\ndI\u0015\nj0P(I\u0015\nj0)\u000e(v\u0015\nn\u0000AP\nj0\u001aj0Ij0)Q\nj0R\ndI\u0015\nj0P(I\u0015\nj0)\u000e(v\u0015n\u0000AP\nj0\u001aj0I\u0015\nj0);\n(68)\nwhereP(I\u0015\nj) are Gaussian probabilities, ( vx\nn;vy\nn) are de-\n\fned asv\u0006\nn= (vx\nn\u0006vy\nn)=p\n2, and the denominator se-\ncures the normalization of the probabilities under the\nconstraints of Eq. (67).Using the integral representation for \u000e-functions\n\u000e(x) =1\n2\u0019Z1\n\u00001ei!xd!; (69)\nmultiple Gaussian integrations of Eq. (68) result in\nhI\u0006\nji=\u001ajv\u0006\nn=(AR2);hIz\nji=\u001ajvz\nn=(AR2); (70)\nwhereRn=P\nj\u001an\njare determined by the spatial depen-\ndence of the electron-nuclear coupling constants. Sub-\nstituting these expressions into Eqs. (64) and (65), we\narrive at the corrections to the nuclear \feld experienced\nby the electron spin during the sweep.\nh\u0001\u0011z\nni=\u0000\u0001IzAR0\n3=R2; (71)\nwhereR0\n3=P\nj\u001a2\nj\u0010j, and the Overhauser \feld mixing its\nSandT+components\nh\u0001vz\nni=\u0000\u0001IzAR3=R2; (72)\nwith \u0001Izof Eq. (26).\nWe see that both the changes in the longitudinal dif-\nference \feld \u0001 vz\nnand the longitudinal average \feld \u0001 \u0011z\nare proportional to the change in the total nuclear spin\n\u0001Iz. It follows from Eqs. (5) and (7) that \u001ajtypically\nhave opposite signs in both dots while \u0010j>0 everywhere,\nhence,R0\n3>0. Therefore, with A>0, the sign ofh\u0001\u0011z\nni\n(the change in the mean Overhauser \feld building in the\ndouble dot) is opposite to the sign of \u0001 Iz, in agreement\nwith Eq. (6). The sign of h\u0001vz\nniis de\fned by the sign R3\nthat depends on the choice of electronic basis functions\n(see Appendix B), therefore, it is not uniquely de\fned\nwith respect to \u0001 Iz.\nThe magnitudes of \u0001 \u0011zand \u0001vz\nnare of the order of\n\u0001IzAn0=Nper cycle, i.e., about \u0001 Iz=p\nNof the mean\nvalues of\u0011zandvz\nn. Forv\u0006\nso= 0, \u0001Iz=\u0000P, hence,\nj\u0001Izj\u00141. However, it is seen from Figs. 2(b) and\n3(b) thatQis an order of magnitude larger than Pwhen\n\r&1. Therefore, when vso6= 0, the conditional ex-\npectation valuesh\u0001\u0011z\nniandh\u0001vz\nnishould experience Q-\nenhancement through the Q-enhancement of \u0001 Iz, and\n\u0011zandvz\nncan change by about 1% per cycle.\nThe mean values of the transverse components of vn,\ncalculated in a similar way from Eq. (65), are\nh\u0001v\u0006\nni=Av\u0006vz\nn\n2v2\n?R3\nR2\u0003\u0006\u0006iAv\u0006\nn\n2v?R0\n3\nR2\u0003z\n\u0007ivz\nn\u0016\u0011(nZ)(Tf\u0000Ti)=~ (73)\nwhere \u0016\u0011(nZ)is a mean value of \u0011j(nZ)over all nuclear\nspecies. Because di\u000berent species are distributed ran-\ndomly at the scale of atomic spacings, they self-average\nin the linear approximation over Tf\u0000Ti, and we accept\nthat all of them have the same absolute values of the an-\ngular momenta, Ij=I. While the \frst term is compara-\nble in the magnitude to Eq. (72), the two last term might\nbe much larger because they increase with the sweep du-\nration. However, Eq. (73) includes changes both in the\namplitude and the phase of \u0001 v\u0006\nn, and the latter might\nnot be essential when solving Eq. (16) that only depends\nonv?. We come back to this term in Sec. VII D.15\nC.ST+-pulses induced interdot shake-ups\nLet us explain the importance of the variance in the\nspin production by considering the total nuclear spins in\nthe left and right dots. Average values of di\u000berent oper-\nators calculated in Sec. VII B were based on the condi-\ntional mean values hI\u000b\njiof nuclear spins I\u000b\njof the order\nofN\u00001=2that are small compared with their root mean-\nsquare values. Therefore, calculating the mean-square\nvalues of all operators and their variances is important\nfor estimating the widths of statistical distributions.\nWe begin with the di\u000berences in the spin polarizations\nof the left and right dots, LandR, that are critical for\nspin manipulation. While division of a double dot into\nits left and right parts holds only when the overlap in-\ntegral is small enough, cf. Appendix B, the results are\ninstructive. Splitting Eq. (4) into sums over LandR, we\nde\fne partial sums\nv\u000b\nnL(R)=AX\nj2L(R)\u001ajI\u000b\nj: (74)\nTheir sums are v\u000b\nnand are a subject to constrains of\nEq. (67). However, their di\u000berences\nu\u000b\nn=v\u000b\nnL\u0000v\u000b\nnR (75)\nare free of any constraints. Using Eq. (25), the change in\nthe left-right polarization di\u000berence is\n\u0001Iz\nLR=\u00001\n2v2\n?(\u0003\u0000v\u0000u+\nn+ \u0003+v+u\u0000\nn): (76)\nWhen averaged over an unpolarized spin reservoir, its\nmean value vanishes, h\u0001Iz\nLRi= 0, and the mean-square\nvalue equals\nh(\u0001Iz\nLR)2i=A2n0\n6v2\n?I(I+ 1)j\u0003j2Z\n\u001a2(R)d3R;(77)\nwith\u001a(R) of Eq. (5) and\nj\u0003j2=P2+Q2: (78)\nA simple estimate of the right hand side of Eq. (77) re-\nsults inj\u0003j2. Therefore, the asymmetry of spin pumping\nof the left and right dots is Q-enhanced whenever Q\u001dP,\nin particular, when vso= 0 andP\u00141. We attribute this\nenhancement to shake-up processes resulting in multiple\nspin \rips per each \\pure\" injected nuclear spin. These\nprocesses are random, and it is not clear for now how\nthey in\ruence inhomogeneous spin distributions.13,15\nThe detailed spatial patterns of spin generation at long\ntime scales are a subtle subject and are related to the\nspatial variation of the electron-nuclear couplings \u001a(Rj)\nand\u0010(Rj) calculated in Appendix B. With mean values\nofI\u0006\njof Eq. (70), spatial distribution of \u0001 Iz\njis related to\n\u0001Izas \u0001I\u0006\nj= (\u001a2\nj=R2)\u0001Iz. The left-right asymmetry\nin\u001a2\njoriginates either from the geometric asymmetry ofthe double dot15or from the L-R-overlap of the electron\ndensity, cf. Appendix B, and produces a regular di\u000ber-\nence in the Izgeneration rate. While the results depend\non the speci\fc distribution of nuclear spins and the S-\nT0mixing,14the mechanism of Q-enhancement is quite\ngeneral whenever \r&1.\nD. Mean-square values and variances\nMean values of Sec. VII B were evaluated over an unpo-\nlarized nuclear spin bath and estimate the mean rates of\nthe change of the di\u000berent parameters. However, the esti-\nmate of the shake-up rate of Sec. VII C demonstrates that\ncalculating variances of these random variables can pro-\nvide additional, and sometimes even more valuable, in-\nformation about the magnitudes of the expected changes\nduring a cycle. The conditional probability distributions\nare so wide that the mean value is not very representa-\ntive. In this section, we evaluate variances of the basic\nnuclear \felds.\nWe begin with calculating the mean-square values. Be-\ncause all nuclear \felds of Eqs. (64) - (66) are linear in the\nmomentaI\u000b\nj, mean values of the quadratic forms in them\ninclude integrals that di\u000ber from Eq. (68) by substitut-\ningI\u0015\njeither by (I\u0015\nj)2or byI\u0015\njI\u0015\nj0withj6=j0. While the\nlatter terms are smaller in the parameter 1 =N\u001c1, they\nhave a higher statistical weight. Summing all terms, one\narrives at length expressions for h(\u0001\u0011z\nn)2iandh(\u0001vz\nn)2i\nthat we do not present here. Instead, using the mean\nvalues of Eqs. (71) and (72), we present the variances\nde\fned as Varf\u0018g=h\u00182i\u0000h\u0018i2\nVarf\u0001\u0011z\nng=j\u0003j2A4\n6v2\n?I(I+ 1)[R0\n4\u0000(R0\n3)2=R2];(79)\nwhereR0\n4=P\nj(\u001aj\u0010j)2, and\nVarf\u0001vz\nng=j\u0003j2A4\n6v2\n?I(I+ 1)[R4\u0000(R3)2=R2]:(80)\nComparison with Eqs. (71) and (72) shows Q-\nenhancement even when vso= 0 (hence, when \u0001 I=\n\u0000P), the e\u000bect that manifested itself already in Eq. (77).\nThis means that the nuclear spins with Iz\njaway from the\nmean conditional expectation values hIz\njirespond to the\nsweeps stronger than the spins with Iz\nj=hIz\nji. Also,\nthis enhanced sensitivity is due to the spatial distribu-\ntion of\u001ajand\u0010jbecause with \u001aj=const and \u0010j=const\nthe brackets in Eqs. (79) and (80) vanish. By the or-\nder of magnitude, both quantities experience changes of\nabout \u0003An0=Nper cycle; with \u0003 \u001910 andN\u0019106, this\nsuggests changes about 1% per cycle. In other words,\naround 10 spins interchange their directions during one\npassage.\nCalculatingh\u0001(v+\nnv\u0000\nn)iresults in a simple equation\nh\u0001(v+\nnv\u0000\nn)i=\u0000IzAvz\nnR3=R2 (81)16\nbecause the contributions of the two last terms of Eq.\n(65) cancel. In absence of spin-orbit coupling, this im-\nmediately suggests h\u0001(v2\n?)i=PAvz\nnR3=R2. Under these\nconditions, large terms in Eq. (65) re\rect only the change\nin the phase of v\u0006\nnthat does not in\ruence dynamical\nequations (16), and the relative change in ( v?\nn)2=v+\nnv\u0000\nn\nis only about N\u00001=2.\nHowever, in presence of spin-orbit coupling the dy-\nnamics of spin amplitudes (~ cS;~cT) is controlled by v\u0006\nrather then v\u0006\nn. Mean value of \u0001( v2\n?), calculated by us-\ning Eqs. (65) and (70), is\nh\u0001(v+v\u0000)i=h\u0001(v+\nnv\u0000\nn)i\n+Avz\nn\n2v2\n?R3\nR2[\u0003\u0000v\u0000\nnv+\nso+ \u0003+v+\nnv\u0000\nso]\n+i(v\u0000\nnv+\nso\u0000v+\nnv\u0000\nso)\u0014\u0016\u0011nB\n~(Tf\u0000Ti)\u0000A\n2v?R0\n3\nR2\u0003z\u0015\n;\n(82)\nwhere \frst term is de\fned by Eq. (81). Physically, sec-\nond and third terms in Eq. (82) take into account the\nangle between v+\nnandv+\nsoin the complex plane, and are\nproportional to the product v?v?\nn. Withv?\u0018v?\nn, rela-\ntive corrections coming from the second term are of the\norder \u0003=p\nNper cycle. The third term is usually much\nlarger because it increases linearly with the pulse dura-\ntion \u0001T=Tf\u0000Ti. It includes two contributions of which\n\frst is due to the Zeeman precession of nuclei and sec-\nond due to the Knight \feld and is proportional to the\nintegral ofj~cT+j2. While the magnitude of the second\ncontribution depends on the shape of the pulse, the ratio\nof these terms is roughly \u0016 \u0011(nB)=(An0=N) and they be-\ncome comparable at B\u00181 mT. This indicates that \frst\ncontribution to the third term usually dominates. With\nv?\u0018v?\nsoand \u0016\u0011(nB)\u001910 mT, the Zeeman term results in\nh\u0001(v+v\u0000)i\u00180:1hv+v\u0000ifor a 0.1\u0016s linear sweep. This\nis much larger than the correction to the same quantity\nestimated in Eq. (81) and to h(\u0001vz\nn)2ihaving the same\nscale. The e\u000bect in InAs should be much larger than in\nGaAs because of the stronger spin-orbit coupling.\nThe above estimates indicate that, because of the\nterms in Eq. (65) linear in the pulse duration, spin-orbit\ncorrections to transverse matrix elements are essentially\nlarger than the corrections to the longitudinal ones.\nIn Eq. (82), Zeeman precession of nuclei manifests itself\ninh(v+v\u0000)ionly through spin-orbit coupling. The e\u000bect\nis much stronger when estimated through the variance of\nv+v\u0000, and we estimate it for vso= 0 when v\u0006=v\u0006\nn.\nDisregarding two \frst terms in Eq. (65), the calculations\nsimilar to those performed above when deriving Eqs. (79)\nand (80) result in\nVarf\u0001(v+\nnv\u0000\nn)g \u0019I(I+ 1)\n6(\u00112\n(nB)\u0000\u0016\u00112\n(nB))\n\u0002(v+\nnv\u0000\nn)A2R2[(Tf\u0000Ti)=~]2;(83)\nwhere\u00112\n(nB)is the mean-square value of \u0011j(nB). It follows\nfrom Eq. (83), the dominating mechanism of changing v?\nnis the nuclear spin precession with a characteristic time\nof about a microsecond at B\u001810 mT. It is about two to\nthree orders of magnitude shorter than the corresponding\ntime forvz\nnestimated above.\nIt is also instructive to compare this estimate with a\nmuch longer time for v?\nnfollowing from Eq. (81). The\nlatter estimate was found with the nuclear con\fguration\nof Eq. (70) that re\rects the mean-values of nuclear spins\nunder the constraints of Eq. (67). In a narrow region\nof the phase space around these mean values dynamics\nofv?\nnis strongly suppressed. The estimate of Eq. (83) is\nmuch more representative because it represents the entire\nphase space compatible with the constraints of Eq. (67).\nA similar type of the behaviour of vz\nnwas discussed above\nas applied to Eq. (80).\nVIII. CONCLUSIONS\nWe have studied the dynamics of the electron and nu-\nclear spins near ST+avoided crossings in double quan-\ntum dots. While adopting the traditional approach based\non the hierarchy of time scales, with a slow nuclear and\nfast electron dynamics, we employed a quantum descrip-\ntion of the electron spin and coherent dynamics of nu-\nclear spins, and investigated the time-resolved patterns\nof single and double Landau-Zener passages through the\nanticrossing point. They are described by two complex\nconjugate functions \u0003\u0006depending on the initial and \f-\nnite times ( Ti;Tf) and the trajectory of the sweep, with\n\u0003\u0000proportional to the integral of the product ~ c\u0003\nS(t)~cT(t)\nof the complex amplitudes of the SandT+states. Their\nreal partsP= Ref\u0003\u0006gare proportional to the S-T+-\ntransition probability and for one-side sweeps oscillate\nat small time scales when the system is close to the\nanticrossing and saturate at long time scales. For lin-\near sweeps, we \fnd the singlet and triplet amplitudes\nin terms of Weber D-functions (parabolic cylinder func-\ntions); the long-time asymptotic limit of Pequals the\nLandau-Zener probability PLZ= 1\u0000e\u00002\u0019\r. For round\ntrips, the system also experiences long-term St uckelberg\noscillations. The \frst sharp oscillations might be uti-\nlized for ultrafast electron spin operation, while the de-\ncay of the oscillations can provide information about de-\nphasing rates. It is important that the imaginary part\nQ= Imf\u0003+gthat acquires contributions from the elec-\ntronic states at a wide time scale and accumulates with\ntime (it diverges logarithmically for linear sweeps) has\na profound e\u000bect on the dynamics of the nuclear spins.\nWhen the Landau-Zener parameter \r&1,Qis typically\none order of magnitude larger than P. Therefore, in pres-\nence of the spin-orbit coupling violating the angular mo-\nmentum conservation, Qmay become the major factor\ncontrolling the angular momentum transfer to nuclei. In\nparticular, this mechanism is e\u000ecient for excursions in-\ncluding a stay near the anticrossing point. Generically,\n\u0003 = (P2+Q2)1=2controls the shake-up processes that\nexchange angular momentum between the left and right17\ndots. With Q\u001dP, it isQthat plays a dominating role\nin these angular-momentum exchange processes. Because\nthe mechanism that plagues many experimental e\u000borts of\nbuilding considerable polarization gradients remains un-\nknown, it is a challenging question whether and how the\nshake-up processes contribute to it; unfortunately, only a\ntheory including multiple passages can resolve it. We also\nestimated changes in the Overhauser \felds during a sin-\ngle cycle and concluded that the transverse components\nare more volatile than the longitudinal ones.\nWe are grateful to B. I. Halperin, C. M. Marcus, I.\nNeder, and M. Rudner for stimulating discussions and\ncomments on the manuscript. E. I. R. was funded by\nIARPA through the Army Research O\u000ece, by NSF under\nGrant No. DMR-0908070, and in part by Rutherford\nProfessorship (Loughborough, UK).\nAppendix A: Spin Operator\nWe use the following convention for the spin-1 operator\nS\nSx=1p\n20\n@0 1 0\n1 0 1\n0 1 01\nA; (A1)\nSy=1p\n20\n@0\u0000i0\ni0\u0000i\n0i01\nA; (A2)\nSz=0\n@1 0 0\n0 0 0\n0 0\u000011\nA: (A3)\nThese operators satisfy the commutation\nrelationsh\n^Si;^Sji\n=i\u000fijk^Sk, where\u000fijkis the Levi-\nCivita tensor, as well as ^S2\nx+^S2\ny+^S2\nz= 2.\nAppendix B: Simple Model\nThe singlet part of the spin wave function is\n\u001fS(1;2) =1p\n2(j\"1ij#2i\u0000j# 1ij\"2i) (B1)\nand the three triplet components of the spin wave func-\ntion are\n\u001fT+(1;2) =j\"1ij\"2i; (B2a)\n\u001fT0(1;2) =1p\n2(j\"1ij#2i+j#1ij\"2i); (B2b)\n\u001fT\u0000(1;2) =j#1ij#2i: (B2c)\nWe will in this section discuss the spatial dependence\nof the hyper\fne coupling constants \u001ajof Eq. (5) and \u0010jof\nEq. (7). In a simple model, the electron wave functionsnear theS-T+anticrossing are\n S(1;2) = cos\u0017  R(1) R(2)\n+sin\u0017p\n2[ L(1) R(2) + L(2) R(1)];(B3)\n T(1;2) =1p\n2[ L(1) R(2)\u0000 L(2) R(1)]; (B4)\nwhereLdenotes the left and Rthe right dot, and the\nangle\u0017depend on the Zeeman energy \u0011Z. The normal-\nization coe\u000ecients in (B3) and (B4) are exact under the\nassumption that the functions  Land Rare orthonor-\nmalized.\nLet us illustrate the spatial dependence of the electron-\nnuclear coupling constants \u001aof Eq. (5) and \u0010of Eq.\n(7) for a simple model of a quantum dot. We assume\nthe electrons are in the lowest orbital harmonic oscillator\nstate. The Cartesian coordinates, the wave function is\n (x;y) = exp\u0002\n\u0000(x2+y2)=l2\u0003\n=(lp\n2=\u0019), wherelis the\nsize of each quantum dot. We have two quantum dots\nthat are separated at a distance d, one atx=\u0000d=2 and\ny= 0 and the other at x=d=2 andy= 0. We form an\northonormal basis set based on the functions  (x\u0000d=2;y)\nand (x+d=2;y). In this basis, we compute \u001a(x;y) and\n\u0010(x;y).\nWe plot in Fig. 5 the electron-nuclear couplings \u001a(x;y)\nand\u0010(x;y) fory= 0 as a function of xwhen\u0017= 0:1\nand\u0017=\u0019=2\u00000:1. The spatial distribution of the\nsinglet-triplet coupling \u001a(x;y) depends on the angle \u0017.\nWhen\u0017is close to \u0019=2, there is a nearly equal prob-\nability for electrons to be located in the left and right\ndot for both the singlet and triplet states. Then the\nsinglet-triplet coupling \u001a(x;y) is nearly antisymmetric\naroundx= 0,\u001a(x;y)\u0019\u0000\u001a(\u0000x;y) [the sign of \u001a(x;y) de-\npends on the sign choice in Eq. (B4)]. When \u0017is small,\nthe electrons are in the singlet state (0 ;2) in the right\ndot, so that \u001a(x;y) passes through zero inside the right\ndot (forx > 0). Therefore, even for two symmetrically\nshaped dots, the S-T+electron-nuclear coupling can be-\ncome asymmetric because of the overlap of the left and\nthe right dot wave functions. The asymmetry depends\non\u0017controlled by the external magnetic \feld.\nThe triplet-triplet electron-nuclear coupling \u0010(x;y)\ndoes not depend on \u0017and is a symmetric function of\nxfor the two symmetric quantum dots.\nAppendix C: Two identities for the parabolic\ncylinderD-functions\nUsing the solution of Eq. (44) for ~ cT+(\u001c) and ~cS(\u001c) and\nthe normalization condition j~cS(\u001c)j2+j~cT+(\u001c)j2= 1, we\narrive at an identity\n\rjD\u00001\u0000i\r(\u0000ei\u0019=4\u001c)j2+jDi\r(ei3\u0019=3\u001c)j2=e\u0019\r=2(C1)\nrelating absolute values of two D-functions at arbitrary\nreal values of \u001cand\r.18\n-100-50 50100xHnmL\n-1.0-0.50.51.0rHx,y=0LHH100nmL-2L\n(a)\n-200-100 100200xHnmL0.20.40.60.81.01.2zHx,y=0LHH100nmL-2L\n(b)\nFIG. 5: The spatial variation of the electron-nuclear couplings\n\u001a(x;y= 0) (a) and \u0010(x;y= 0) (b). In (a), the red (full)\ncurve is for \u0017=\u0019=2\u00000:1 and the blue (dashed) curve is\nfor\u0017= 0:1. The size of the dots is l= 50 nm and the\nseparation between the dots is d= 100 nm. The overlap\nintegral between the left and the right oscillator wave function\nis 0:1. It is the most striking feature that the overlap between\nthe wave functions induces asymmetry of the \u001a(x;y) even in\ngeometrically symmetric double dots. 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The Zeeman coupling between the field and the electron spin \nleads to a spin dependent force, and to spin accumulation at the edges. We compare the effect of \nthis relativistic correction to the electr on dynamics to the features induced by the spin -orbit \ninteraction. The effect of current induced magnetic fields on t he spin Hall effect can be comparable \nto the extrinsic contribution from the spin -orbit interaction, although it does not require the \nprese nce of heavy elements with a strong spin -orbit interaction. The induced spin s are oriented \nnormal to the metal slab.  \n \nIntroduction.  The passage of an electric current in a metallic sheet leads to the accumulation of spin at its edges, \nthe Spin Hall Effect (SHE). This phenomenon, related to the accumulation of c harge when a current is induced in a \nferromagnet, the Anomalous Hall Effect (AHE) has opened new ways of manipulating spins in metals.  A natural \nexplanation of the SHE and AHE can be given in terms of the spin -orbit interaction1. For the SHE, either spin d ependent \nscattering due to impurities2, or the action of an intrinsic interaction on accelerating carriers3,4  can lead to the deflection \nof spin currents, and to spin accumulation at  the edges, see Fig.[1]. The nature of the SHE and the AHE are discussed \nin detail in refs.4,5. \n \nIn the following, we analyze an additional contribution to the SHE, arising from the existence of non -homogeneous \nmagnetic fields within the metallic sheet, see Fig.[2]. These fields give rise to a spatially dependent Zeeman energy,  \nand to lateral forces on the carriers, whose sign depends on the spin. As a result, a gradient in the spin density within \nthe sample is induced. It will be shown that this mechanism gives rise to spin imbalances comparable to those derived \nfrom the extrin sic and intrinsic mechanisms mentioned earlier.  \n \nOur analysis does not rely explicitly on the existence of a strong spin orbit coupling, so that it can be generalized \nto metals made from light elements (note, however, that the Zeeman energy can be conside red a relativistic effect which \narises from the Dirac equation which describes the electrons).  The main features of the model are described next. Then, \nsemi quantitative estimates of the spin accumulation induced by inhomogeneous magnetic fields are made, and \ncompared to the results derived from the contributions to the SHE from extrinsic and intrinsic spin -orbit interactions. \nFinally, the main conclusions and possible extensions will be presented.  We use CGS units throughout the manuscript. \nThis choice hel ps us to highlight the way in which relativistic corrections enter in the analysis.  \n \nFigure 1. Sketch of the spin currents induced by the magnetic field created by an electrical current in a Hall bar.  \n \nInhomogeneous magnetic fields in current carrying metall ic sheets.  We consider a thin metallic sheet, of \ndimensions ℓ𝑥 ,ℓ𝑦 and ℓ𝑧, with ℓ𝑧≪ℓ𝑥 ,ℓ𝑦, see Fig.[1] . We assume a constant current density per unit area, 𝑗𝑦 along \nthe 𝑦 direction. This current induces at the plane of the  sheet, a magnetic field along the 𝑧 direction. In the limit  ℓ𝑦≫\nℓ𝑥,ℓ𝑧 and ℓ𝑧≪ℓ𝑥,  the field is6 (see also the Supplementary Information)  \n𝐵𝑧𝐼(𝑥)=2𝑗𝑦ℓ𝑧\n𝑐log(ℓ𝑥2⁄+𝑥\nℓ𝑥2⁄−𝑥)                 (1) \nThe label  𝑗 stands for the current density, 𝑗𝑦=𝐼(ℓ𝑥ℓ𝑧) ⁄ , where  𝐼 is the total current . Throughout most of the sheet \nthe field can be well approximated by function proportional to the 𝑥 coordinate6,  \n𝐵𝑧𝐼(𝑥)≈8𝑗𝑦𝑥\n𝑐ℓ𝑧\nℓ𝑥          (2) \nyxz\nlylxlzjyelectronThis is the field measured in a frame of refere nce at rest. In the frame of reference where the current carriers are at \nrest a positive current, associated with the ions, can be defined. This current induces an opposite magnetic field, which \nis felt by the carriers. The Zeeman energy associated to this  field is  \n𝐸𝑍𝐼=∫𝑑3𝑟⃗𝜇𝐵𝐵𝑧(𝑟⃗)[𝑛↑(𝑟⃗)−𝑛↓(𝑟⃗)]             (3) \nWhere 𝜇𝐵=(𝑒ℏ)(2𝑚𝑐) ⁄  is Bohr’s magneton. As a result, a force acting on the spin of the carriers arises  \n𝐹𝐼≈±8𝜇𝐵𝑗𝑦\n𝑐ℓ𝑧\nℓ𝑥=±8𝜇𝐵𝐼\n𝑐ℓ𝑥2                (4)  \nThis force is sketched in Fig.[2].  \n \nFigure 2. Sketch of the inhomogeneous magnetic field created by a current, and the associated Zeeman forces.  \n \nNumerical estimates.  Magnetic field.  The magnetic field induced by the cu rrent, given in eq.(1), is maximum at \nthe edges of the sample, 𝑥≈ℓ𝑥2⁄,  \n𝐵𝑚𝑎𝑥 ≈8𝑗𝑦ℓ𝑧\n𝑐         (5) \nFor typical current densities, 𝑗𝑦≈107 A×m−2 and ℓ𝑧≈100  nm we obtain 𝐵𝑚𝑎𝑥 ≈10−2G. This field is much \nlower th an typical fields used in experiments. We assume that the orbital  polarization induced by a magnetic field of \nthis magnitude is negligible.  As mentioned before, we focus on the spin dependent forces induced by the gradient of \nthis field.  \n Numerical estimat es. Hall angle. The spin Hall angle, obtained from eq.(4), is described in the Supplementary \nInformation. An approximation based on a three dimensional nearly free electron model for the carriers gives, see \neq.(A5):  \n𝛼≈8\n1372𝑘𝐹𝑎𝐵\n3𝜋2𝑘𝐹 ℓℓ𝑧\nℓ𝑥    (6) \nWhere 𝛼 is the Hall angle, 𝑘𝐹 is the Fermi wavevector, 𝑎𝐵≈0.53 Å is the Bohr radius, and ℓ is the mean free path. \nThe factor 1372 in the denominator of the r.h.s. in eq.(6)  contains the fine structure constant, and it highlights the \nrelativistic origin of the effect.  \nEq.(6) applies to metallic samples with light elements, where only impurities with heavy elements can lead to a \nspin Hall effect. Experiments with Al samples w ere reported in6. We assume that 𝑘𝐹𝐴𝑙≈1.7 Å−1 , 𝑘𝐹𝐴𝑙 ℓ≈102 and \nℓ𝑧ℓ𝑥⁄ ≈15⁄. We obtain 𝛼≈2.3×10−4. This value is of the same order of magnitude as the results reported in8,9 \n(see also10,11). The estimate in eq.(6) is valid for nearly free electron systems , where the effective mass is approximately \nequal to the free electron mass, 𝑚𝑒𝑓𝑓≈𝑚𝑒. The value of 𝛼 in eq.(6) is enhanced by a factor 𝑚𝑒𝑓𝑓 𝑚𝑒⁄ if the two \nmasses differ significantly, as in transition metals.  \nxz\nl  x / 2 -l x / 2\nF FF FB\nB\nelectronA similar analysis to  the one leading to eq.(6), but for a quasi -two dimensional electron gas where only one subband \nis occupied gives, see eq.(A6)  \n𝛼2𝐷≈8\n13721\n2𝜋𝑘𝐹 ℓ𝑎𝐵\nℓ𝑥   (7) \nThe dependence of the angle on the fine structure constant implies that , 𝛼∝𝑐−2, where 𝑐 is the velocity of  light. \nThis is the same as the one expected for the extrinsic contribution from skew scattering effects, see below. This may \ncomplicate the experimental separation of the contribution from the field gradient discussed here from that of skew \nscattering . A plot of the depend ence of 𝛼 on elastic scattering and on the aspect ratio of the sample, using. Eq.(6), is \nshown in Fig.[3].  \nRef.[2] suggested the measurement of the Spin Hall Effect through the voltage induced in a strip connecting the \nedges of the sample. This setup does not nec essarily work for the detecting the contribution discussed here, as the \ncurrent induced magnetic field can also affect the electrons in the strip , see discussion in  the Supplementa ry \nInformation . \n  \nFigure 3. Spin Hall angle, 𝛼, due to the current induced magnetic field, as function of the product of the Fermi \nwavelength and the mean free path, 𝑘𝐹ℓ and the aspect ratio of the sample, ℓ𝑧ℓ𝑥⁄. See equation (6).  \nNumerical estimates. Spin accumulation at the edges. The net spin at the edges, if  the spin diffusion length is \nsmaller than the width of the sample, ℓ𝑥, is determined by the d ifference in the chemical potential for the spin up and \nspin down electrons, 𝜇↑−𝜇↓≈2𝜇𝐵𝐵𝑧(𝑥≈±ℓ𝑥\n2) . The expression for the field  is given in eq.(1). This  expression has \nbeen obtained in the limit ℓ𝑧≪ℓ𝑥 and it breaks close to the edges, when  |ℓ𝑥\n2−|𝑥||≤ℓ𝑧. We take the average of 𝐵𝑧 \nover distances fr om the edge comparable to ℓ𝑧, and obtain 〈𝐵𝑧〉≈2𝑗𝑦ℓ𝑧\n𝑐log(ℓ𝑥\nℓ𝑧). We finally obtain that the spin \naccumulation at the edges , in units of electrons per area is   \n𝑛↑−𝑛↓≈4𝜇𝐵𝑗𝑦ℓ𝑧\n𝑐log(ℓ𝑥\nℓ𝑧)×𝑛(𝜖𝐹)      (8) \nWhere 𝑛(𝜖𝐹) is the density of states of the metal at the Fermi energy. We assume ℓ𝑥=1𝜇m, ℓ𝑧=0.1𝜇m, 𝑗𝑦=\n103Amm2⁄  and 𝑛(𝜖𝐹)=1eV−1Å−3. Inserting these parameters in to eq.(8) we obtai n 𝑛↑−𝑛↓≈1018cm−3. \nComparison to the skew scattering contribution to the Spin Hall Effect.  We first est imate the combined effect of \nthe spin scattering interaction and defect scattering following ref.[2]. We assume that the effect of the impurities is \nroughly equivalent to the effect of a magnetic field equal to the magnetization due to the carriers with a given spin, \n𝐵𝑒𝑓𝑓≈𝑒ℏ\n𝑚𝑐𝑛↑ where 𝑛↑ is the density per unit volume of carriers with spin up. The associated force is  \n𝐹𝑒𝑥𝑡𝑆𝐻≈v𝑒𝐵𝑒𝑓𝑓\n𝑐≈v𝑒2ℏ\n𝑚𝑐2𝑛↑       (9) \nWhere v is the carrier velocity. Using 𝐼≈𝑒v(𝑛↑+𝑛↓)ℓ𝑥ℓ𝑧 , we obtain  \n𝐹𝑒𝑥𝑡𝑟𝑆𝐻≈±ℏ𝑒𝐼\n𝑚𝑐2ℓ𝑥ℓ𝑧≈±𝜇𝐵𝐼\n𝑐×1\nℓ𝑥ℓ𝑧  (10) \nWhich leads to a spin Hall angle  \n𝛼𝑒𝑥𝑡𝑟 ≈1\n1372𝑘𝐹𝑎𝐵\n3𝜋2𝑘𝐹ℓ    (11) \nSo that, using eq.(6), we obtain  \n𝛼𝐼\n𝛼���𝑥𝑡𝑟≈8ℓ𝑧\nℓ𝑥     (12) \nNote that this estimate of 𝐹𝑒𝑥𝑡𝑆𝐻  in eq.( 10) gives an upper bound to its actual value, as the analysis assumes that the \nspin dependent scattering induced by impurities leads to an effective field comparable to  the one generated by the total \nspin of the carriers  in the metal . A general discussion of skew scattering effects, following the original ideas12,13,14, can \nbe found in15. A different estimate of 𝐹𝑒𝑥𝑡𝑆𝐻 based on a two dimensional nearly free electron model a nd the Boltzmann \nequation is given in the supplementary information, see eq.(A17 ), and it gives  \n𝛼𝑒𝑥𝑡𝑟2𝐷 ≈(𝑘𝐹𝑎𝐵)2\n1372  ℓ\nℓ𝑆𝑂𝐼               (13) \nWhere ℓ𝑆𝑂𝐼 is a mean free path associated  to the impurities  which ind uce the skew scattering. Using this estimate, \nand comparing to eq.(7), we find  \n𝛼𝐼2𝐷\n𝛼𝑒𝑥𝑡𝑟2𝐷≈8\n𝑘𝐹𝑎𝐵ℓ𝑆𝑂𝐼\nℓ𝑥    (14) \nIt is finally interesting to note that three dimensional scattering by impurities with spin -orbit c oupling, described \ngenerically by terms such as 𝑉𝑖𝑚𝑝(𝑟⃗)≈𝑉0𝐿⃗⃗𝑠⃗𝛿(𝑟⃗), do not favor a particular spin orientation. This scattering is expected \nto dominate in metallic systems where many subbands are occupied, such that ℓ𝑧≫𝑘𝐹−1. In these samples, there is no \npreferred spin orientation which spontaneously will accumulate at the edges (note this in not the case for experiments \nwhere a spin polarized current is injected). On the other hand, the magnetic field induced by the current considered \nhere selects a spin orientation, and always leads to a spin accumulation at the edges.  \nOther contributions to the Spin Hall Effect.  A similar analytical semi -quantitative comparison between the current \ninduced SHE discussed here and the intrinsi c contribution of the spin orbit interaction is not possible. Simple estimates \nbased on the Rashba interaction give vanishing effects16, once leading effects due to impurities are taken into \naccount4,5,6. More involved calculations using the Berry curvatur e in complex, spin dependent, models for the bands of \ntransition metals suggest the the SHE conductivity can be written as17,18 \n𝜎𝑥𝑦𝑧≈𝑒\n4𝑎×〈𝑙⃗ 𝑠⃗〉𝐹𝑆\nℏ2               (15) \nHere, 𝑎 is the lattice constant, and 〈𝑙⃗ 𝑠⃗〉𝐹𝑆  is the average over the Fermi surface of the product of the orbital angular \nmomentum and the spin, which depends on the strength of the spin orbit interaction. The value of 〈𝑙⃗ 𝑠⃗〉𝐹𝑆 is determined \nby the intrinsic spin orbit interaction in the mater ial, which typically scales as 𝑐−2. Hence, the current contrib ution to \nthe spin Hall voltage, the contribution from skew scattering, and the intrinsic contribution all arise from relativistic \neffects, and show the same dependence on the vel ocity of light . Note that the intrinsic contribution changes the \nconductivity , see eq.(15).  \nThere is finally a contribution to the spin Hall effect from side jump scattering. To our knowledge, there are not \nreliable techniques to extract simple estimates of the magnitude of these pro cesses. We cannot compare the role of the \ncurrent induced field considered here to that of side jump scattering.  \nConclusions.   \nWe have analyzed the contribution to the spin Hall effect of the magnetic field associated to the current flowing \nthrough a metal lic sample. The spin accumulation at the edges arises from the Zeeman coupling to the inhomogeneous \nmagnetic field which exists within the sample. The gradient of the Zeeman coupling acts like a force which accelerates \nthe carriers in the direction perpend icular to the current, in opposite way for spin up and spin down electrons, where \nthe spins are ori ented along the magnetic field.  This effect is relativistic in nature, and its contribution to the spin Hall \nangle depends on the value of the velocity of li ght as 𝛼∝𝑐−2, similarly to the (extrinsic) contribution from skew \nscattering processes.  The current induced spin accumulation is linearly proportional to the scattering time or the transport mean free \npath, also like the contribution from skew scatte ring. Hence, it should dominate in clean samples, with 𝑘𝐹ℓ≫1. \nEstimates from simple models suggest that the effect of the current induced magnetic field should be at least comparable \nto that of skew scattering.  \nThe current induced spin Hall effect dep ends on the aspect ratio of the sample, through the ratio ℓ𝑧ℓ𝑥⁄, where ℓ𝑧 \nis the thickness, and ℓ𝑥 is the width. Hence, it is not a unique property of the material. This can explain the broad \ndistribution of Hall angles for the same mater ial reported in the literature.  In contrast with the effect of skew scattering, \nthe current induced spin Hall effect does not require the existence of a strong spin -orbit interaction, either due to heavy \nion impurities or intrinsic to the material. Hence, it should be present in any sample, and play a significant role in light \nmetal materials, such as Al.  \nSamples which are far from being quasi -two dimensional, 𝑘𝐹ℓ𝑧≫1, typically do not have a preferred spin \norientation, so that transverse spin curre nts due to skew scattering will average to zero. The current induced magnetic \nfield provides a preferred spin direction, enhancing the robustness of the spin Hall effect.  \nAcknowledgements  \nThis work was supported by the Spanish Ministry of Innovation, Scien ce and Technology and Spanish Ministry of \nEconomy and Competitiveness through Researc h Projects MAT2015 -67557 -C2-1-P, MAT2017 -86450 -C4-1-R, \nS2013/MIT -2850 NANOFRONTMAG and by the European Commission  AMPHIBIAN (H2020 -NMBP -2016 -\n720853).  \nReferences  \n \n1 M. I. Dya konov, and V. I. Perel, JETP Lett.  13 467 (1971) . \n2 J. E. Hirsch , Phys. Rev. Lett.  83, 1834  (1999).  \n3 R. Karplus, and J. M. Luttinger, Phys. Rev. 95, 1154 (1954).  \n4 J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett.  92, 126603 \n(2004).  \n5 J. Sinova, S. O. Valenzuela, J.  Wunderlich, C. H. Back and  T. Jungwirth , Rev. Mod. Phys . 87, 1213 (2015).  \n6 M. Lin iers, V. Madurga, M. Vázquez and A. Hernando,  Phys. Rev. B  31, 4425 (1985).  \n7 S. O. Valenzuela, M. Tinkham, Nature  442, 176 (2006).  \n8 Y. Kato , R. C. Myers , A. C. Gossard , D. D. Awschalom, Science  306 1910  (2004) . \n9 K. Ando, and E. Saitoh, Nature Comm.  3, 629 (2012).  \n10 H.L. Wang, C.H. Du, Y. Pu, R. Adur, P.C. Hammel, and F.Y. Yang Phys. Rev. Lett.  112, 197201  (2014).  \n11 A. Hoffmann, IEEE Trans. On Magnetics , 49, 5172 (2013).  \n12 N.F. Mott Proc. Roy. Soc. A  124 425 (1929) . \n13 J. Smit, Physica  21, 877 (1955).  \n14 J. Smit, Physica  24, 39 (1958).  \n15 N. A. Sinitsyn, Journal of Phys.: Cond. Matt.  20, 023201 (2008).  \n16 J. Ionue, G. E. W. Bauer, and L. W. Mol enkamp, Phys. Rev. B  70, 041303 (2004).  \n17 H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett. 102, 016601 (2009).  \n18 E. Saitoh, M. Ueda, H. Miyajima, G. Tatara.  Appl. Phys. Lett.  88 182509 (2006).  \n \n \n \nSupplementary Information.  \n \nCalculation of the magnetic field in a thin slab. We analyze a slab which is infinite in the 𝑦 direction, and with \ndimensions ℓ𝑥,ℓ𝑧 in the 𝑥,𝑧 directions. Then, the field at a position (𝑥,𝑦,𝑧) inside the slab, −ℓ𝑥2⁄≤𝑥≤\nℓ𝑥2⁄ ,−ℓ𝑧2⁄≤𝑧≤ℓ𝑧2⁄ can be written, using the Biot Savart law, as  \n \n        𝐵 ⃗⃗⃗⃗(𝑥,𝑦,𝑧)=𝑗𝑦\n𝑐∫ 𝑑𝑦′∫ 𝑑𝑥′ℓ𝑥2⁄\n−ℓ𝑥2⁄∞\n−∞∫ 𝑑𝑧′(𝑧−𝑧′)𝑥̂−(𝑥−𝑥′)𝑧̂\n[(𝑥−𝑥′)2+(𝑦−𝑦′)2+(𝑧−𝑧′)2]32⁄ℓ𝑧2⁄\n−ℓ𝑧2⁄                                  (A1)  \n \nWhere 𝑥̂ and 𝑧̂ are unit vectors along the 𝑥 and 𝑧 directions. Integrating over 𝑥′ and 𝑦′, we obtain  \n \n𝐵𝑥(𝑥,𝑦,𝑧)=2𝑗𝑦\n𝑐∫ 𝑑𝑧′tan(𝑥−𝑥′\n𝑧−𝑧′)|\n𝑥′=−ℓ𝑥2⁄𝑥′=ℓ𝑥2⁄−ℓ𝑧2⁄\n−ℓ𝑧2⁄≈2𝜋𝑗𝑦\n𝑐∫ 𝑑𝑧′sign (𝑧−𝑧′)≈±2𝜋𝑗𝑦ℓ𝑧\n𝑐−ℓ𝑧2⁄\n−ℓ𝑧2⁄             (A2)  \n \n𝐵𝑧(𝑥,𝑦,𝑧)=−2𝑗𝑦\n𝑐∫ 𝑑𝑧′1\n2log[(𝑥−𝑥′)2+(𝑧−𝑧′)2]|𝑥′=−ℓ𝑥2⁄𝑥′=ℓ𝑥2⁄ −ℓ𝑧2⁄\n−ℓ𝑧2⁄≈2𝜋𝑗𝑦ℓ𝑧\n𝑐log|𝑥+ℓ𝑥2⁄\n𝑥−ℓ𝑥2⁄|                 (A3) \n Where, in ord er to obtain the last result , the assumption ℓ𝑧≪ℓ𝑥 has been made.  The two signs in the expression for \n𝐵𝑥 refer to the top and bottom surfaces of the slab. Note that the integral of the field along a contour defined as  a \nsection of the stripe of width 2𝑥 is given by  \n \nΦ(𝑥)≈2∫𝑑𝑥𝑥\n−𝑥𝐵𝑥(𝑥,𝑦,𝑧)≈8𝜋𝑗𝑦𝑥ℓ𝑧\n𝑐                            (A4) \n \n \n \nSpin Hall angle due to the current induced magnetic field.  The spin Hall angle can be written as  \n𝛼≈𝐹𝐼\n𝑒𝐸∥        (𝐴5) \nWhere 𝐹𝐼 is given in eq.( 4) of the main text, and 𝐸∥ is the field which induces the longitudinal current  \n \n𝑗∥=𝑛𝑣 ≈𝑛𝑒𝐸∥𝜏\n𝑚=𝑛𝑒𝐸∥ℓ\n𝑚𝑣𝐹         (𝐴6) \nWhere 𝑛 is the carrier density, 𝑣 is the drift velocity, 𝜏 is the scattering time, 𝑣𝐹 is the Fermi velocity, and ℓ is the \nmean free path. For a three dimensional metal and using a nearly free electron description, we have  \n \n𝑛=𝑘𝐹3\n3𝜋2               \n𝑣𝐹=ℏ𝑘𝐹\n𝑚     (𝐴7) \nWhere 𝑘𝐹 is the Fermi wavevector. From eqs.(A5), (A6), and (A7)  we obtain  \n𝛼≈8𝑒2\n𝑚𝑐2𝑘𝐹2\n3𝜋2ℓℓ𝑧\nℓ𝑥≈8\n1372𝑘𝐹𝑎𝐵\n3𝜋2𝑘𝐹 ℓℓ𝑧\nℓ𝑥   (𝐴8) \nWhere  𝑎𝐵≈0.53 Å is the Bohr radius.  \nFor a two dimensional system where only one subband is occupied, the carrier density is  \n \n𝑛=𝑘𝐹2\n2𝜋ℓ𝑧         (𝐴9) \nThe spin Hall angle becomes  \n𝛼2𝐷≈8\n13721\n2𝜋𝑘𝐹 ℓ𝑎𝐵\nℓ𝑥       (𝐴10)   \nSkew scattering and mean free path . The contribution from skew scattering to the spin Hall conductivity arises from \nthe spin dependence of the scattering rates in the presence of the spin -orbit interaction3,4,5. We use a nearly free electron \napproximation to the electron band, and assume that the impurity induced matrix elements between scattering states \ncan be written as  \n〈𝑘⃗⃗′,𝑠′|𝑉|𝑘⃗⃗,𝑠〉=𝑉𝑘.𝑘′[𝛿𝑠,𝑠′+𝑖ℏ2\n4𝑚2𝑐2(⟨𝑠′|𝜎⃗|𝑠⟩×𝑘⃗⃗′)∙𝑘⃗⃗]     (𝐴11) \nWe analyze a quasi -two dimensional system, wh ere the Fermi surface can be approximated by a circle of radius  𝑘𝐹. \nThe momenta 𝑘⃗⃗ and  𝑘⃗⃗′ lie in the 𝑥−𝑦 plane. The transition rates near the Fermi surface can be written as function of \nthe angle 𝜃 between 𝑘⃗⃗ and 𝑘⃗⃗′. Using eq.(A11 ), and considering higher order scattering processes, we approximate the \ntransition rates by  \n𝑊𝜃𝑠𝑧≈1\n𝜏+1\n𝜏𝑆𝑂𝐼ℏ2𝑘𝐹2𝑠𝑧\n𝑚2𝑐2sin(𝜃)             (𝐴12) \nwhere 𝑠𝑧=±1\n2 describes the spin of the electron, and  we have distinguished two scattering processes described by two \nscattering rates, 𝜏, which describe symmetric scattering, and 𝜏𝑆𝑂𝐼 which describes skew scattering induced by impurities \nwith spin -orbit coupling. The transport properties can be compute d from the Boltzmann equation, which can be written \nas 𝜕𝑓𝜃𝑠𝑧\n𝜕𝑡=0=𝑒𝐸⃗⃗𝑣⃗𝜕𝑓0(𝜖)\n𝜕𝜖+∑𝑊𝜃−𝜃′𝑠𝑧(𝑓𝜃𝑠𝑧−𝑓𝜃′𝑠𝑧)\n𝜃′   (𝐴13) \nwhere 𝑓0(𝜖) is the unperturbed Fermi -Dirac distribution and 𝑣⃗=ℏ𝑘⃗⃗\n𝑚. There are two independent equations which \ndescribe th e distribution of electrons with 𝑠𝑧=±1\n2. We split 𝑓𝜃 into a 𝑠𝑧 independent term, 𝑓̅𝜃, and a term which \ndescribes the skew scattering, ±𝛿𝑓𝜃, where the sign refers to the spin. We assume that |𝑓̅𝜃̅|≫|𝛿𝑓𝜃| and obtain  \n𝑓̅𝜃≈−ℏ𝑘𝐹𝑒|𝐸⃗⃗|𝜏cos(𝜃)\n𝑚𝜕𝑓0(𝜖)\n𝜕𝜖    (𝐴14) \nThe shift in the carriers’ velocity is  \n𝑣∥≈𝑒|𝐸⃗⃗|𝜏\n𝑚         (𝐴15) \nInserting this expression into the Boltzmann equation, we obtain  \n0=ℏ𝑘𝐹𝑒|𝐸⃗⃗|cos(𝜃)\n𝑚𝜕𝑓0(𝜖)\n𝜕𝜖+1\n𝜏(𝑓̅𝜃±𝛿𝑓𝜃)±ℏ2𝑘𝐹2\n2𝑚2𝑐2𝜏\n𝜏𝑆𝑂𝐼ℏ𝑘𝐹𝑒|𝐸⃗⃗|sin(𝜃)\n𝑚𝜕𝑓0(𝜖)\n𝜕𝜖   (𝐴16) \nThis equation can be understood as if the function 𝑓̅𝜃±𝛿𝑓𝜃 is induced by a field which is rotated with respect to the \napplied field by an angle  \n𝛼≈±ℏ2𝑘𝐹2\n2𝑚2𝑐2 𝜏\n𝜏𝑆𝑂𝐼 ≈±(𝑘𝐹𝑎𝐵)2\n1372  ℓ\nℓ𝑆𝑂𝐼        (𝐴17) \nWhere 𝑎𝐵 is the Bohr radius, and ℓ𝑆𝑂𝐼 is a mean free path related to the impurities which induce the skew scattering.  \nNote that the  estimate in eq.(A17 ) is valid only when one two dimensional subband is occupied. In the limit ℓ𝑧−1≪𝑘𝐹 \nthe sample must be approximated using a three dime nsional Fermi surface.  In that case, the ansatz for the spin -orbit \nmatrix element s defined in eq.(A5 ) does not favor a particular spin direction, when scattering processes are averaged \nover the Fermi surface. As a result, an unpolarized current will not l ead to a net spin polarization at the edges of the \nsample.  \nEffect of Lorentz force due to the field produced by the primary current j x. The field associated with the current \n𝐵𝑧(𝑥)exerts a force on the carriers that move along the x axis. This force is given by 𝐹𝑥(𝑥)=𝑒𝑣𝐵𝑧(𝑥)𝑐⁄ and induces \na perturbation of the uniform density of carriers. However the value of the maximum Lorentz force for the density \ncurrents considered in this article (Hall effect with an applied field of ~1 G)   is four orders of magnitude smaller than \n𝐹𝑥(𝑥)=𝜇𝐵𝜕𝑥𝐵𝑧(𝑥). Therefore, the charge imbalance generated by the magnetic field induced by the current can be \nneglected.  \nExperimental techniques capable of measuring the contribution of the magnetic field induced by the current to the spin \nHall effect.  \nRef.(2) in the main text argues that the magnetic field induced by a current will not generate a voltage on a metallic \nbridge which contacts both edges of a Hall bar. This argument is correct, and it is base d on the fact that, when the bridge \nis close to the Hall bar, the magnetic field induced by the current will thread both the bar and the bridge, preventing \nany spin current to flow along the bar, and to induce a voltage across it. On the other hand, the co mbination of a spin \ngradient and spin dependent scattering by impurities will lead to the voltage discussed in ref.(2).  \nThis situation changes if the magnetic field threading the bridge is different from the field threading the bar. This can \nbe achieved by  changing the distance between the bridge and the Hall bar. Any technique which does not require a \ncontact between the edges of the bar will be sensitive to the spin accumulation induced by the field associated to the \ncurrent. Possible experimental techniq ues are. I) injection of a spin polarized current using a ferromagnet, refs.(7), and \n(9) of the main text, ii) spin pumping from a magnetic insulator, ref.(10) of the main text, iii) measurement of the Kerr \nrotation by optical means, ref. (8) of the main t ext.  A general overview of techniques used to measure the spin Hall \neffect can be found in the reviews cited in the main text, refs.(5) and (11).  \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "2001.04615v1.Simulating_fermions_in_spin_dependent_potentials_with_spin_models_on_an_energy_lattice.pdf",        "content": "Simulating fermions in spin-dependent potentials with spin models on an energy\nlattice\nMichael L. Wall\nJILA, NIST and University of Colorado, 440 UCB, Boulder, CO 80309, USA and\nThe Johns Hopkins University Applied Physics Laboratory, Laurel, MD, 20723, USA\n(Dated: January 15, 2020)\nWe study spin-1/2 fermions in spin dependent potentials under the spin model approximation , in\nwhich interatomic collisions that change the total occupation of single-particle modes are ignored.\nThe spin model approximation maps the interacting fermion problem to an ensemble of lattice\nspin models in energy space, where spin-spin interactions are long-ranged and spin-anisotropic. We\nshow that the spin model approximation is accurate for weak interactions compared to the harmonic\noscillator frequency, and captures the collective spin dynamics to timescales much longer than would\nbe expected from perturbation theory. We explore corrections to the spin model, and the relative\nimportance of corrections when realistic anharmonic potential corrections are taken into account.\nAdditionally, we present numerical techniques that are useful for analysis of spin models on an energy\nlattice, including enacting a change of single-particle basis on a many-body state as an e\u000bective time\nevolution, and \ftting of spatially inhomogeneous long-range interactions with exponentials. This\nlatter technique is useful for constructing matrix product operators for use in DMRG analyses, and\nmay have broader applicability within the tensor network community.\nI. INTRODUCTION\nLattice models of quantum magnetism, which describe\nthe interactions between quantum mechanical spins on\na regular array, are the oldest and still most prevalent\nrealizations of strongly correlated quantum many-body\nsystems [1]. While realizations of quantum magnetism\noccur in a variety of solid state and other condensed mat-\nter settings, modern advances in atomic, molecular, and\noptical (AMO) systems have opened the possibility for\nengineering models in which the dimensionality, strength,\nand even range of spin-spin interactions are all amenable\nto experimental control [2{6]. In addition, AMO experi-\nments have at hand unique tools for microscopically prob-\ning and characterizing magnetic order [7, 8]. In many\nAMO platforms, interactions are short range and spin-\nindependent, and so quantum magnetism arises from a\ncombination of atomic motion in a trap and quantum\nstatistics through, e.g., the superexchange mechanism [9{\n11] or through direct spin exchange [12{14]. Because of\nthis complex interplay, properly understanding magnetic\nphenomena in AMO systems requires an approach which\ntreats motion, interactions, and quantum statistics on a\ncommon footing.\nIn this work, we explore such an approach which is\napplicable to weakly interacting two-component parti-\ncles in tight traps. The key approximation in our ap-\nproach, whose validity we investigate in detail, is that the\nsingle-particle motional states are not changed by inter-\nactions, but interactions can a\u000bect spin dynamics though\nexchange. While this approach is only formally valid in\nthe regime in which the characteristic interaction energy\nper particle Uis weak compared to the trap energy scale\n~!, many relevant quantities, such as the demagneti-\nzation time of a polarized ensemble, can be evaluated\nwith much greater accuracy [15]. If the single-particle\nmode quantum numbers of the trap are thought of as\nnxnynx\nyx$$(a)(b)EnergyEnergyFIG. 1: (color online) Energy Lattices. The single-particle\nquantum numbers of a trap can be thought of as enumerating\nthe sites of a regular lattice in energy space, which we call\nan energy lattice. For D-dimensional harmonic traps, the\nnumber of quanta along each Cartesian direction enumerates\naD-dimensional cubic energy lattice. Examples are given for\n(a) a 1D Harmonic trap and (b) a 2D Harmonic trap.\nenumerating positions on a regular spatial array which\nwe call the energy lattice (Fig. 1), then the Hamiltonian\nwith this approximation takes the form of a spin model,\nand hence this is known as the spin model approxima-\ntion[16, 17]. In contrast to most spin models formulated\nin real space, spin models on the energy lattice feature\nlong-range and inhomogeneous spin-spin interactions and\ne\u000bective magnetic \felds. Additionally, while the interac-\ntion Hamiltonian with this approximation takes the form\nof a spin model, the Hilbert space it acts upon is still that\nof spin-1/2 fermions, and so the single-particle dynamics\nof arbitrarily correlated motional states can be exactly\naccounted for.arXiv:2001.04615v1  [cond-mat.quant-gas]  14 Jan 20202\nThe key advantage of the above prescription is that it\nprovides a description of the quantum dynamics of in-\nteracting fermions model in terms of a spin model, as\ntools for simulating the dynamics of quantum spin mod-\nels are, in general, better developed than their fermionic\ncounterparts [18]. The resulting spin models have spin-\nspin interactions which are long-ranged, which enables\nanalytical understanding through the use of collective or\nnearly-collective models [19, 20]. In addition, as shown in\nRef. [21], the combined e\u000bects of motion and spin dynam-\nics in the lowest order can be captured within the spin\nmodel approximation by simulating a collection of spin\nmodels, each de\fned on a di\u000berent energy lattice. These\ndi\u000berent energy lattices, each of which has its own set of\nspin model couplings, represent di\u000berent motional con-\n\fgurations which are involved in the dynamics. Hence,\nthe coherences between di\u000berent energy lattice dynamics\ncarry information about the interplay of spin and motion.\nIn addition to exploring the foundations and validity of\nthe spin model approach, we also present numerical tech-\nniques that are useful when dealing with dynamics on the\nenergy lattice. In particular, we present an account of\nmethods that were used in Ref. [21] to study spin model\ndynamics with matrix product state (MPS) simulations.\nMPSs, which are the underlying variational framework of\nthe density-matrix renormalization group (DMRG) algo-\nrithm [22, 23], have a long history in quantum spin sys-\ntems [24, 25], and continue to be important in the fron-\ntiers of quantum magnetism, including in long-range spin\nsystems [26] and in higher dimensions [27]. While some\nof our methodologies are speci\fc to MPSs, others, such\nas global transformations between energy lattice repre-\nsentations, have broader applicability. Some of the MPS\ntools we develop to study the present spin models, such\nas the representation of long-range, non-translationally\ninvariant interactions, may also have broader applicabil-\nity in other DMRG applications, such as in MPS-based\napproaches to quantum chemistry [28, 29]. In addition,\nthe spin model approximation, which we only rigorously\nbenchmark in 1D (see also Ref. [15]), is expected to be\nvalid in higher dimensions, a claim supported by recent\n3D experiments [30]. This will enable the use of ap-\nproximate techniques for spin systems, such as the trun-\ncated Wigner approximation [18, 31] to study the out-of-\nequilibrium dynamics of fermions in dimensions greater\nthan one, where no e\u000ecient, unbiased algorithms cur-\nrently exist.\nII. STATES, INTERACTIONS, AND\nOBSERVABLES ON THE ENERGY LATTICE\nA key component of our approach is that of an en-\nergy lattice , which is a discrete set of points indexed by\nsingle-particle energy. While there can be considerable\nfreedom in choosing how the lattice sites are arranged\nand indexed, for (near-)harmonic traps it is natural to\nchoose the energy lattice to be a regular cubic latticewith the same dimensionality as the trapped system, as\nin Fig. 1. Throughout, we will denote the trap mode in-\ndex with Roman letters n, with the understanding that\nthis can be immediately generalized to a vector of indices\nnin higher-dimensional scenarios, and the spin state with\ngreek letters \u001b.\nGiven a set of these possibly spin-dependent trap states\nf n\u001b(x)gwhich spans the low-energy part of Hilbert\nspace that is of interest to us, we can expand the\n\feld operator ^ (x) in this set. Then, substituting this\nexpansion into the second-quantized representation of\nthe many-body Hamiltonian generates an e\u000bective low-\nenergy model, in analogy with the procedure commonly\nused to derive lattice Hubbard models using Wannier\nfunctions [32, 33]. For spin-1/2 fermions experiencing s-\nandp-wave interactions modeled by contact pseduopo-\ntentials [34], the resulting many-body model is\n^H=X\nn;\u001b~!n\u001b^nn;\u001b (1)\n+1\n2X\nn1;n2;n0\n2;n0\n1X\n\u001b\u001b0I\u001b\u001b0\nn1;n2;n0\n2;n0\n1^ay\nn1\u001b^ay\nn2\u001b0^an0\n2\u001b0^an0\n1\u001b\nwhere the \frst line is the single-particle energy of trap\nstatejniin spin state \u001band the second line represents the\ninteraction Hamiltonian. In this expression, ^ an\u001bdestroys\na particle in mode nand spin state \u001b, ^nn;\u001b= ^ay\nn\u001b^an\u001b,\nand the interaction matrix elements are\nI\u001b\u001b0\nn1;n2;n0\n2;n0\n1=4\u0019~2as(1\u0000\u000e\u001b\u001b0)\nMS\u001b\u001b0\nn1;n2;n0\n2;n0\n1\n+6\u0019~2b3\n\u001b\u001b0\nMP\u001b\u001b0\nn1;n2;n0\n2;n0\n1(2)\nwhereasis thes-wave scattering length, b3\n\u001b\u001b0are thep-\nwave scattering volumes, and the geometrical integrals S\nandPare given as\nS\u001b\u001b0\nn1;n2;n0\n2;n0\n1=Z\ndr ?\nn1\u001b(r) ?\nn2\u001b0(r) n0\n2\u001b0(r) n0\n1\u001b(r);\n(3)\nP\u001b\u001b0\nn1;n2;n0\n2;n0\n1=Z\ndr\n\u0002\u0002\n ?\nn1\u001b(r)\u0000\nr ?\nn2\u001b0(r)\u0001\n\u0000\u0000\nr ?\nn1\u001b(r)\u0001\n ?\nn2\u001b0(r)\u0003\n\u0001\u0002\n n0\n1\u001b(r)\u0000\nr n0\n2\u001b0(r)\u0001\n\u0000\u0000\nr n0\n1\u001b(r)\u0001\n n0\n2\u001b0(r)\u0003\n:(4)\nWe use discrete variable representations for the represen-\ntation and manipulation of position-space wavefunctions\nin this work, including the evaluation of these integrals,\nbecause of their simplicity, \rexibility, and exponential\nconvergence [35].\nTranslating observables to the energy lattice represen-\ntation also proceeds straightforwardly by using the ex-\npansion of the \feld operator in terms of the trap states.\nFor example, the density of the collective spin raising\noperator ^S+(x) can be written as\n^S+(x) =X\nnm n\"(x) m#(x) ^ay\nn\"^am#: (5)3\nIntegrating this density, we \fnd the collective raising op-\nerator as\n^S+=X\nnm\rnm^ay\nn\"^am#; (6)\nwhere\rnm=R\ndx n\"(x) m#(x). For spin-independent\ntraps,\rnm=\u000enm, but for spin-dependent traps this ma-\ntrix is non-diagonal, with its non-zero elements having\nsubunity modulus. Hence, coherences between di\u000berent\npositions on the energy lattice are important for captur-\ning the spatial dependence of the fermionic spin density.\nThe spin model approximation [16, 17] consists of re-\nstricting the single-particle modes to a \fxed set, and\nonly allowing for interaction terms I\u001b\u001b0\nn0m0;mnwhich do\nnot change the single-particle modes participating in the\ncollision. Namely, this approximation keeps the direct\nI\u001b\u001b0\nnm;mnand exchange I\u001b\u001b0\nmn;mnterms. With this restric-\ntion, the Hamiltonian Eq. (1) reduces to\n^H=X\nn;\u001b~!n\u001b^nn;\u001b+X\nnI\"#\nnnnn^nn\"^nn#+X\nn6=n0Knn0^nn^nn0\n+X\nn6=n0[J?\nnn0\u0010\n^SX\nn^SX\nn0+^SY\nn^SY\nn0\u0011\n+JZ\nnn0^SZ\nn^SZ\nn0\n+Cnn0^SZ\nn^nn0+Dnn0^nn^SZ\nn0] (7)\nThe general form of the spin model is that of a long-\nrange XXZ model in with inhomogeneous longitudinal\n\felds (the terms with CandDcoe\u000ecients). Here, the\nparameters appearing in the spin model are de\fned as\nKnn0=1\n2(2I\"\"\nnn0n0n+ 2I##\nnn0n0n+I\"#\nnn0n0n+I#\"\nnn0n0n);\nJ?\nnn0=\u0000I\"#\nnn0nn0; (8)\nJZ\nnn0=1\n2(2I\"\"\nnn0n0n+ 2I##\nnn0n0n\u0000I\"#\nnn0n0n\u0000I#\"\nnn0n0n);\nCnn0=1\n2(2I\"\"\nnn0n0n\u00002I##\nnn0n0n\u0000I\"#\nnn0n0n+I#\"\nnn0n0n);\nDnn0=1\n2(2I\"\"\nnn0n0n\u00002I##\nnn0n0n+I\"#\nnn0n0n\u0000I#\"\nnn0n0n);\nwhere we have used the fact that I\"#\nnn0nn0=I#\"\nnn0nn0and\nI\u001b\u001b\nnn0nn0=\u0000I\u001b\u001b\nnn0n0n. We note that the coe\u000ecients Cand\nDare only nonzero in the case that the trap is spin-\ndependent. In the context of optical lattice clocks where\nhigh precision measurements are possible and tempera-\ntures correspond to a thermally averaged number of trap\nquanta\u001850 [16], both s- andp-wave collisions play an\nimportant role in the spin model. However, in ultracold\nor lower-precision scenarios, we can neglect the p-wave\ncomponents and keep only the s-wave, resulting in thesimpler coe\u000ecients\nKnn0=1\n2(I\"#\nnn0n0n+I#\"\nnn0n0n); (9)\nJ?\nnn0=\u0000I\"#\nnn0nn0; (10)\nJZ\nnn0=\u00001\n2(I\"#\nnn0n0n+I#\"\nnn0n0n); (11)\nCnn0=1\n2(I#\"\nnn0n0n\u0000I\"#\nnn0n0n); (12)\nDnn0=1\n2(I\"#\nnn0n0n\u0000I#\"\nnn0n0n): (13)\nBefore proceeding, we would also like to note that the\nessential idea of the spin model, which is to restrict the\nHamiltonian to energy-conserving direct and exchange\nprocesses, is quite old and has appeared in many con-\ntexts. Perhaps the earliest relevant use of this approach\nfor studying quantum e\u000bects in collisions of spinful par-\nticles is from Lhuillier and Lalo e [36, 37]. Many appli-\ncations of the spin model idea to cold atomic systems\narrived in the 2000s, following experiments at JILA [38]\nand Duke [39], which focused on fully collective spin sys-\ntems [40{42], kinetic theory approaches [43, 44], and,\nlater, on interaction e\u000bects in atomic clocks [45, 46]. The\nspin model and related approaches have been applied to\nthe recent experiments reported in Ref. [20, 30].\nPerhaps the most signi\fcant di\u000berence of our approach\ncompared to those listed above are that we only enact\nthe spin model approximation at the level of the Hamil-\ntonian, Eq. (7). In particular, our spin model framework\nplaces no restrictions whatsoever on the state, which is\nstill de\fned on the Hilbert space of spin-1/2 fermions in\ngeneral, and so can have arbitrary correlations between\nspins or between spin and motional degrees of freedom.\nThis fact is essential for applications to spin-dependent\nquantum quenches [21]. The other signi\fcant di\u000berence\nwith many of the works above is that our approach is\nfully quantum; that is, we do not study the spin model\nwithin a mean-\feld or semiclassical approach, but instead\nuse tools developed for strongly correlated quantum sys-\ntems.\nA. Behavior of the spin model parameters for\nspin-dependent traps\nPlots of the s-wave spin model parameters Eq. (10)-\n(12) for a 1D harmonic trap are given in Fig. 2. Here\nand throughout this work, these parameters are mea-\nsured in units of the s-wave interaction strength U\u0011\n4\u0019~2a(1D)\ns=(MaH), witha(1D)\nsthe e\u000bective 1D scattering\nlength obtained by integrating along tight con\fnement in\nthe transverse directions1. We will \frst look at panels\n1Note that a(1D)\ns has units of inverse length.4\nFIG. 2: (color online) E\u000bective spin-spin interactions on the energy lattice. Interaction parameters for (a)-(d) harmonic traps\nwith a spin-dependent displacement \u0006x0, (e)-(f) spin-independent harmonic traps, and (g)-(l) harmonic traps with a spin-\ndependent harmonic trapping frequencyp\n!2\u0006!2\nB. Panels (a), (b), (e), (f), (j), (k), and (l) are surface plots of interaction\nparameters as functions of the sites of the energy lattice, showing the long-range, non-translationally invariant character of the\ninteractions. Additionally, we see that introducing spin dependence to the trap a\u000bects the direct interactions Jzgenerally only\nslightly, but has a drastic e\u000bect upon the exchange coe\u000ecients Jz, reducing their magnitude and introducing negative values.\nThis is also seen in panels (c), (d), (g), (h), and (i), which are histograms giving the probability P[J] of obtaining the value\nJfor an interaction parameter in the \frst 50 modes as functions of the trap spin dependence.5\n(e) and (f), which are plots of Jz\nnn0andJ?\nnn0for spin-\nindependent traps ( Cnn0=Dnn0= 0 for spin-independent\ntraps). First, we note that interactions between di\u000ber-\nent modes on the energy lattice are long ranged even\nthough the underlying interactions are short-ranged in\nreal space due to the fact that the single-particle motional\nstates have signi\fcant spatial overlap. A rough estimate\nof the asymptotic decay of these interactions for spin-\nindependent traps is 1 =pn\u0000m. Next, we note that the\ninteractions are not translationally invariant on the en-\nergy lattice, which is to say that, e.g., Jz\nnn06=f(jn\u0000n0j),\nin contrast to interactions in real space.\nFor spin-independent traps, Jz\nnn0=J?\nnn0, resulting in\na Heisenberg model with SU(2) spin-rotation symmetry;\nthis re\rects the underlying symmetry of s-wave inter-\nactions. When the trap becomes spin-dependent this\nis no longer the case, as we have explicitly broken the\nsymmetry between spin states. Fig. 2 shows the interac-\ntion parameters computed for two special cases of spin-\ndependent traps. The \frst, corresponding to the upper\npanels (a)-(d), are for a trap with a spin-dependent dis-\nplacement and the second, corresponding to the lower\npanels (g)-(l), are for a trap with a spin-dependent trap-\nping frequency. The eigenstates for the former are shifted\nharmonic oscillator states,  \u001bn(x) =\u001en\u0010\nx+\u001bx0\naH\u0011\n,\nwith\u001en(x) the harmonic oscillator eigenstates, \u001b= +=\u0000\nfor\"=#,x0the displacement of the traps, and aHthe\nharmonic oscillator length, and the states for the lat-\nter are \u001bn(x) = (1 +\u001b!2\nB\n!2)1=8\u001en(x(1 +\u001b!2\nB\n!2)1=4), cor-\nresponding to spin-dependent trapping frequencies !\u001b= p\n!2+\u001b!2\nB.\nFrom Eq. (3), we see that the parameters JZinvolve\nthe integral of a product of two densities, and hence are\npositive de\fnite. On the other hand, the exchange coef-\n\fcientsJ?involve integrating a product of terms of the\nform \u001bn(x) \u001bn0(x), which themselves integrate to zero\nand so are not positive de\fnite. As the traps for spin up\nand spin down are displaced in opposite directions, the\nJZs generally decrease in magnitude, but this decrease is\nsmall for shifts small compared to a harmonic oscillator\nlength. This is shown in Fig. 2(c), which is a histogram of\ntheJ?s for the \frst 50 modes as a function of the spin-\ndependent displacements. On the other hand, as shown\nin Fig. 2(d), the exchange coe\u000ecients decay much more\nquickly with displacement, and also take on negative val-\nues. Snapshots of the complete mode dependence for the\nlargest displacement are given in panels (a) and (b); the\noscillatory behavior of J?with mode number is clearly\nvisible. We note that, because the traps have the same\nshape but are simply displaced, CandDremain zero for\nanyx0.\nWe now turn to the traps with spin-dependent fre-\nquency!\u001b=p\n!2+\u001b!2\nB, shown in Fig. 2(g)-(l). Sim-\nilar to the displaced traps in panels (c) and (d), we see\nthat the direct interactions JZare relatively insensitive\n(statistically) to the spin-dependence of the trap, while\nspin dependence causes the exchange terms to drop inmagnitude and take on negative values. In addition, the\nC=\u0000Dterms (panel (i)) become nonzero in this case,\nbut are generally quite small. Finally, panels (j)-(l) show\na snapshot of the interaction parameters for the largest\ndisparity in frequencies.\nIII. NUMERICAL TECHNIQUES\nIn this section we present two methods that are useful\nfor simulating spin systems on an energy lattice. The\n\frst is to recast global transformations of a state be-\ntween two energy lattices, as occurs when the single-\nparticle potential is abruptly modi\fed, in terms of long-\nrange time evolution under a non-interacting Hamilto-\nnian. Here, we benchmark our methods against two cases\nwhere the transformation is known analytically, namely,\nharmonic oscillators subject to sudden spin-dependent\ndisplacement or change in frequency. The second numer-\nical tool we provide is to represent non-translationally\ninvariant interactions, such as occur on the energy lat-\ntice, in terms of sums of exponentials with site-dependent\nweighting and exponential decay coe\u000ecients. Such a rep-\nresentation is essential for building matrix product oper-\nator (MPO) representations of spin model Hamiltonians\nto use in MPS calculations. We will not cover the basics\nof MPS calculations here, as there are many excellent re-\nviews devoted to the subject [47, 48]. More information\nabout MPOs and how they can be used to enhance MPS\nalgorithms may be found in Refs. [49{52]. We note that\nour numerical methodologies are by no means restricted\nto MPS algorithms, but may be used anywhere global\ntransformations or inhomogeneous interactions may be\nfound. In addition, within the MPS/DMRG community,\nthese methods may \fnd uses in other model applications,\ne.g., quantum chemistry [28, 29].\nA. Transforming between global representations\nA ubiquitous approach in ultracold gas experiments\nis to probe a system by suddenly quenching some pa-\nrameter in the Hamiltonian and observing the ensuing\nnon-equilibrium dynamics; this is the basis of Ramsey\nspectroscopy, for example. When the quenched param-\neter is related to the trapping potential, the dynamics\nwithin the spin model approach is calculated by project-\ning the initial state onto a new energy lattice de\fned by\nthe quenched trap and then evolving the spin dynam-\nics with Eq. (7) while keeping the mode occupations in\nthe new trap \fxed. The transformation between states\non di\u000berent energy lattices is generally a very complex,\nhighly non-local operation on the energy lattice. How-\never, as we show here, there is a well-de\fned procedure by\nwhich we can cast this quench as a time evolution under\na simulated \\Hamiltonian\" which consists of long-range\nand inhomogeneous free-particle hopping. Hence, this\nquench procedure can be simulated numerically using any6\nmethod capable of long-range time evolution. An explicit\nmatrix product operator (MPO) form for this inhomoge-\nneous, long-range Hamiltonian, which is useful for time\nevolution within the matrix product state (MPS) formal-\nism, can be obtained using the methods of the next sec-\ntion. Many MPS-based methods for time evolution un-\nder long-range Hamiltonians exist, including Krylov sub-\nspace methods [52{54], the time-dependent variational\nprinciple [55], and the local Runge-Kutta method [56]\nused in Ref. [21]. A nice feature of our e\u000bective time-\nevolution approach is that this change of basis can be\napplied to any arbitrary state; it is not restricted to prod-\nuct states.\nA transformation between the two single-particle bases\nfjnigandfj~nigis provided by the matrix\nUn~n=hnj~ni: (14)\nAs we let the number of states jniandj~nitend to in\fnity,\nthis becomes a unitary matrix. Postponing temporarily\nquestions about the unitarity of this matrix for \fnite-\ndimensional representations, we note that this unitary\noperator can be written as the exponential of a Hermitian\noperator ^U= exp(\u0000i^H), which has the interpretation of\ntime evolution for unit time with an ersatz \\Hamilto-\nnian,\" which we will call the change of basis Hamiltonian\n(COBH). Further, as the COBH generates a transforma-\ntion between single-particle Hilbert spaces, its second-\nquantized counterpart is a non-interacting Hamiltonian\nwith, in general, long-range and inhomogeneous hopping.\nThere are two important situations in which the COBH\ncan be determined exactly, which are the cases of a har-\nmonic oscillator subject to a sudden shift in trap center\nand a sudden change of trap frequency. In the case of\na shifted oscillator, the unitary transformation may be\nwritten as\n^Ushifted =e\u0015(^a\u0000^ay)=p\n2; (15)\nwhere ^ais the lowering ladder operator of the unshifted\nharmonic oscillator and \u0015speci\fes how far the state is\nshifted in oscillator units, i.e. hxj^Ushiftedjni=\u001en(x+\u0015)\nfor any harmonic oscillator eigenfunction hxjni=\u001en(x).\nWe can re-express the action of the lowering operator in\nmode space, ^ ajni=pnjn\u00001i, in terms of energy lattice\noperators as\n^a!X\njp\nj^by\nj\u00001^bj: (16)\nWith this, the COBH for a shifted harmonic trap is\n^Hshifted =ilog^Ushifted; (17)\n=iX\nj\u001b\"r\nj\n2\u0015\u001b^by\nj\u00001;\u001b^bj;\u001b\u0000H:c:#\n; (18)\nwritten in terms of the annihilation operators of the\nenergy lattice ^bj;\u001bfor sitejand spin\u001band the spin-\ndependent displacements \u0015\u001b. This COBH is a tight bind-\ning model with pure imaginary hopping coe\u000ecients vary-\ning aspjwith lattice site.In the case of a change in trap frequency, the transfor-\nmation is\n^Udilated =elog\u0015(^a2\u0000(^ay)2)=2; (19)\nwhere\u0015is the scale parameter such that hxj^Udilatedjni=p\n\u0015\u001en(\u0015x). Expressed in terms of the new trapping fre-\nquency ~!,\u0015=p\n~!=!. The COBH for a general spin-\ndependent change of trapping frequency !!!\u001bis hence\n^Hdilated =ilog^Udilated; (20)\n=iX\nj\u001b\"p\nj(j\u00001) log!\u001b=!\n4^by\nj\u00002;\u001b^bj;\u001b\u0000H:c:#\n;\n(21)\nwhere we have again mapped mode operators to the en-\nergy lattice using Eq. (16). Eq. (21) is a model involving\nonly next-nearest neighbor hopping with pure imaginary\nhopping coe\u000ecients.\nIn the general case, one can \fnd the COBH numeri-\ncally asH=ilogU, withUde\fned in Eq. (14). Uni-\ntary operators are normal, and so an appealing method\nto \fnd the logarithm is to compute the spectral decom-\nposition using numerical eigensolver routines. Standard\neigensolver approaches for general (i.e. non-symmetric)\nmatrices, such as ZGEEV in Lapack [57], cannot guaran-\ntee orthogonality of the eigenvectors when eigenvalues are\nnear-degenerate. Instead, it is desirable to compute the\ncomplex-valued Schur decomposition U=QRQywith\nQunitary and Rupper triangular. Routines for com-\nputing this form, such as ZGEES, do guarantee orthog-\nonality of the vectors in Q. Further, since Uis nor-\nmal,Rmust be diagonal with the eigenvalues as entries,\nand soHn~n=iP\nmQnmlog (Rmm)Q?\n~nm. As mentioned\nabove, the transformation matrix Eq. (14) is generally\nnot unitary for a \fnite-dimensional set of states. Even\nif this operator is \\approximately unitary\" for some sub-\nset of statesfjqigin the sense thatP\nmUqmU?\nqm\u00191\nfor these states, the Schur form of this operator will be\nnon-diagonal due the other states not in this set. Hence,\nit is essential as an intermediate step to compute the\nunitary matrix nearest to Uin the least-squares sense,\nwhich is obtained as U=UVy, whereU\u0006Vyis the sin-\ngular value decomposition of U, and use this matrix as\ninput for the Schur decomposition. While this operator\nwill not act appropriately on the entire set of basis states,\nit will reproduce the appropriate unitary action on the\nstatesfjnigwhich were near-unitary in the sense de\fned\nabove. Hence, by increasing the basis size, we can con-\nstruct the correct unitary transformation on any desired\nsubset of the basis states.\nExamples of the COBH matrix elements Hnn0are given\nin Fig. 3 for the cases of a shifted trap and a dilated\ntrap, using a trap displacement of 0 :2aH,aHthe har-\nmonic oscillator length, and an increase in the trapping\nfrequency by 0 :1!, respectively. The left panels show the\nresults for a harmonic oscillator, and display the band-\ndiagonal structure speci\fed above with matrix elements7\n 5 10 15 5 10 15\n 0.0001 0.001 0.01 0.1 1\n’HO_Dilation.dat’ u 1:2:(abs($3)+1e-4)\n 5 10 15 5 10 15\n 0.0001 0.001 0.01 0.1 1\n’Gauss_Dilation.dat’ u 1:2:(abs($3)+1e-4)\n 5 10 15 5 10 15\n 0.0001 0.001 0.01 0.1 1nn0|Imhn|ˆHdilated|n0i|10\u0000410\u0000410\u0000210\u0000211\n|Imhn|ˆHdilated|n0i|(a)(b)HarmonicGaussian\n 5  10  15 5 10 15\n 0.0001 0.001 0.01 0.1 1\n10\u0000410\u000021|Imhn|ˆHshifted|n0i|10\u0000410\u000021\n|Imhn|ˆHshifted|n0i|n\nnnn0\nn0n0(c)(d)HarmonicGaussian\nFIG. 3: (color online) Change of basis Hamiltonians for\nshifted and dilated traps. The absolute magnitude of the ma-\ntrix elements of the change of basis Hamiltonian (COBH) H\nare shown versus eigenstate index. The top panels (a)-(b)\nare for dilated traps and the bottom panels (c)-(d) for shifted\ntraps. The left panels (a), (c) show the results for a har-\nmonic oscillator, demonstrating the strictly banded structure\nof Eqs. (18) and (21). The right panels (b), (d) show the re-\nsults for a deep Gaussian trap, in which anharmonicity smears\nout the banded structure. The smearing is greater for higher\neigenstates, which sample more of the anharmonic regions of\nthe trap.\nincreasing with n. The right panels show the numeri-\ncally obtained COBH for a 1D Gaussian trap of the form\nVe\u00002x2=`2. Here, we take the depth V=E`= 800, where\nE`=~2=(2m`2) is the energy associated with the trap\nlength scale `. The shift and increase in trap frequency\nused in the COBH calculation use the harmonic approxi-\nmations obtained by a quadratic expansion near the trap\nminimum: ~!=p8E`VandaH=`=(2V=E`)1=4. The\nplots show the expected behavior: the dominant COBH\nstructure of the Gaussian trap is the same as the cor-\nresponding harmonic trap. However, there are more\nnonzero elements, and the magnitudes of these elements\nincrease as the eigenstate energy increases, as these eigen-\nstates sample more of the anharmonic regions of the trap.\nB. Representation of Inhomogeneous long-range\ninteractions\nAn important intermediate step in applying MPS\nmethods to a model with long-range interactions is to\nwrite the Hamiltonian as a matrix product operator. For\na translationally invariant, decaying interaction of the\nform\nX\ni<jf(j\u0000i)^Ai^Bj; (22)\na well-established method exists, known as the Hankel\nsingular value decomposition, which amounts to least-squares \ftting the interaction function f(r) by a sum\nof (possibly complex) exponentials [58, 59]. That is, the\nansatz\n~f(r) =nexpX\nn=1Jn\u0015r\nn; (23)\nis optimized via the functionalPrmax\nr=1\f\f\ff(r)\u0000~f(r)\f\f\f2\nwith the maximum range rmaxand the number of ex-\nponentialsnexpas convergence parameters.\nThis approach no longer holds for interactions which\nare not translationally invariant. While interactions rep-\nresented in real space are translationally invariant, in-\nteractions on the energy lattice are strongly inhomoge-\nneous (see Fig. 2), and so alternative methods are re-\nquired. Non-translationally-invariant interactions also\nappear in other contexts, such as quantum chemical cal-\nculations with DMRG [29] and in the representation of\n2D systems\u0000even systems with translationally invariant\ninteractions\u0000by mapping to a 1D chain [60].\nThe most direct method of constructing an MPO rep-\nresentation is to variationally optimize the elements of\nthe MPO itself, which can be done using the meth-\nods developed for MPSs by mapping this operator to\na state on a lattice with d2\nidimensional local Hilbert\nspaces. While these methods have been successfully used\nin some cases [51], they su\u000ber from potential drawbacks.\nFirst, while MPO constructions of known Hamiltonian\nare extremely sparse, this variational method does not\npreserve sparsity and in general produces fully dense\nMPOs (though this can be alleviated in some cases, see,\ne.g. [61]). Even in the case that the bond dimension of\nthe MPO, which is often used as a proxy for its com-\nplexity, is small, the total number of operations required\nto apply this MPO to an MPS may be larger than for\na sparse MPO with larger bond dimension. In addition,\nvariational \ftting of MPOs can be numerically costly,\namounting to an MPS simulation with high-dimensional\nlocal Hilbert spaces. This optimization must be per-\nformed whenever any change is made to the operator,\nand so previous optimizations or results cannot be re-\nused easily.\nHere, we propose an alternative approach, in which the\nparameters appearing in the exponential \ft, Eq. (23), are\nallowed to vary in space. Namely, given a generic interac-\ntion of the formP\ni<jf(i;j)^Ai^Bj, we \ft the terms with\ni<j with the ansatz\n~f(i;j) =nexpX\nn=1Ji;nj\u00001Y\nk=i\u0015k;n: (24)\nThe ansatz Eq. (24) is a natural extension of Eq. (23)\nin which the coupling constants fJigand decay param-\netersf\u0015kgare allowed to vary along the chain. To\n\fnd these site-dependent coe\u000ecients, we solve the least-8\nsquares minimization problem\nmin\nJ1:::JL;\u00151:::\u0015L\u00001L\u00001X\np=1L\u0000pX\ni=1\f\f\f\f\f\ff(i;i+p)\u0000nexpX\nn=1Ji;ni+p\u00001Y\nj=i\u0015j;n\f\f\f\f\f\f2\n:\n(25)\nA direct non-linear least squares minimization of\nEq. (25) using the Levenberg-Marquardt algorithm [62]\nhas proven to be ill-conditioned due to strong cross-\ncorrelations between variations in the \u0015parameters. Us-\ning standard regularization procedures, e.g., Tikhonov\nor SVD regularization [63], to improve the conditioninghas the unfortunate consequence of introducing an imag-\ninary component to ~fon the order of the regularization\nparameter.\nTo avoid the above drawbacks of direct non-linear\nleast-squares minimization of Eq. (25), we instead use\nan alternating least-squares approach which is very simi-\nlar to the sweeping method used in DMRG [47, 64]. The\nmotivation for this is the fact that the functional Eq. (25)\nwith all parameters held \fxed except for one (either a \u0015j\nor aJj) is a quadratic form in that parameter. For ex-\nample, we can write the functional as a quadratic form\nin\u0015qas\nL\u00001X\np=1L\u0000pX\ni=1n\njf(i;i+p)j2\u0000[i\u0014q\u0014i+p\u00001]f(i;i+p)Jy\nii+p\u00001Y\nj=i0\n\u0003?\nj\u0015?\nq\u0000[i\u0014q\u0014i+p\u00001]f?(i;i+p)JT\nii+p\u00001Y\nj=i0\n\u0003j\u0015q\n+0\n@\u0015y\nqi+p\u00001Y\nj=i0\n\u0003?\njJ?\ni1\nA0\n@JT\nii+p\u00001Y\nj0=i0\n\u0003j\u0015q1\nAo\n; (26)\nwhere the prime on the products indicates that j=q\nis omitted, \u0003 denotes the diagonal matrix with \u0015as en-\ntries, and the Iverson bracket [ a] = 1 ifais true and0 otherwise. Taking the derivative with respect to \u0015?\nq\nand setting it equal to zero, this quadratic form de\fnes\na linear system of equations for minimization\n0\n@L\u00001X\np=1L\u0000pX\ni=1[i\u0014q\u0014i+p\u00001]i+p\u00001Y\nj=i0\n\u0003?\njJ?\niJy\nii+p\u00001Y\nj0=i0\n\u0003j1\nA\u0015q=L\u00001X\np=1L\u0000pX\ni=1[i\u0014q\u0014i+p\u00001]f(i;i+p)Jy\nii+p\u00001Y\nj=i0\n\u0003?\nj:(27)\nThe right-hand side is a constant vector, and parentheses\non the left-hand side denotes the matrix of correlations\nbetween changes in \u0015qand all other parameters. Solving\nthis equation in the least-squares sense, i.e. applying the\npseudoinverse of the matrix on the left hand side to the\nright hand side, gives the optimal exponential decay pa-\nrameters \u0015qfor minimizing our functional with all other\nparameters held \fxed. We will refer to the minimiza-\ntion of our functional with respect to a single parameter\nwith all others held \fxed as local minimization. This is\nexactly in analogy with the local minimization in varia-\ntional MPS algorithms [47], in which the matrices cor-\nresponding to a single site are optimized with all others\nheld \fxed. The local minimization with respect to Jiisgiven by the linear system of equations\n0\n@L\u0000iX\np=1i+p\u00001Y\nj=i\u0003?\nji+p\u00001Y\nj0=i\u0003j1\nAJq\n=L\u0000iX\np=1[i\u0014q\u0014i+p\u00001]f(i;i+p)\u0015y\nii+p\u00001Y\nj=i+1\u0003?\nj:(28)\nThe complete procedure is to \\sweep\" across parameters,\nperforming local minimization on the \u0015s andJs in turn,\nuntil convergence is reached. This algorithm is much\nmore stable than a direct global least-squares minimiza-\ntion, and also produces real approximations ~f.\nAs our proposed scheme is essentially variational, the\nquality of our \fnal state and the rate of convergence are\ngreatly improved by having a good initial guess. We do\nso by \fnding the nearest translationally invariant inter-\naction, and then \ftting that interaction using the Hankel\nsingular value decomposition. Finding the nearest trans-9\n 1e-06 1e-050.0001 0.001 0.01 0.1 1\n 0.01 0.1 1 10Iteration numberResidual|J?\u0000˜J?|2\n10\u0000610010110\u0000110\u0000410\u0000210\u00002100\nFIG. 4: (color online) Exponential Fitting of non-\ntranslationally invariant interactions. The error of the non-\ntranslationally invariant MPO \ftting procedure, as quanti-\n\fed by the (Frobenius norm) residual jJ?\u0000~J?j2between\nthe exact interaction matrix J?and the approximate matrix\n~J?constructed as in Eq. (24), is shown as a function of the\noptimization e\u000bort, quanti\fed by number of optimization it-\nerations. Here, one iteration refers to optimization of each Ji\nand\u0015ionce.\nlationally invariant interaction is equivalent to \fnding\nthe nearest matrix to f, denotedfT, which has constant\ndiagonals (i.e. a Toeplitz matrix). This is\nfT(jpj) =PL\u0000p\ni=1f(i;i+p)\nL\u0000p: (29)\nFitting this with the Hankel singular value decomposi-\ntion to obtain ~Jand ~\u0015, we can initialize Ji=~Jand\n\u0015i=~\u0015. We stress that our method optimizes the coef-\n\fcient matrix frather than a many-body operator, and\nso produces e\u000ecient, re-usable approximations for any\noperators ^Aand ^B, and does not have to perform any\noperations in a many-body Hilbert space.\nTo show how this algorithm works in practice, we will\n\ft the exchange interaction matrix in the case that the\nparticles are subject to a harmonic trap and a magnetic\n\feld gradient of strength m!2x2\n0[21],\nJ?\nnm=Z\ndx n(x\u0000x0) n(x+x0)\n\u0002 m(x\u0000x0) m(x+x0): (30)\nAn error-e\u000bort plot of the optimization procedure is\nshown in Fig. 4. The nearest-translationally invariant\ninteraction matrix, which is the input to the optimiza-\ntion procedure, has an error \u0018O (1), while running the\noptimization procedure for 50 \\sweeps\" produces an er-\nror nearly six orders of magnitude smaller. Although\nour number of variational parameters is much greater\ncompared to the translationally invariant case, the struc-\nture and sparsity of the MPO representation have not\nchanged.IV. VALIDATION OF THE SPIN MODEL AND\nIMPORTANCE OF CORRECTIONS\nThe spin model is, at face value, a rather severe ap-\nproximation. In particular, taking the expectation of\nthe spin model Hamiltonian for a non-interacting state\nis equivalent to \frst-order perturbation theory in the in-\nteraction Hamiltonian. However, since our approxima-\ntion is made at the operator level rather than the ex-\npectation value level, dynamics obtained with the spin\nmodel Hamiltonian includes contributions to all orders in\nperturbation theory for certain terms in the interaction\nHamiltonian, while completely neglecting the e\u000bects of\nother terms. That is to say, the spin model is not equiv-\nalent to perturbation theory in the interaction Hamil-\ntonian for dynamics. In spite of its seeming severity,\nthe spin model performs extraordinarily well in predict-\ning some quantities, such as the decay of the contrast in\nRamsey spectroscopy following a sudden spin-dependent\nquench of the trapping parameters [21]. The purpose of\nthis section is to numerically benchmark the spin model\nagainst solutions of the full Hamiltonian Eq. (1) for small\nsystems where computing the full dynamics is possible.\nIn addition, we consider what the dominant corrections\nto the spin model are for weak to moderate interactions\ncompared to the trapping frequency, and how the impor-\ntance of these corrections is modi\fed by adding realistic\nanharmonicity to the trap.\nIn what follows, we will consider a system with Ns= 5\nspin-1/2 fermions experiencing only s-wave interactions.\nOur initial state is the motional ground state in a spin-\nindependent harmonic trap with all spins pointing along\nthexdirection. As all of the particles are prepared iden-\ntically, they do not experience the s-wave interactions.\nWe then suddenly quench on a spin-dependent potential\nin the form of a spin-dependent displacement or dilation\nof the traps. This change in the trapping potential re-\nsults in spin-dependent motion in the trap, which in turn\nleads tos-wave collisions.\nOne of the most striking consequences of s-wave colli-\nsions is coherent demagnetization of the system, evinced\nby decay ofhSxito zero. Examples of this demagneti-\nzation behavior for an interaction strength U=~!=p\n8\nand trap displacements of x0= 0:1aHandx0= 0:3aH\nare shown in Fig. 5, with the solutions of the full model\ncorresponding to solid red lines and the spin model pre-\ndiction as dashed blue lines. On a coarse scale, we see\nthat the spin model does an excellent job of capturing the\noverall envelope and timescale of the collapse of magne-\ntization. However, a closer analysis (insets) shows that\nthere are small oscillations on top of this envelope which\ncorrespond to spin-dependent motion in the trap which\narises from both single-particle and interaction e\u000bects.\nThe spin model only captures the single-particle compo-\nnent of this motion, and ignores interaction e\u000bects, and\nso only reproduces the exact result at short times.\nThe full model Eq. (1), which fully incorporates the\ne\u000bects of interactions on motion in the trap, contains in10\n 0 0.5 1\n 0 200 400 60010.50\n 0 0.5 1\n 0 200 400 60010.500200400600x0=0.1aH\nx0=0.3aH\n 0.5 1\n 0 60\n 0 0.5 1\n 0hSxi/(Ns/2)hSxi/(Ns/2)!t\nFIG. 5: (Color online) Demagnetization following a spin-\ndependent displacement. The exact (red solid) and spin-model\n(blue dashed) dynamics of the collective spin for an initially\npolarized state subject to a spin-dependent displacement of\nharmonic traps by x0= 0:1aH(top panel) or x0= 0:3aH(bot-\ntom panel) and interaction strength U=~!=p\n8. The overall\ndemagnetization and revival timescales are well-captured by\nthe spin model approach, while smaller-scale features due to\ninteraction-modi\fed motion in the trap are not perfectly cap-\ntured. Note that non-interacting motion in the trap is exactly\ncaptured within the spin model framework.\nprincipleO\u0000\nN4\nmodes\u0001\nparameters for a \fxed set of Nmodes\nsingle-particle modes. In contrast, the spin model con-\ntains onlyO\u0000\nN2\nmodes\u0001\nparameters. Clearly, not all of the\nneglected parameters contribute equally, and so a natural\nquestion is which of these parameters are most impor-\ntant for capturing interaction e\u000bects on motion. Moti-\nvated by perturbation theory, we expect that the parame-\ntersIn0\n1;n0\n2;n2;n1which preserve single-particle energy, i.e.\nn0\n1+n0\n2=n1+n2, will be the most relevant, followed\nby those parameters which change the single-particle en-\nergy by\u00061 quantum,\u00062 quanta, etc. Further, within the\nset of parameters which do not change the single-particle\nenergy, those which involve modes n0\n1andn0\n2which are\nclosest ton1andn2(up to exchange) will have a larger\nmatrix element. Hence, we can classify Hamiltonians in-\ncluding corrections beyond the spin model with two num-\nbers (\u0001n;d), where \u0001n= max(n0\n1+n0\n2\u0000(n1+n2)) is\nthe di\u000berence in the single-particle energy of the initial\nand \fnal con\fgurations and dis the maximum \\mode\n 0.0001 0.001 0.01 0.1 1\n 0 200 400 600200400600!t110\u0000210\u000040(0,0)(0,1)Inﬁdelity (1\u0000F)(0,1)(0,2)FIG. 6: (Color online) Convergence of spin model (0; d)with\nrange d.Time evolution of the in\fdelity following a spin-\ndependent displacement with x0= 0:1aH. The curves from\ntop to bottom are labeled in terms of increasing mode dis-\ntance d, with the top curve being the spin model result and\nthe bottom curve accounting for all interaction processes that\npreserve single-particle energy. Results converge rapidly with\nincreasing din this case.\ndistance\"\nM(n0\n1;n0\n2;n1;n2) = min(jn0\n1\u0000n1j+jn0\n2\u0000n2j;\njn0\n1\u0000n2j+jn0\n2\u0000n1j) (31)\nwhich accounts for exchange. In terms of this notation,\nthe spin model is the Hamiltonian (0 ;0), involving no\ndi\u000berence in single-particle energy and also no \\mode\nseparation\" between the initial and \fnal con\fgurations.\nAs a strict measure of how well the spin model and cor-\nrections to it perform, we will consider the \fdelity of the\nstate evolved under such an approximate Hamiltonian\nwith the full evolution under the Hamiltonian Eq. (1),\nF=jh exactj approx:ij. An example for x0= 0:1aHand\nU= 0:1~!is shown in Fig. 6. All curves show models\nin which single-particle energy is conserved (\u0001 n= 0),\nwhile the curves from top to bottom show the results\nas the mode distance dincreases from zero (only mode-\npreserving direct and exchange terms, i.e. the spin model)\nto arbitrary distances. In this case, the spin model has\na roughly 10% in\fdelity at long times, while the d!1\ncase improves this to roughly 1% in\fdelity. The in\f-\ndelity rapidly saturates with dbeyondd= 1 in this case;\nd!1 is essentially indistinguishable from d= 2 on the\nscale of this plot.\nThe top panel of Fig. 7 again considers the x0= 0:1aH,\nU= 0:1~!scenario, but considers two new features.\nThe \frst is given in the inset, where we show the er-\nror in the collective magnetization per spin for the spin\nmodel and (0 ;1) model. While the spin model has de-\nviations with non-negligible amplitude, we see that the\ndeviations are centered around zero, which is to say that\nthe average, collective behavior is well-captured (see also\nFig. 5). In addition, we see that the (0 ;1) model has spin\ndeviations which are barely perceptible on the scale of11\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1\n 0 200 400 6001\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1\n 0 200 400 600\n200400600x0=0.1aH\nx0=0.3aH\n!t10\u0000210\u0000410\u00006110\u0000210\u0000410\u000060(0,0)(0,1)(1,1)(2,1)(0,0)(0,1)(1,1)(2,1)Inﬁdelity (1\u0000F)Inﬁdelity (1\u0000F)\n-0.2 0 0.2\n 0 100\n-0.20.20100\n-0.2 0 0.2\n 0 100-0.201000.2\u0000hSxi/(Ns/2)\u0000hSxi/(Ns/2)\nFIG. 7: (Color online) In\fdelity of spin model in a displaced\nharmonic trap with and without corrections. Time evolution\nof the in\fdelity and collective magnetization per spin (inset)\nfollowing a spin-dependent displacement with x0= 0:1aH\n(top panel) and x0= 0:3aH(bottom panel). The spin dy-\nnamics are well-captured by mode-preserving terms, while the\ninclusion of mode non-conserving terms increases the \fdelity.\nthis plot. Hence, single-particle energy-preserving mode\nchanges are the dominant source of the collective spin dy-\nnamics. The second feature of Fig. 7 is that models with\nenergy non-conserving terms (\u0001 n>0) are included, and\ndisplay further improvements in the \fdelity. The bottom\npanel of Fig. 7 present the same analysis but for a larger\ndisplacement of x0= 0:3aH, and demonstrates that the\nbehavior seen for x0= 0:1aHis generic.\nFrom the above, we see that the dominant correc-\ntions relevant for capturing the behavior of low-order\ncorrelation functions come from single-particle-energy-\npreserving terms. In a harmonic potential, there are a\nlarge number of such resonances due to the linear spac-\ning of energy levels. In an anharmonic potential these\nresonances are no longer exact, and the only exactly res-\nonant collisions are the direct and exchange ones kept by\nthe spin model. To better understand the e\u000bects of an-\nharmonicity on the \fdelity of the spin model approach,\nwe consider a Gaussian potential \u0000Vexp\u0000\n\u00002x2=`2\u0001\n. The\nbest harmonic approximation to this potential, given by\nmatching the local curvature near x= 0, yields the har-\nmonic frequency ~!=p8E`VwithEl=~2=(2m`2) and\nthe harmonic length aH=`=(2V=E`)1=4. Treating the\nquartic-order expansion of the potential in \frst-order per-\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1\n 0 200 400 600 800 1000 12001\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1\n 0 200 400 600 800 1000 1200\n400x0=0.1aH\nx0=0.3aH\n!t10\u0000210\u0000410\u00006110\u0000210\u0000410\u000060(0,0)(0,1)(1,1)(2,1)(0,0)(0,1)(1,1)(2,1)Inﬁdelity (1\u0000F)Inﬁdelity (1\u0000F)8001200FIG. 8: (Color online) In\fdelity of spin model in a dis-\nplaced gaussian trap with and without corrections; weak inter-\nactions. The in\fdelity of various models in a Gaussian trap\n(dark lines) compared to a harmonic trap (faint lines), all at\nU= 0:02~!, which nearly matches the smallest anharmonic-\nity. Anharmonicity is shown to suppress processes outside the\nspin model.\nturbation theory about the harmonic solution, we \fnd\nthe energies\nEn\u0019\u0000V+~!\u0012\nn+1\n2\u0013\n+3\n2~!p\n8\u0016V\u0012\nn2+n+1\n2\u0013\n;\n(32)\nwhere \u0016V=V=E`. Hence, if we consider an interaction\nof modesnandmscattering into ( n+d) and (m\u0000d),\nthat would be energy conserving in a harmonic trap, this\nprocess is o\u000b-resonant by an amount\n\u0001En;m;d\u00193~!p\n8\u0016Vd[d\u0000(m\u0000n)]; (33)\n= 3E`d[d\u0000(m\u0000n)]: (34)\nAs expected, this energy di\u000berence vanishes for d= 0 or\nd= (m\u0000n), corresponding to no change in the modes\nor a mode swap.\nFor the purposes of understanding the relative impact\nof anharmonicity on the \fdelity of the spin model, it\nwould be most useful to work at \fxed interactions and\nparticle number and modify the anharmonicity through\nthe well depth. However, there is a direct connection be-\ntween the depth of the well, and hence the anharmonicity,12\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1\n 0 200 400 6001\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1\n 0 200 400 600\n200400600x0=0.1aH\nx0=0.3aH\n!t10\u0000210\u0000410\u00006110\u0000210\u0000410\u000060(0,0)(0,1)(1,1)(2,1)(0,0)(0,1)(1,1)(2,1)Inﬁdelity (1\u0000F)Inﬁdelity (1\u0000F)\nFIG. 9: (Color online) In\fdelity of spin model in a dis-\nplaced gaussian trap with and without corrections; strong in-\nteractions. The same as Fig. 8 but for interaction strength\nU= 0:1~!which is nearly 5 times the smallest anharmonic-\nity. Here, the results for the Gaussian and harmonic traps\nbehave similarly.\nand the number of bound states. A WKB approximation\nfor the number of bound states yields\nNb\u0019r\u0016V\n\u0019+1\n2: (35)\nHence, for \fxed \u0016U=U=~!, the ratio of anharmonicity\nto interactions scales as\n\u0001En;m;d\n\u0016U~!\u00193\n\u0016Up\n8\u0019(Nb\u00001=2)d[d\u0000(m\u0000n)];(36)\nwhich shows that the ratio can only be changed by\nchangingUat \fxed number of bound states. Taking\nNb\u001830 bound states gives \u0016V\u00183000, and so for\nthe interactions to be on the same order as the anhar-\nmonicity requires U\u00190:02~!. Fig. 8 shows the ana-\nlog of Fig. 7 in a Gaussian trap with \u0016V= 3000 and\nU= 0:02~!, where att= 0 the two spin states feel po-\ntentials\u0000Vexp\u0010\n\u00002 (x\u0006x0)2=`2\u0011\n. That is to say, the\npotential itself is shifted rather than a constant gradient\napplied (recall the two operations are identical for a har-\nmonic trap). The dark lines are the results for the Gaus-\nsian trap, and the faint lines are the corresponding har-\nmonic trap results. We see a very marked increase in the\n\fdelity of the spin model for the Gaussian trap compared\nto the harmonic well by more than an order of magnitude\n-1-0.5 0 0.5 1\n 0 200 400 600\n-1-0.5 0 0.5 1\n 0 200 400 6000200400600!t\u00001\u000010011hSxi/(Ns/2)hSxi/(Ns/2)!B=0.1!\n!B=0.3!FIG. 10: (Color online) Demagnetization following a spin-\ndependent change in trap frequency. The exact (red solid) and\nspin-model (blue dashed) dynamics of the collective spin for\nan initially polarized state subject to a spin-dependent change\nof harmonic trap frequency characterized by !B= 0:1!\n(top panel) or !B= 0:3!(bottom panel) and interaction\nstrength U= 0:4~!. The overall demagnetization and re-\nvival timescales are well-captured by the spin model approach,\nwhile smaller-scale features due to interaction-modi\fed mo-\ntion in the trap are not captured. Note that non-interacting\nmotion in the trap is exactly captured within the spin model\nframework.\non average over the times considered, with more modest\ngains for the models that include \u0001 n6= 0 ord6= 0. As\ninteractions are increased relative to the anharmonicity,\nthe Gaussian and harmonic oscillator results again be-\ncome comparable, as shown in Fig. 9 for U= 0:1~!,\nroughly 5 times the anharmonicity. Similar conclusions\nare expected to hold in higher dimensions, where the den-\nsity of degenerate harmonic oscillator modes grows more\nrapidly. We note that current optical lattice clocks oper-\nate in the regime of interactions .anharmonicity [17].\nWe now turn to the case in which the particles experi-\nence a sudden spin-dependent change in trap frequency\nto new frequencies !\u001b=p\n!2+\u001b!2\nB. Here, Fig. 10 is\nan analog of Fig. 5 for the displaced trap case, show-\ning coherent demagnetization due to interactions with\nweak and strong spin-dependent trapping changes. As\ndiscussed further in Ref. [21], the top panel is indicative\nof a regime whose spin dynamics is well-described by a\nglobal precession of the collective spin in the XY plane,13\n0.00000100.00001000.00010000.00100000.01000000.10000001.0000000\n 0 200 400 6000.00000100.00001000.00010000.00100000.01000000.10000001.0000000\n 0 200 400 600\n0200400600!t110\u0000210\u0000410\u00006110\u0000210\u0000410\u00006Inﬁdelity (1\u0000F)Inﬁdelity (1\u0000F)(0,0)(0,1)(1,1)(2,1)(2,1)(1,1)(0,1)(0,0)!B=0.1!\n!B=0.3!\nFIG. 11: (Color online) In\fdelity of spin model in a depth-\nmodulated harmonic trap with and without corrections. Time\nevolution of the in\fdelity following a spin-dependent dilation-\nwith !B= 0:1!(top panel) and !B= 0:3!(bottom panel).\nThe interaction strength is U= 0:4~!. The dynamics are\ncaptured roughly equally well by the spin model and the con-\nsiderably more complex (2 ;1) model.\nwith the potential inhomogeneity driving oscillations be-\ntween the manifold of fully collective \\Dicke\" states and\nthe neighboring \\spin-wave\" manifold. In this regime,\nthe spin model does an exceptional job of reproducing the\nexact results. The bottom panel shows the breakdown of\nthis picture when potential inhomogeneity is increased\nto become on the same order of the interactions, and de-\nviations of the spin model from the exact result due to\ninteraction-induced mode changes are evident. This is\nalso formalized through the in\fdelity in Fig. 11, which is\nanalogous to Fig. 7 for the displaced trap case with larger\ninteractions U= 0:4~!. In contrast to the displaced case,\nincreasing the mode distance ddoes not produce a signif-\nicant decrease in the in\fdelity; rather, the spin model has\nqualitatively similar \fdelity to the more complex (2 ;1)\nmodel.\nFinally, we consider the impact of potential anhar-\nmonicity on the case of a dilated trap, again consider-\ning a Gaussian trap of the form Ve\u00002x2=`2. The most\nnatural means of changing the e\u000bective trapping fre-\nquency would be to change the depth of the potential\nin a spin-dependent fashion, i.e., V!V+\u001b\u0001Vwith\n0.00000100.00001000.00010000.00100000.01000000.10000001.0000000\n 0 400 800 12000.00000100.00001000.00010000.00100000.01000000.10000001.0000000\n 0 400 800 1200\n0400!t110\u0000210\u0000410\u00006110\u0000210\u0000410\u00006Inﬁdelity (1\u0000F)Inﬁdelity (1\u0000F)(0,0)(0,1)\n(1,1)(2,1)(2,1)(1,1)\n(0,1)(0,0)!B=0.1!\n!B=0.3!\n8001200FIG. 12: (Color online) In\fdelity of spin model in a depth-\nmodulated gaussian trap with and without corrections; weak\ninteractions. The in\fdelity of various models in a Gaussian\ntrap (dark lines) compared to a harmonic trap (faint lines),\nall at U= 0:02~!, which nearly matches the smallest an-\nharmonicity. Anharmonicity suppresses processes outside the\nspin model for small !B=!(top panel), but not signi\fcantly\nfor larger !B=!(bottom panel).\n\u0001V=~2!2\nB=(8E`) =V!2\nB=!2. However, such a change\nin depth introduces a homogeneous spin-dependent en-\nergy o\u000bset of\u0018\u0001Vthat suppresses interaction e\u000bects,\nsee Eq. (32). Hence, in our simulations, we add a homo-\ngeneous potential of V+\u001b\u0001Vto best match the spec-\ntrum to the harmonic spectrum. As before, we study the\ncases ofUcomparable to the anharmonicity and Ularge\ncompared to anharmonicity, and take \u0016V\u00183000. In the\nformer case, shown in Fig. 12, where U= 0:02~!, we\nsee that anharmonicity suppresses terms outside of the\nspin model at small !B= 0:1!. However, for the case of\nlarger!B= 0:3!the anharmonic trap result has worse\n\fdelity than the harmonic trap one, though the \fdeli-\nties of both are quite good, at the 1 \u000010\u00004level. The\ncase of strong interactions compared to anharmonicity,\nU= 0:1~!, is shown in Fig. 13. Here, similar to the\ncase of the displaced traps in Fig. 9, the harmonic and\nanharmonic trap results behave similarly.\nV. CONCLUSIONS\nWe discussed the spin model approximation for\nfermions in spin-dependent potentials, in which interac-14\n0.00000100.00001000.00010000.00100000.01000000.10000001.0000000\n 0 200 400 6000.00000100.00001000.00010000.00100000.01000000.10000001.0000000\n 0 200 400 600!B=0.1!\n0200400600!t110\u0000210\u0000410\u00006110\u0000210\u0000410\u00006Inﬁdelity (1\u0000F)Inﬁdelity (1\u0000F)(0,0)(0,1)\n(1,1)(2,1)(2,1)(1,1)\n(0,1)(0,0)\n!B=0.3!\nFIG. 13: (Color online) In\fdelity of spin model in a depth-\nmodulated gaussian trap with and without corrections; strong\ninteractions. The in\fdelity of various models in a Gaussian\ntrap (dark lines) compared to a harmonic trap (faint lines),\nall at U= 0:1~!, which is roughly \fve times the smallest\nanharmonicity. The harmonic and anharmonic traps behave\nsimilarly in this case.tion processes that change single-particle states are ne-\nglected, for both harmonic and anharmonic potentials\nsubject to sudden displacements or changes in depth.\nThe parameters appearing in these spin models were an-\nalyzed for a range of trap displacements and dilations.\nNumerical procedures for transforming many-body states\nbetween single-particle representations using long-range\ntime evolution under an e\u000bective single-particle Hamil-\ntonian and \ftting of non-translationally invariant inter-\nactions to sums of decaying exponentials were presented,\nboth of which may \fnd other applications in tensor net-\nwork algorithm applications. 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The quantum spin ﬂuctuations in these systems can shift the optical resonances by more\nthan the homogeneous linewidth and produce huge Faraday rotation noise. We demonstrate, that\nthe resonance shift spin noise spectroscopy gives access to the high order spin correlators, which\ncontain complete information about the spin dynamics in contrast with the second order correlator\nmeasured by conventional Pauli-blocking spin noise spectroscopy. The high order quantum spin cor-\nrelators manifest themselves as a comb of peaks in the Faraday rotation noise spectra in transverse\nmagnetic ﬁeld. This eﬀect is closely related with the multispin ﬂip Raman scattering observed in\nthe Mn-doped nanostructures.\nI. INTRODUCTION\nThe quantum spin ﬂuctuations were ﬁrst predicted by\nFelix Bloch back in 1946 [1]. With the development of\nthe experimental techniques, the optical spin noise spec-\ntroscopyappearedandeventuallybecameapowerfultool\nfor the spin dynamics investigation in a broad class of\nparamagnetic media, from atomic gases to semiconduc-\ntors [2, 3]. In typical experiments, the spin ﬂuctuations\nwithin the small volume of a paramagnetic material pro-\nduce a stochastic Faraday rotation of the linearly polar-\nized light, which probes the system, and the spin noise\nspectra are obtained by the Fourier transformation of the\ntime-dependent Faraday rotation.\nIn a magnetic ﬁeld perpendicular to the probe beam,\nthe spin noise spectrum shows peaks at the Larmor fre-\nquencies of the studied spins, similar to the optically de-\ntected magnetic resonance. Quantum-mechanically the\nspinnoisesignalcanbeconsideredasaresultoftheinter-\nference of the probe beam with the light emission caused\nby spin-ﬂip forward scattering of the probe light [4, 5].\nIn atomic gases, the dependence of the light scatter-\ning amplitude on the spin state of an atom is provided\nby the Pauli blocking of the optical transitions in certain\npolarizations, deﬁned by the probed spins orientation.\nThis scenario is also realized for electrons and holes in\nsemiconductors with the pronounced spin-orbit interac-\ntion, such as GaAs or CdTe [6]. In the charged quantum\ndots (QDs), for example, for the electron spin-up or spin-\ndown state the optical transitions to the singlet heavy\nhole trion state are possible for \u001b+or\u001b\u0000-polarized light\nonly, respectively. In thermal equilibrium, the number of\nspin-up and spin-down electrons is the same on average,\nbut stochastic spin ﬂuctuations produce a weak Gaussian\nFaraday rotation noise, which is measured. This type\nof experiments can be called “Pauli-blocking spin noise\nspectroscopy”.\n\u0003smirnov@mail.ioﬀe.ruThe small Faraday rotation angles in Pauli-blocking\nspinnoisespectroscopymakeitdiﬃculttodetectthespin\ncorrelation functions of the orders higher than two [7, 8].\nIn the same time, the complete information about the\nspin dynamics including its intrinsic quantum properties\ncan be obtained from the complete set of the spin corre-\nlators of all orders only [9–11]. As a minimal extension of\nthestandardtheoriestheweakmeasurementsofthethird\nand fourth order spin correlators were described [12, 13].\nAnalternativeconnectionbetweenthespinsystemand\nthelightpolarizationisrealizedindilutedmagneticsemi-\nconductors [14]. In this case the probed spins belong to\nthed-shell electrons of Mn2+ions, embedded into the\ncrystal lattice. Mn atoms do not create localized charge\ncarrier states, and their spins do not aﬀect interband op-\ntical transitions directly. However, their spins are cou-\npled to the spins of the conduction electrons and holes\nby thesp\u0000dexchange interaction. The correspond-\ning coupling constant is very large, of the order of 1\neV. Due to this interaction, ﬂuctuations of the magnetic-\nion spins modulate the energies of the interband opti-\ncal transitions (most often involving localized excitons),\nand this creates a polarization noise of the probe light,\nso we call this type of experiments “resonance shift spin\nnoise spectroscopy”. Atomic-like hyperﬁne structure of\nMn2+spin levels was resolved in such experiments with\nthe very diluted CdMnTe quantum wells in weak mag-\nnetic ﬁelds [15]. The same mechanism is responsible for\nthe observation of the nuclear spin noise in GaAs [16, 17].\nMoreover the resonance shift spin noise spectroscopy can\nbe applied to any impurities in the semiconductors or to\nthe spins of nuclei of free atoms and molecules.\nThe general arguments, which followed the ﬁrst Pauli-\nblocking spin noise measurement [4] establish the rela-\ntion between the spin noise spectrum and the spin ﬂip\nRaman spectrum. In the case of the resonance shift\nspectroscopy, this relation apparently breaks down. In-\ndeed, there exists a phenomenon of the multispin ﬂip\nscattering, when the resonant Raman spectrum of Mn\ndoped nanostructures shows a comb of up to 15 equally\nspaced peaks [18, 19]. These spectra are explained by thearXiv:2001.09060v1  [cond-mat.mes-hall]  24 Jan 20202\nscattering via the virtual magnetic-polaron states, and\nthe observed phenomenon is therefore essentially quan-\ntum [20, 21]. On the other hand, the Larmor precession\nof the spin ﬂuctuation of one, several or many Mn2+ions\ninduces the peak in the spin noise spectrum at the single\nLarmor frequency only, and not at its multiplies.\nIn this work, we show that the multispin ﬂip Raman\nscattering is a counterpart of the high order quantum\nspin noise spectra, which can be observed by means of\nresonance shift spin spectroscopy in semi-magnetic struc-\ntures. We develop a general theory of the resonance shift\nquantum spin noise spectroscopy and describe in a uni-\nﬁed way the spin noise and Raman spectra. We demon-\nstrate, that the shape of the spectra is diﬀerent for the\nthermal and quantum spin noise. Detection of the high\norder spin correlators allows one to completely describe\nthe spin dynamics, and to distinguish between Gaussian\nand non-normal spin ﬂuctuations. In particular, we show\nthat for deep impurities in semiconductors and in atomic\nsystems the spin noise spectra strongly diﬀer from the\nGaussian noise spectra.\nThe paper is organized as follows: In Sec. II we present\na model and derive the general expression for the Fara-\nday rotation noise spectrum in the framework of the res-\nonance shift spin noise spectroscopy. In what follows,\nwe focus on the semimagnetic quantum wells and QDs,\nwhere we anticipate the fastest experimental measure-\nment of the higher order spin correlators. In Sec. III\nwe establish the relation between the Faraday rotation\nnoise and Raman spin-ﬂip spectra in diﬀerent polariza-\ntions. In Sec. IV we calculate and describe the spectra\nfor the Gaussian spin noise and in Sec. V we describe the\nnon-normal spin noise. Finally, we discuss the applica-\ntions of our theory to the diﬀerent spin systems from the\nsolid state to free atoms and summarize our ﬁndings in\nSec. VI.\nII. MODEL\nAs a model system for the resonance shift spin noise\nspectroscopy we consider a II-VI semiconductor doped\nwith manganese. The spins of Mn2+atoms can be op-\ntically monitored via a localized exciton resonance. We\nassume, thattheexcitonsarelocalizedatdefects, inQDs,\nor at imperfections of the interfaces of a quantum well.\nThe general form of the Hamiltonian is\nH(t) =H0+Hexc+Hint+V(t): (1)\nHereH0is the Hamiltonian of Mn2+spin system,Hexcis\nthe exciton Hamiltonian, Hintdescribes the interaction\nbetween exciton and Mn2+spins, andV(t)stands for the\ncoherent optical excitation of excitons.\nThe spin dependent part of the interaction of magnetic\natoms with excitons stems from the exchange interaction\nwith electron and hole in the exciton. The general formof this interaction is [22]\nHint=~X\ni\"\n!e\nex;iSeIi+X\n\u000b!h;\u000b\nex;iSh\n\u000bIi;\u000b#\n;(2)\nwhereienumerates Mn2+spinsIi,SeandShare the\nelectron and hole spins in the given exciton, respectively,\n!e\nex;iand!h;\u000b\nex;iare the corresponding exchange interac-\ntion constants with the Cartesian index \u000b=x;y;z. Due\nto the diﬀerent symmetry of the electron and hole Bloch\nwave functions in the \u0000valley, the electron exchange in-\nteraction is isotropic, while for the heavy hole it is not.\nThis anisotropy plays an important role for this system,\nand has the same origin as the anisotropy of the eﬀective\ng-factor.\nWe consider the optical detection of Mn2+spins by\nresonant laser probe light, which is described by the term\nV=\u0000PyEe\u0000i!pt+ H:c:; (3)\nin the Hamiltonian (1). Here Pis the dipole moment\noperator (in Schr¨ odinger representation), !pis the probe\nfrequency and Eis the amplitude of the incident electric\nﬁeld. The probe light induces an exciton dipole polariza-\ntion, which is described by the Heisenberg time depen-\ndentoperator P(t). InAppendixAwedemonstrate, that\nit has the form P(t) =P (t), where (t)is a dimension-\nless operator, which satisﬁes the transparent equation\nd (t)\ndt=i\n~PyEe\u0000i!pt\u0000i\n~h\nHexc+~Hint(t)i\n (t)\n\u0000\r (t):(4)\nHere we introduced the interaction Hamiltonian in the\ninteraction representation\n~Hint(t) = eiH0t=~Hinte\u0000iH0t=~(5)\nand an optical transition dephasing rate \r. The equa-\ntion (4) can be formally integrated, and the result for\nthe time dependent exciton polarization reads\nP(t) =i\n~P1Z\n0e\u0000i!p(t\u0000\u001c)\u0000\r\u001c\n\u0002Texp2\n4\u0000i\n~\u001cZ\n0H0\nexc(t\u0000\u001c0)d\u001c03\n5d\u001c(PyE):(6)\nHereTexpdenotes the normal time ordered exponential\n(later times on the left) and H0\nexc(t) =Hexc+~Hint(t)\nis an eﬀective time dependent exciton Hamiltonian. It\nis this part of the expression that contains information\nabout parameters of Mn2+spin dynamics (H0). The ob-\ntained general expression allows one to describe various\nphysical systems and experimental conditions. Below we\nconsider a speciﬁc case, when this expression is greatly\nsimpliﬁed.3\nFigure 1. Quantum ﬂuctuations of the Mn2+spins (magenta\narrows) in the exciton localization volume (green) randomly\nshift the exciton transitions energies. The states are charac-\nterized by the heavy hole spin (red arrow) Jz=\u00063=2, while\nthe projection of the electron spin (blue arrow) on the mag-\nnetic ﬁeld does not change.\nWe focus on the Voight geometry, when external mag-\nnetic ﬁeld is applied along xdirection, perpendicular to\nthe optical axis z. The Mn2+spin Hamiltonian takes a\nform\nH0=~\nLIx; (7)\nwhere\nI=NX\ni=1Ii (8)\nis the total spin of NMn atoms in the exciton localiza-\ntion volume and \nL=g\u0016BB=~is the Larmor precession\nfrequency in the magnetic ﬁeld Bwithgbeing theg-\nfactor and \u0016Bbeing the Bohr magneton. We assume\nthe Mn concentration to be small enough to neglect their\nexchange interaction. This corresponds to the Mn con-\ncentration of a few percent or less.\nWe stress, that we consider here the optical transitions\nto the localized exciton state, while other types of tran-\nsitions, e.g., to the trion or to the biexciton state can be\ndescribed in a similar way. Let us make some other sim-\nplifying assumptions, which make the theory transparent\nand the results very illustrating. First, we neglect the\ntransverse hole g-factor, which is usually very small [23].\nSecond, we assume, that the xprojection of the electron\nspindoesnotchange. Thismeans,thatthemagneticﬁeld\nis not weak, and the electron Zeeman energy exceeds the\ninteraction strength with the random spin components\nIi;yandIi;z. In fact, Mn2+spins can be partially polar-\nized along the magnetic ﬁeld at low temperatures, and\ncan create the eﬀective exchange magnetic ﬁeld along the\nsame direction, which can exceed the external magnetic\nﬁeld for electrons. Below for simplicity we consider only\none electron spin state (say, Sx= +1=2). This implicitlyassumes, that the splitting of the electron spin sublevels\nexceeds the homogeneous and inhomogeneous widths of\nthe optical resonance. Under these assumptions we ar-\nrive to the optical V-scheme, which is shown in Fig 1.\nHere the exciton vacuum state jgican be excited by \u001b+\nof\u001b\u0000polarized light to the exciton state with the heavy\nhole spinsJz=\u00063=2, respectively.\nWithout the exchange interaction with the magnetic\nimpurities the two excitonic states are degenerate, so the\nexciton Hamiltonian reads\nHexc=~!0nexc; (9)\nwhere!0is the resonance frequency and nexcis the oc-\ncupancy of the both exciton states. The exchange inter-\naction leads to the splitting of the two resonances, which\nwe describe by\nHex=~!ex2\n3Sh\nzIz: (10)\nHere we neglect the total shift of the two resonances due\nto the exchange interaction with electron ( !e\nex;i= 0) and\nconsider the hole exchange interaction along the zaxis\nonly [22]. Also for the simplicity we use the box model\nandsetequalexchangeinteractionconstantsforallMn2+\nspins:!h;\u000b\nex;i= (2=3)!ex\u000e\u000b;z.\nUnder these assumptions, the two excitonic states are\nnot mixed. As a results, the circularly polarized \u001b\u0006\nprobe light induces the exciton dipole polarization with\nthe same helicity P\u0006(t) = [\u0007Px(t)\u0000iPy(t)]=p\n2. From\nEq. (4) we ﬁnd, that\ndP\u0006(t)\ndt=\u0000i [!0\u0006!exIz(t)\u0000i\r]P\u0006(t)+i\n~jdj2E\u0006(t)e\u0000i!pt;\n(11)\nwheredis the optical transition dipole moment (see Ap-\npendix A). This equation clearly shows, that the spin\npolarization Iz(t)shifts the exciton resonance energy !0,\nwhich allows one to optically monitor Mn2+spin ﬂuc-\ntuations. In fact, it follows from Eqs. (5) and (7), that\nIz(t) = cos(\n Lt)Iz+ sin(\nLt)Iy, but we prefer to keep\nthe general notation Iz(t), which is valid for arbitrary\nspin Hamiltonian H0.\nIn the speciﬁc system under study, the general expres-\nsion (6) for the polarization reduces to\nP\u0006(t) = ijdj2\n~E\u0006e\u0000i!pt1Z\n0ei(!p\u0000!0)\u001c\u0000\r\u001c\n\u0002Texp2\n4\u0007i!ex\u001cZ\n0Iz(t\u0000\u001c0)d\u001c03\n5d\u001c:(12)\nHere the inner integral can be solved as described in Ap-\npendix A, but for the calculation of the spectra of the\nsecondary emitted light this expression is more conve-\nnient.\nIt is useful to analyze this expression in the adiabatic\napproximation. Provided \nL\u001c\rthe Mn spin dynamics4\nis slow as compared with the exciton polarization relax-\nation. In this case one can replace Iz(t\u0000\u001c0)withIz(t)\nand solve the ordered exponential. Then the outer inte-\ngral can be solved as well, which yields\nP\u0006(t) =jdj2E\u0006\n~e\u0000i!pt\n!0\u0006!exIz(t)\u0000!p\u0000i\r:(13)\nThis expression shows that: (i) Mn spin polarization\nshifts the exciton resonance frequency, and (ii) the rela-\ntion between Mn spin polarization and the exciton dipole\npolarization is nonlinear. The latter makes it possible to\ndetect high order spin correlators using the resonance\nshift spin spectroscopy. Crucially, Iz(t)should be con-\nsidered here as the Heisenberg operator. Only when Iis\na large classical vector, the corresponding operator can\nbe replaced with its expectation value.\nIII. GENERAL RELATIONS BETWEEN SPIN\nCORRELATION FUNCTIONS AND\nRESONANCE SHIFT OPTICAL RESPONSE\nThe spin noise spectra are typically measured using\nthe linearly polarized light,\nE=E0ex; (14)\nwhereE0is an amplitude of the probe light, and exis a\nunit vector along xaxis. The optical Faraday and ellip-\nticity signals,FandE, respectively, are measured in the\ntransmission or reﬂection geometry. They are given by\nthe real and imaginary parts of the equality [24]\nF\u0000iE=E0\u0003\nxE0\ny: (15)\nHereE0is the amplitude of the emitted (or scattered)\nlight. It consists of the contribution from the elastic scat-\ntering and the secondary emission by the exciton dipole\npolarization:\nE0=aE+bP; (16)\nwhereaandbare complex coeﬃcients, which depend on\nthe geometry of the structure [17]. In the next section we\ncalculate the Faraday rotation noise spectra, which can\nbe detected using the resonance shift spin spectroscopy.\nThe Faraday rotation angle of the probe polarization\nplane and the ellipticity angle can be calculated as\n\u0012F\u0000i\u0012E=F\u0000iE\nI; (17)\nwhereI=jE0\nxj2is proportional to the intensity of the\ndetected light. These expressions assume, that the an-\ngles are small \u0012F;E\u001c1or equivalentlyjE0\nyj \u001c jE0\nxj.\nUsually the scattering is weak, bPx\u001caE0, so by virtue\nof Eq. (16) we arrive to\n\u0012F+ i\u0012E=E0\ny\nE0x=b\naE0Py: (18)This expression shows, that the spin signals are propor-\ntional to the exciton polarization along yaxis.\nThe optical signals noise spectra are deﬁned as the\nFourier transform of the correlation functions:\n(\u00122\nF;E)\n=1Z\n\u00001h\u0012F;E(t)\u0012F;E(t+\u001c)isei\n\u001cd\u001c:(19)\nHere the angular brackets denote quantum mechanical\naveraging and the subscript “ s” denotes the symmetrized\ncorrelation function\nh\u0012(0)\u0012(\u001c)is=h\u0012(0)\u0012(\u001c) +\u0012(\u001c)\u0012(0)i\n2:(20)\nIn the steady state, the averages do not depend on time\nt, so for the rest of the paper we set t= 0in the cor-\nrelation functions. The symmetrization is related with\nthe fact, that the detected light is almost classical and\ntheopticalspinsignalsareself-homodyned[25–27]. From\nEqs. (19) and (18) one can see, that that the Faraday and\nellipticity noise spectra are determined by the Fourier\ntransform of the correlation functions of the components\nof the exciton polarization. In Pauli-blocking spin noise\nspectroscopy this correlator is simply proportional to the\nspin noise spectrum. However, in the case of the res-\nonance shift spin noise spectroscopy, there is no direct\nrelation between them.\nIn the same time the Raman spectrum of the scattered\nlight in polarization \u000bis given by\nStot(!) =1Z\n\u00001hE0\u0003\n\u000b(t)E0\n\u000b(t+\u001c)iei!\u001cd\u001c:(21)\nIn the general case the Raman spectrum consists of a \u000e-\npeak at!=!p, which does not carry information about\nthe spin system, and the rest of the spectrum S(!). It\ncan be presented as\nS(\n +!p) =jbj21Z\n\u00001hP\u0003\n\u000b(0)P\u000b(\u001c)iei\n\u001cd\u001c:(22)\nThus we arrive again at the Fourier transform of the ex-\nciton polarization correlation function.\nThe exciton polarization is given by Eq. (12), which\nimplicitlydependsonMn2+spinﬂuctuations. Nowletus\nestablish the general relations between the spectra of dif-\nferent polarization components and the multi-order spin\ncorrelation functions.\nA. Spectrum in circular polarization\nIn this section we consider an auxiliary problem of \u001b+\nincident light, so the scattered light has the same polar-\nization,\u000b= +. We calculate the spectra of the dimen-\nsionless exciton polarization\np(t) =\u0000~\r\njdj2E0P(t)ei!pt; (23)5\nwhich in this case are proportional to the Raman spin ﬂip\nspectrum in \u001b+polarization, see Eq. (22). The spectrum\nin\u001b\u0000polarization is the same.\nIt is convenient to rewrite Eq. (12) as\np+(t) =1Z\n0e\u0000i\u000ek\u0000kTexp [\u0000iJ(t)] dk;(24)\nwhere\n\u000e= (!0\u0000!p)=\r; (25)\nis a dimensionless detuning and\nJ(t) =kZ\n0m(t\u0000k0=\r)dk0(26)\nwith\nm(t) =!ex\n\rIz(t) (27)\nbeing a dimensionless splitting of the resonance. Fur-\nther, we note that for the localized excitons, the inhomo-\ngeneous broadening usually exceeds by far the homoge-\nneous one (see, e.g., Ref. 18). Therefore the spin noise\nand Raman spin ﬂip spectra should be averaged over the\ndetuning as\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1\n2\u00191Z\n\u00001\np\u0003\n+(0)p+(\u001c)\u000b\nd\u000e:(28)\nHere we introduced the factor 1=(2\u0019)to shorten the fol-\nlowing expressions. We substitute here the exciton polar-\nization from Eq. (24) and obtain the averaged correlation\nfunction\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1Z\n0e\u00002kDh\nTeiJ(0)ih\nTe\u0000iJ(\u001c)iE\ndk;\n(29)\nwhereTdenotes the reversed time ordering. The corre-\nlator in this expression can be calculated using the cu-\nmulant expansion.\nGenerally, the quantum noise statistics is completely\ndescribed by the series of cumulants of the random vari-\nable [28–30]. In Appendix B we obtain the general ex-\npressions for the polarization correlation function and\nsimplifyitfortheGaussianspinnoise. Toobtainasimple\nexpression for the polarization correlator let us consider\nagain the adiabatic approximation, \nL\u001c\r. In this case\none can replace m(t\u0000k0=\r)in Eq. (26) with m(t). Then\nfrom Eq. (29) we obtain\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1X\nn=02nX\nl=0(\u00001)n+l\n22n+1\u00122n\nl\u0013\nml(0)m2n\u0000l(\u001c)\u000b\n;\n(30)where\u00002n\nl\u0001\nis the binomial coeﬃcient. This expression\nshows, that the polarization correlator and Faraday rota-\ntion noise spectra are determined by the spin correlation\nfunctions of all orders. Thus, the resonance shift spin\nnoise spectroscopy measures high order spin correlators\nin addition to the standard second order correlator. This\nsurprising result originates in the nonlinear relation be-\ntween the exciton polarization and the total Mn2+spin\nin the limit Iz\u001d1[see, e.g., Eq. (13)].\nFor Gaussian spin noise using Eq. (B7) we obtain\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1Z\n0e\u0000k2\u0001m2(\u001c)e\u00002kdk;(31)\nwhere we introduced\n\u0001m2(\u001c) =\nm2\u000b\n\u0000hm(0)m(\u001c)i: (32)\nThis integral can be solved as\n\np\u0003\n+(0)p+(\u001c)\u000b\n=\n1\n2r\u0019\n\u0001m2(\u001c)e1=\u0001m2(\u001c)erfc \n1p\n\u0001m2(\u001c)!\n:(33)\nIf exchange interaction is weak, m(t)\u001c1one can use\nthe asymptotic expansion\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1\n21X\nn=0\u0014\n\u0000\u0001m2(\u001c)\n2\u0015n\n(2n\u00001)!!:(34)\nThis expression directly relates the spin correlation func-\ntion\u0001m2(\u001c)with the polarization correlator.\nB. Faraday rotation noise spectra\nand Raman spectra in linear polarizations\nLet us return to the linearly polarized probe light,\nEq. (14). Similarly to Eq. (24), the dimensionless ex-\nciton polarization in this case reads\npx(t) =1Z\n0e\u0000i\u000ek\u0000kTcos [J(t)] dk;(35a)\npy(t) =\u00001Z\n0e\u0000i\u000ek\u0000kTsin [J(t)] dk; (35b)\nwhere we introduced the notations\nTcos(x) =Teix+Te\u0000ix\n2;Tsin(x) =Teix\u0000Te\u0000ix\n2i:\n(36)\nThen one can perform the calculations following the lines\nof the previous subsection: average the correlation func-\ntionhp\u0003\n\u000b(0)p\u000b(\u001c)iover the detuning, express it through\nthe cumulants and in the limit N\u001d1neglect all the\ncumulants of high orders. For clarity we give the ﬁnal\nresult for the adiabatic limit [c.f. Eq. (31)]:6\n\"hp\u0003x(0)px(\u001c)i\n\np\u0003y(0)py(\u001c)\u000b#\n=1Z\n0dke\u00002k\u0000k2hm2i\"\nch\u0000\nk2hm(0)m(\u001c)i\u0001\nsh\u0000\nk2hm(0)m(\u001c)i\u0001#\n: (37)\nThese expressions along with Eq. (22) allow one to di-\nrectly calculate the Raman spectrum for the given Mn\nspin correlation function. Here the integrals can be ex-\npressed through the error function, and its asymptotic\nexpansions can be found. However, these expressions are\ncumbersome, and will not be needed below.\nAs a ﬁrst step towards the relation between Raman\nand spin noise noise spectrum, we consider the Ra-\nman spectrum in crossed ( y) polarization in the limit\nhm(t)m(t+\u001c)i\u001c 1. In this case the relation between\nthe exciton dipole polarization and Mn spin polarization\nit linear, so from Eq. (37) we obtain the spectrum\nS(!) =\f\f\f\fbjdj2E0!ex\n~\r2\f\f\f\f2\n(I2\nz)!\u0000!p; (38)\nwherethespinnoisenoisespectrumofMn2+ionsisgiven\nby\n(I2\nz)\n=1Z\n\u00001hIz(0)Iz(\u001c)iei\n\u001cd\u001c: (39)\nAn analogous relation between the Raman spin ﬂip and\nthe spin noise spectra was derived by Gorbovitskii and\nPerel for Pauli-blocking spin noise spectroscopy [4]. Be-\nlowwedemonstrate, thatasimilarrelationholdsbetween\nmultispin ﬂip spectrum and the noise spectrum of optical\nsignals for any strength of the exchange interaction.\nIndeed, from Eq. (18) we obtain the correlation func-\ntions of the Faraday rotation and ellipticity angles\nh\u0012F(0)\u0012F(\u001c)i=\f\f\f\fjdj2b\n~\ra\f\f\f\f2\np00\ny(0)p00\ny(\u001c)\u000b\ns;(40a)\nh\u0012E(0)\u0012E(\u001c)i=\f\f\f\fjdj2b\n~\ra\f\f\f\f2\np0\ny(0)p0\ny(\u001c)\u000b\ns;(40b)\nwhere one and two primes denote the real and imagi-\nnary parts of the polarization, respectively. Similarly to\nEq. (29), we average the correlation functions of p0\ny(\u001c)\nandp00\ny(\u001c)over the detuning making use of Eq. (35b),\nand obtain\n\np00y(0)p00y(\u001c)\u000b\ns=\np0y(0)p0y(\u001c)\u000b\ns=1\n2\np\u0003y(0)py(\u001c)\u000b\ns:(41)\nThis expression diﬀers from the second line of Eq. (37)\nby a factor and symmetrization. Thus the noise spec-\ntra of the Faraday rotation and ellipticity angles can be\ncalculated as a symmetrized Raman spectrum in crossed\nlinear polarizations. In the next section we calculate and\ndescribe these spectra.IV. FARADAY ROTATION NOISE SPECTRUM\nIN VOIGT GEOMETRY\nA. Spin correlation functions\nThe general expressions (37) relate the Faraday rota-\ntion noise and Raman spin ﬂip spectra with the spin cor-\nrelators. Here we calculate the correlation functions in\nexternal magnetic ﬁeld described by Eq. (7).\nThe average Mn spin is oriented along Band equals\nto\nhIxi=NsBs\u0012g\u0016BBs\nkBT\u0013\n; (42)\nwheres= 5=2is a single Mn2+spin,Bs(x)is the Bril-\nlouin function, kBis the Boltzmann constant, and Tis\nthe temperature. Using the commutation relations for\nthe spin components we ﬁnd also the correlators\nhIyIzi=\u0000hIzIyi=i\n2hIxi; (43a)\n\nI2\nz\u000b\n=\nI2\ny\u000b\n=N\n2\u0002\ns(s+ 1)\u0000\ns2\nx\u000b\u0003\n:(43b)\nNotably, the ﬁrst of these two equations is responsible\nfor the quantum part of the spin correlation functions.\nIndeed, forclassicalnoisethecorrelationfunctionsdonot\ndepend on the order, in which the ﬂuctuating quantities\nare multiplied. Moreover, Eq. (43b) shows, that even\nat zero temperature, when\ns2\nx\u000b\n=s2, zero-point spin\nﬂuctuations\nI2\nz\u000b\n=Ns=2are present.\nThetimecorrelationfunctionsfor \u001c >0obeytheequa-\ntions\nd\nd\u001chIz(0)Iz(\u001c)i= \nLhIz(0)Iy(\u001c)i\u0000hIz(0)Iz(\u001c)i\n\u001cs;(44a)\nd\nd\u001chIz(0)Iy(\u001c)i=\u0000\nLhIz(0)Iz(\u001c)i\u0000hIz(0)Iy(\u001c)i\n\u001cs;\n(44b)\nwhere\u001csis a transverse spin relaxation time (we assume\nthat ~=\u001cs\u001ckBT). The solution of these equations with\nthe initial conditions (43) reads\nhIz(0)Iz(\u001c)i=1\n2\u0014\u0012\nI2\nz\u000b\n+hIxi\n2\u0013\nei\nL\u001c\n+\u0012\nI2\nz\u000b\n\u0000hIxi\n2\u0013\ne\u0000i\nL\u001c\u0015\ne\u0000j\u001cj=\u001cs:(45)7\nThen using the deﬁnition (27) we ﬁnd the dimensionless\ncorrelation function\nhm(0)m(t)i=\u0000\n\u0016+e\u0000i\nL\u001c+\u0016\u0000ei\nL\u001c\u0001\ne\u0000j\u001cj=\u001cs;(46)\nwhere we introduced\n\u0016\u0006=!2\nex\n2\r2\u0012\nI2\nz\u000b\n\u0007hIxi\n2\u0013\n: (47)\nThe correlatorhm(0)m(t)iultimately deﬁnes the noise\nspectra of Faraday rotation and ellipticity, which we an-\nalyze in the next subsection.\nB. Faraday rotation noise and Raman spectra\nSimilarly to Sec. III it is convenient to start from\nthe analysis of the exciton dipole polarization correla-\ntion function in circular polarizations, which deﬁne the\ncorresponding Raman spectra.\nTo shorten the notation we introduce the scaled spec-\ntrum [c.f. Eq. (22)]\nS++(\n) =1Z\n\u00001\np\u0003\n+(0)p+(\u001c)\u000b\nei\n\u001cd\u001c:(48)\nFrom Eqs. (34) and (46) one can see, that the spectrum\nconsists of the peaks at frequencies n\nL, wherenis an\ninteger. The general form of the spectrum of circularly\npolarized exciton dipole polarization is\nS++(\n) =1X\nn=1X\n\u0006P\u0006\nn(\n\u0007n\nL);(49)\nwhereP\u0006\nn(\n)are even functions peaked at zero and we\nneglect the peak at zero frequency.\nAnothergeneralpropertyfollowsfromtheratioofpref-\nactors in Eq. (46). From the deﬁnition (39) (without\nsymmetrization) one can see, that\n(I2\nz)\nL\n(I2z)\u0000\nL= e\u0000~\nL=(kBT): (50)\nGenerally, this relation is inherited by the polarization\nspectra [Eq. (48)] in the form\nS++(\n)\nS++(\u0000\n)= e\u0000~\n=(kBT); (51)\nwhich is well known for the Raman spectra.\nThe Raman spin ﬂip spectra in \u001b+and in\u001b\u0000polar-\nizations coincide and are given by Eq. (49):\nS\u0000\u0000(\n) =S++(\n): (52)\nWerecall, thatthetwocircularpolarizationsareindepen-\ndent in the lowest order in the incident electric ﬁeld, sothe cross-polarized spectra in circular polarizations van-\nish:S\u0006\u0007(\n) = 0.\nTheRamanspinﬂipspectrainlinearpolarizationsand\nFaraday rotation and ellipticity noise spectra can be cal-\nculated using Eqs. (37) and (41) for the adiabatic regime.\nSimilarly to Eq. (49) the Raman spectra take the form\nSxx(\n) =Syy(\n) =1X\nk=1X\n\u0006P\u0006\n2k(\n\u00072k\nL);(53a)\nSxy(\n) =Syx(\n) =1X\nk=0X\n\u0006P\u0006\n2k+1(\n\u0007(2k+ 1)\nL):\n(53b)\nThe Faraday rotation (and ellipticity) noise spectra are\ngiven by\nSFR(\n) =Sxy(\n) +Sxy(\u0000\n)\n2; (54)\nwhich diﬀers from (\u00122\nF)\n(and (\u00122\nE)\n) by a factor.\nIn the illustrative case of m(t)\u001c1one can substitute\nEq. (45) in the asymptotic Eq. (34), which yields\nS++(\n) =1X\nn=1X\n\u0006(2n\u00001)!!\n2n\u0016n\n\u0006\u001cs=n\n1 + [(\n=n\u0007\nL)\u001cs]2:\n(55)\nOne can see, that the spectrum consists of Lorentzian\npeaks at the frequencies \u0007n\nLwith the areas (divided\nby2\u0019)\nA\u0006\nn=(2n\u00001)!!\n2n+1\u0016n\n\u0006; (56)\nrespectively. The widths of these peaks are n=\u001cs. We\nstress, that the second order spin correlation func-\ntion (45) contains the spin precession frequencies \u0006\nL\nonly. Therefore the appearance of the overtones is a ﬁn-\ngerprint of the contributions of the higher order spin cor-\nrelators.\nFor high order contributions, n\u001d1, Eq. (55) diverges,\nandtheaboveanalysisisinapplicable. Generally, onehas\nto start from Eq. (31), where\n\u0001m2(\u001c) =X\n\u0006\u0016\u0006\u0010\n1\u0000e\u0007i\nL\u001ce\u0000j\u001cj=\u001cs\u0011\n(57)\naccording to Eq. (46). Thus we obtain\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1Z\n0e\u00002ke\u0000k2(\u0016++\u0016\u0000)\n\u0002exph\nk2\u0000\n\u0016+e\u0000i\nL\u001c+\u0016\u0000ei\nL\u001c\u0001\ne\u0000j\u001cj=\u001csi\ndk:(58)\nThe exponent in the second line can be decomposed in8\nthe Taylor series as\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1Z\n0e\u00002ke\u0000k2(\u0016++\u0016\u0000)\n\u00021X\nn=0k2n\nn!\u0000\n\u0016+e\u0000i\nL\u001c+\u0016\u0000ei\nL\u001c\u0001ne\u0000nj\u001cj=\u001csdk:(59)\nAgain, one can see, that the correlation function contains\nharmonics/e\u0007in\nL\u001c, so the spectrum consists of the\npeaks at frequencies \u0006n\nL.\nIn Fig. 2 we compare the diﬀerent spectra in the limit\nof zero temperature T= 0and strong exchange inter-\nactionp\nN!ex\u001d\r. All the spectra show the comb\nof peaks at frequencies, which are multiples of the Lar-\nmor spin precession frequency. The peaks in the Raman\nspectra correspond to the multiple spin ﬂips mediated\nby the excitonic state, as shown in the top of the ﬁg-\nure. Importantly, multiple spin ﬂips take place in one\nprocess due to the RKKY-type exchange interaction be-\ntween Mn2+spins mediated by the heavy hole spin. The\nsame mechanism can also lead to the double spin ﬂips\nof donor bound electrons [31] and of electrons conﬁned\nin the nanoplatelets [32]. In the Faraday rotation noise\nspectra, the peaks at frequencies \u0006n\nLreﬂect the con-\ntributions of quantum spin noise correlation functions of\nthe order 2n. We recall, that the average spin polar-\nization along zaxis is absent, so the correlators of the\nodd orders vanish. Observation of high order spin corre-\nlators is possible due to the nonlinear relation between\ndipole polarization and the total Mn2+spin. Similarly,\nnoise of the linear birefringence was recently shown to\nproduce the peak at the double Larmor frequency for\ncesium atoms [33].\nThe Raman spin ﬂip spectra in Fig. 2 are asymmetric\n(contain only the peaks at negative frequencies) because\nthe energy can not be absorbed from the zero-point spin\nﬂuctuations. Alternatively one can say, that the Mn2+\nspins are all oriented along xaxis and can be ﬂipped\nonly in the opposite direction. The Raman spectra in\nlinear polarizations are similar, but co-polarized (blue\ncurve) and cross-polarized (red curve) spectra consist of\nthe peaks at even and odd frequencies only, respectively,\nsee Eqs. (53). In the same time, the Faraday rotation\nnoise spectrum (black curve) is symmetric and contains\nthe odd peaks only, as it follows from Eq. (54).\nInthelimitofzerotemperature, theexpressionsforthe\nspectra are particularly simple even beyond the adiabatic\napproximation. From the spin correlation function (46)\nand Eqs. (B8) we ﬁnd that\nhJ(0)J(\u001c)i=\nJ2(0)\u000b\nsei\nL\u001c\u0000j\u001cj=\u001cs;(60a)\n\nJ2(0)\u000b\ns=Ns!2\nex\n\n2\nL[1\u0000cos(\nLk=\r)];(60b)\nwhere we took into account that \u001cs\r\u001d1. Substi-\ntution of these expressions in the polarization correla-\nFigure 2. The noise spectrum of Faraday rotation, SFR(\n),\n(black line), and multispin ﬂip Raman spectra Sxx(\n)(blue\nline with ﬁlling), Sxy(\n)(red line with ﬁlling), and S++(\n)\n(green line), calculated after Eqs. (54), (53), and (49), re-\nspectively. The parameters of the calculation are T= 0,p\nN!ex=\r\u001d1, and\u001cs\nL= 20. The peaks correspond to the\nmultiple ﬂips of Mn2+spins (magenta arrows in the sketch)\nmediated by the exchange interaction with the heavy hole\nspin (red arrow) in the exciton localization area (green).\ntion function (B7) yields the areas of the peaks in the\nform [18, 20, 21]\nA\u0000\nn=1\n2\u001c01Z\n0e\u0000t=\u001c0\u0001In\nx(t)\nn!e\u0000\u0001Ix(t)dt:(61)\nHere we introduced the notations t=k=\r,\u001c0= 1=(2\r)\nand [34]\n\u0001Ix(t) =I!2\nex\n\n2\ntot[1\u0000cos(\n tott)] (62)\nwithI=Nsand\ntot=p\n!2ex+ \n2\nL. Physically, \u0001Ix(t)\nis the change of Ixduring the spin precession in the sum\nof the exchange and external magnetic ﬁelds for the time\nt. The integrand in Eq. (61) has a form of the probabil-\nity of the change of Ixbynin the Poisson distribution.\nThe integration describes the average of this probability\nduring the exponential exciton decay described by e\u0000t=\u001c0.\nFig. 3 shows the spectra for diﬀerent temperatures,\nor equivalently for diﬀerent magnetic ﬁelds. For bet-\nter visibility we focus on the Raman spectrum S++(\n),\nwhile the Faraday rotation noise spectrum can be ob-\ntained by selecting the odd numbered peaks and sym-\nmetrizing them in frequency, see Eq. (54). At high tem-\nperature (red curve), the spectrum is symmetric, which9\n-4 -2 0 2 4012345\nFigure 3. Circular dipole polarization noise spectra, S++(\n),\ncalculated after Eq. (49) in the limit of strong exchange in-\nteraction (p\nN!ex=\r\u001d1) for\u001cs\nL= 20and for diﬀerent\ntemperatures, as indicated in the plot.\ncorresponds to purely thermal spin ﬂuctuations. In the\nlimit!exp\nN\u001d\r, the peaks are very broad and strongly\noverlap. Athighfrequenciesthespectrumisdescribedby\nS++(\n) =\u0019\r\n!exs\n3\u001cs\n2Ns(s+ 1)j\nj:(63)\nThe fact thatS++(\n)decreases with increase of !exis\ncaused by the large splitting of the two excitonic reso-\nnances in this limit and a small region of values of \u000eIz,\nwhich produces sizable Faraday rotation, see Eq. (13).\nWith decrease of the temperature the spectrum be-\ncomes asymmetric, which evidences the increasing role\nof non-commutativity of spin components. In the limit\nof zero temperature the spectrum contains the Stokes\ncomponents only. In this limit \u0016+= 0andA\u0000\nndecay\nvery slowly obeying the power law:\nA\u0000\nn/1=pn: (64)\nC. Favorable conditions for the measurement of\nthe high order correlators\nTo successfully apply the resonance shift spin noise\nspectroscopy the exchange interaction should be quite\nstrong. Indeed, in the opposite limit of weak interaction,\n\u0016\u0006\u001c1, the area of the n-th peak is given by Eq. (56)\nand is proportional to !2n\nex. Therefore, the exchange in-\nteraction should be strong, which is easily realized in\nsemimagnetic semiconductors and many other systems,\nsee Sec. VI.\nExperimentally it is easier to measure the Fara-\nday rotation noise spectra in the sub-GHz frequency\nrange, which corresponds to the weak magnetic ﬁelds\nB.40mT. In this case the average spin polarization\n1 3 5 7 9 1101234561 3 5 710-210-1100101Figure 4. Faraday rotation noise spectra calculated after\nEq. (C3) for the typical experimental parameters: T= 2K,\nN= 50,Bexch =~!ex=(\u0016Bg) = 1:5T withg= 2,\n\r= 0:33meV, and \nL\u001cs= 20. The black and red curves\ncorresponds to the strong ( B= 6T,\nL= 168GHz) and\nmoderate ( B= 40mT, \nL= 1:1GHz) external magnetic\nﬁeld, respectively. The inset compares Gaussian noise spec-\ntrum(blackcurve)withtheFaradayrotationnoiseforasingle\nspinI= 1=2(blue curve) and I= 5=2(magenta curve) in the\nlimitT!0and!ex!1for\nL\u001cs= 30.\nis small, and we ﬁnd\nhm(0)m(\u001c)i=35N!2\nex\n12\r2cos(\nL\u001c)e\u0000j\u001cj=\u001cs:(65)\nThe areas of the peaks decay quickly in this limit even\ndespite the strong exchange interaction. The Faraday ro-\ntation noise spectrum for the typical experimental con-\nditions for B= 40mT is shown in Fig 4 by red curve.\nOne can see, that the peak at the frequency 3\nLis much\nsmaller, than the peak at \nL, and the peak at 5\nLis\nhardly visible in this limit.\nMore favorable conditions for the measurement of the\nhigh order spin correlations are realized in strong mag-\nnetic ﬁelds, when the spin polarization is large. The spin\nnoise spectrum for B= 6T is shown by the blue curve\nin Fig. 4, where one can distinctly see many peaks at the\nodd multiples of \nL. In this case we obtain from Eq. (46)\nthe dimensionless spin correlation function\nhm(0)m(\u001c)i=5N!2\nex\n4\r2ei\nL\u001ce\u0000j\u001cj=\u001cs:(66)\nThe areas of the peaks decay slowly in this case as de-\nscribed by Eq. (64). This limit corresponds to the domi-\nnance of the quantum spin ﬂuctuations over the classical\nones, and was not reached nor approached yet. To mea-\nsure the Faraday rotation noise in the high frequency\nrange one has to use special techniques, such as pulse\ntrains [35] or heterodyne detection [36]. Nevertheless, we\nbelieve that this experimental challenge will be under-\ntaken in the nearest future.10\nV. NON-GAUSSIAN SPIN NOISE\nIn the previous sections we described the Faraday ro-\ntation noise spectra, and demonstrated, that they give\naccess to the high order quantum spin correlation func-\ntions. However, under the assumption of many indepen-\ndent Mn2+spins in the exciton localization volume, the\nhighorderspincorrelationfunctionsallcanbereducedto\nthe second order correlator. So it is interesting to go be-\nyond this approximation and to study the non-Gaussian\nspin noise.\nGenerally, the noise spectrum is deﬁned by Eq. (19),\nwhere the Faraday rotation and the ellipticity angles are\nrelated to the exciton polarization by Eq. (18). Using the\naveraged polarization correlation function (41) we ﬁnd\nthe normalized Faraday rotation noise spectrum in the\nform\nSFR(\n) =1Z\n\u00001\np\u0003y(0)py(\u001c)\u000b\nsei\n\u001cd\u001c;(67)\nwhere the dimensionless polarization is given by\nEq. (35b). In the adiabatic approximation, \nL\u001c\r,\nusing the deﬁnition (26) we obtain\npy(t) =\u00001Z\n0e\u0000i\u000ek\u0000kM(t)dk; (68)\nwhere we introduced the operator\nM(t) = sin[km(t)]: (69)\nThen we perform averaging over the detuning, as deﬁned\nin Eq. (29), and obtain the spectrum\nSFR(\n) =1Z\n\u00001d\u001cei\n\u001c1Z\n0dke\u00002khM(0)M(\u001c)is:(70)\nFor non-Gaussian spin noise, the cumulants of m(0)and\nm(t)allow one to calculate this correlation function sim-\nilarly to Sec. IIIA.\nFor example, in the presence of the resident charge car-\nriers, Mn2+spins are coupled with the carrier-mediated\nexchange RKKY interaction, which may eventually lead\nto the transition into the ferromagnetic phase [37]. For\nthe Mn2+concentration approaching the paramagnetic-\nferromagnetic transition, their spins are no longer inde-\npendent, and the spin noise in non-Gaussian. The spin\nﬂuctuationsinthiscasecanbedescribedtheoreticallyus-\ning the Landau theory [38, 39], eﬀective polaron Hamil-\ntonian [40, 41], dynamical mean ﬁeld theory [42, 43],\nor using more sophisticated approaches [44, 45]. In the\nvicinity of the phase transition the eﬀective Larmor fre-\nquency decreases [46] and role of higher order cumulants\nincreases [47].However, in view of the application of the resonance\nshift spin noise spectroscopy to other systems we con-\nsider another situation. Namely, let us study the Fara-\nday rotation noise induced by a single spin I(N= 1)\ncoupled to the optical resonance. In this case all the cu-\nmulants are equally important, see Eq. (B5), so the spin\nnoise is strongly non-Gaussian. This limit can be real-\nized, e.g., for deep impurities or atomic systems, see the\nnext section.\nFor a single spin it is easier to calculate the Faraday\nrotation noise spectrum directly using the spin density\nmatrix formalism, than using the cumulant expansion.\nWe ﬁnd the operator M(t)from the equation of motion\ndM(\u001c)\nd\u001c=i\n~[H0;M(\u001c)] +LfM(\u001c)g;(71)\nwhereH0is deﬁned in Eq. (7) and Lis the Lindblad\noperator, describing the spin relaxation. Provided the\ntransverse spin relaxation time \u001csis much shorter than\nthe longitudinal one ( T1), we write the Lindblad operator\nin the form\nLfM(\u001c)g=1\n\u001cs\u0002\n2IxM(\u001c)Ix\u0000I2\nxM(\u001c)\u0000M(\u001c)I2\nx\u0003\n:\n(72)\nThe kinetic equation has a trivial initial condition\nM(0) = sin(km), wheremis the Schrodinger operator\ndeﬁned in Eq. (27). Finally, the correlation function in\nEq. (70) should be calculated using the steady state den-\nsity matrix\n\u001a= e\u0000H0=(kBT).\nTr\u0010\ne\u0000H0=(kBT)\u0011\n:(73)\nAs an example let us consider I= 1=2. In this case\nsin(km) = 2 sin\u0012k!ex\n2\r\u0013\nIz: (74)\nThen the solution of Eq. (71) simply reads\nM(\u001c) = 2 sin\u0012k!ex\n2\r\u0013\n[Izcos(\nL\u001c) +Iysin(\nL\u001c)] e\u0000\u001c=\u001cs:\n(75)\nFor any temperature the correlation function is the same:\nhM(0)M(\u001c)is= sin2\u0012k!ex\n2\r\u0013\ncos(\nL\u001c)e\u0000j\u001cj=\u001cs:(76)\nSubstituting this function in Eq. (70) we ﬁnd the non-\nGaussian Faraday rotation noise spectrum for I= 1=2:\nSFR(\n) =1\n8!2\nex\n!2ex+ 4\r2[P1(\n) +P1(\u0000\n)]:(77)\nThis spectrum is shown by the blue curve in the Fig. 4.\nNote, thatitsshapeformallycoincideswiththespectrum\nof spin ﬂuctuations, which is usually measured by the\nPauli-blocking spin noise spectroscopy.11\nGenerally, sin(km)can be presented as a linear com-\nbination of the operators In\nzwith oddn\u00142I. As a\nresult, the high order spin correlators can be reduced to\na few lower orders, and the spectrum consists of a ﬁ-\nnite number of peaks. The maximum peaks’ number is\nnmax= 2[I\u00001=2] + 1, where square brackets denote the\ninteger part. Similarly, in the multispin ﬂip Raman spec-\ntra in circular polarization the maximum peaks’ number\nis2I, which corresponds to the fact, that a single spin\ncan not be ﬂipped more than 2Itimes in one direction.\nThe Faraday rotation noise spectrum for a single spin\nI= 5=2(in the limit T= 0) is shown in the inset in Fig 4\nby a magenta curve. One can see, that the maximum\npeaks’ number is 5, and the peaks are much broader,\nthan for the Gaussian spin noise. This indicates, that\nthe higher order spin correlators generally contain more\ninformation than the common second order one.\nVI. DISCUSSION AND CONCLUSION\nTo detect the higher order spin correlators, the ratio\nbetween the exchange broadening of the exciton reso-\nnancep\nN!exshould be comparable to or larger than the\nhomogeneous linewidth \r, as shown in Sec. IVC. This\ncondition is easily satisﬁed in Mn-doped QWs [18] and\nQDs [19], where up to 15peaks in the Raman spin ﬂip\nspectra are visible. For these structuresp\nN!ex\u00182meV\nand\r\u00181meV.\nFor Mn-doped nanosystems, the number of the probed\nspins is typically very large N&100, and the total spin\nnoise is almost Gaussian. However with increase of the\nMn2+concentration a limited number of closely located\npairs of magnetic atoms appears. The strength of the\nexchange interaction in a pair can be of the order of\n0:5meV [48–50], and the ground state of the pair is the\nsinglet spin state. The diﬀerence between the interaction\nconstants with the heavy hole in the localized exciton for\nthe two spins in a pair leads to the mixing between sin-\nglet and triplet states. We expect, that it will manifest\nitself as another comb of peaks with the frequencies a\nbit larger than n\nLin Faraday rotation noise spectrum.\nDue to the small number of pairs their contribution is\nnon-Gaussian, and therefore contains detailed informa-\ntion about the spin dynamics of the pair of strongly cou-\npled spins.\nImportantly, the resonance shift quantum spin noise\nspectroscopy can be applied to a very broad class of spin\nsystems. For example, it can be applied to measure the\nnuclearspinﬂuctuationsinQDs. Inthiscasethemainre-\nquirement for the detection of high order spin correlators\nis the sizable hyperﬁne interaction strength as compared\nwith the inverse lifetime of the excited state. For exam-\nple, in GaAs-based QDs the hyperﬁne interaction con-\nstant for electrons is A\u0019100\u0016eV [17, 51], so for small\nQDs withN\u0018104one hasp\nN!ex\u0018A=p\nN\u00181\u0016eV,\nwhich is the typical exciton homogeneous linewidth [52].\nThus we expect, that the higher order nuclear spin cor-relators can be measured for small QDs as well as for the\nsmall colloidal nanocrystals.\nResonance shift spin noise spectroscopy is particularly\nuseful, when the number of probed spins is small, be-\ncause in this case the high order spin correlators can not\nbe reduced to the lower orders. To measure these corre-\nlation functions, the spin-related (e.g. hyperﬁne) struc-\nture of individual optical transitions should be visible.\nThere are many examples of such systems: For NV\u0000cen-\nters in diamond the hyperﬁne interaction constant with\nthe nearest C13atom can reach 0:1\u0016eV [53], which is\napproximately two times larger, than the homogeneous\nlinewidth at liquid helium temperatures [54]. For rare\nearth ions, Acan reach 10\u0016eV, while the homogeneous\nlinewidthisafewtimessmaller[55]. Inthepastfewyears\nthe van der Waals heterostructures are under intense in-\nvestigation. For localized spatially indirect excitons, the\nhyperﬁne interaction induced spin relaxation time is pre-\ndicted to be T\u0003\n2\u0018~=(p\nN!ex)\u00181ns [56, 57], which is\nan order of magnitude shorter, than the exciton lifetime\n1=\r= 10ns [58], so the nuclear related broadening of\nthe optical transition exceeds its linewidth by an order\nof magnitude thanks to the small exciton Bohr radius.\nThe similar situation is also realized for the lead halide\nperovskites where T\u0003\n2is comparable with 1=\r[59]. So\nthese systems are prominent for the resonance shift nu-\nclear spin noise spectroscopy. Apart from the solid state\nphysics, the hyperﬁne structure of optical transitions is\nquite routinely observed for atoms, such as K, Na, Rb,\nCs; and for simple molecules, such as I 2[60].Therefore\nthese systems are also promising for the resonance shift\nspin noise spectroscopy.\nIn conclusion, we have developed a theory of a class\nof optical phenomena that occur in optically transpar-\nent solids with localized spins (e.g. Mn2+spins in di-\nluted magnetic semiconductors), forming a basis for a\nset of experimental methods, which can be generically\ncalled resonance shift spin noise spectroscopy. The dis-\ntinctive feature of these phenomena is that the spins do\nnot directly participate in the probed optical transitions\n(e.g. excitonic ones), but they shift such transitions via\nthe spin-spin interactions. To demonstrate the univer-\nsality and power of this approach, we obtained the ex-\npressions for multispin ﬂip Raman spectra in diluted-\nmagnetic quantum wells and calculated the Faraday ro-\ntationnoisespectra. Wepredictmultipleovertonesofthe\nLarmor frequency in the spectra, which reﬂect the contri-\nbutions of the high order correlation functions of the spin\nﬂuctuations. Our predictions open a way for the exper-\nimental investigation of high order spin noise, including\nquantum noise. Our approach is directly extendable to a\nwide range of solid-state and atomic systems.\nACKNOWLEDGMENTS\nWegratefullyacknowledgethe fruitfuldiscussionswith\nM. M. Glazov and D. Scalbert. D.S.S. was supported by12\nthe Russian Science Foundation Grant No. 19-72-00081\nand the Basis Foundation. K.V.K. was supported by a\ngrant from Saint-Petersburg State University and DFG\nID 39411635 (Project No.40.65.62.2017).\nAppendix A: Calculation of the polarization\nWe start from the general form of the Hamiltonian (1).\nIts Hilbert space is a direct product of the states of the\nspin system and the excitonic states including exciton\nvacuum state. For heavy hole excitons there are four\nstates, which can be labeled by the electron spin projec-\ntionSe\nz=\u00061=2and the hole spin Sh\nz=\u00063=2. The exci-\ntonic states can be denoted as jki, wherek= 1;2;:::.\nIn the ﬁrst order in the incident ﬁeld amplitude, one\ncan consider a single exciton states only, so the exciton\nHamiltonian has the form\nHexc=X\nkHk0k\nexccy\nk0ck; (A1)\nwhereck(cy\nk) are the annihilation (creation) operators\nfor the statesjki. This Hamiltonian describes the ﬁne\nstructure of the excitonic levels and exciton interaction\nwith the external magnetic ﬁeld.\nThe coherent exciton generation is described by\nEq. (3), where\nP=X\nkdkck (A2)\nwithdkbeing the dipole moments of the excitonic states.\nIn the particular model, which is used in the derivation\nof Eq. (11), there are two excitonic states with the dipole\nmoments d\u0006=d(\u0000ex\u0007iey)=p\n2, where e\u000bare the unit\nvectors along the corresponding axes.\nThe Hamiltonian H0describes the magnetic spin sys-\ntemonlyanddoesnotcontainoperators ckandcy\nk. More-\nover, the Hamiltonian of the spin-exciton exchange inter-\naction has the form\nHint=X\nk;k0;iIiHkk0\nintcy\nkck0: (A3)\nNote, that this Hamiltonian contains oﬀ diagonal terms\n(withk6=k0) and coincides with Eq. (2).\nThe operator of the system evolution is\nU=Texp\u0014\n\u0000i\n~Zt\n0H(t0)dt0\u0015\n; (A4)\nand the Heisenberg polarization operator is\nP(0)(t) =UyPU: (A5)\nWeareinterestedinthecontribution P(t)toP(0)\n\u000b(t)only,\nwhich is linear in the amplitude of the probe light E.ThisoperatoractsonlyintheHilbertspaceoftheexciton\nvacuum state, so it is given by\nP(t) = ei\n~H0tPtZ\n0e\u0000i\n~(H0+Hexc+Hint)\u001c\n\u0002i\n~\u0000\nPyE\u0001\ne\u0000i\n~H0(t\u0000\u001c)d\u001c:(A6)\nSince the spin Hamiltonian H0commutes with P, this\nexpression can be written as\nP(t) =i\n~PtZ\n0\b(t;\u001c)d\u001c\u0000\nPyE\u0001\n;(A7)\nwhere\n\b(t;\u001c) = ei\n~H0te\u0000i\n~(H0+Hexc+Hint)\u001ce\u0000i\n~H0(t\u0000\u001c):(A8)\nTo simplify this expression we note that\n@\b(t;\u001c)\n@\u001c=\u0000i\n~\b(t;\u001c)h\nHexc+~Hint(t\u0000\u001c)i\n(A9)\nwith ~Hint(t)given by Eq. (5). One can readily see, that\n\b(t;0) = 1, so the solution of this equation is\n\b(t;\u001c) =Texp8\n<\n:\u0000i\n~\u001cZ\n0h\nHexc+~Hint(t\u0000\u001c0)i\nd\u001c09\n=\n;:\n(A10)\nSubstituting this expression in Eq. (A7) we see, that\nP(t) =P , where\n =i\n~tZ\n0Texp8\n<\n:\u0000i\n~\u001cZ\n0h\nHexc+~Hint(t\u0000\u001c0)i\nd\u001c09\n=\n;d\u001c\n\u0002\u0000\nPyE\u0001\n(A11)\nin agreement with Eq. (6). One can readily check, that\nit satisﬁes Eq. (4) indeed.\nAppendix B: Cumulant expansion\nThe generating function for the cumulants of J(0)and\nJ(\u001c)can be taken in the following form:\nK= lnD\nei\u000bJ(0)\u0000i\fJ(\u001c)E\n: (B1)\nIts Tailor series deﬁnes the cumulants\n\u0014\u0000\nJl(0);J(n\u0000l)(\u001c)\u0001\nas\nK=1X\nn=1nX\nl=0\u0012n\nl\u0013\n\u000bl(\u0000\f)n\u0000l\u0014\u0010\nJl(0);J(n\u0000l)(\u001c)\u0011\n:(B2)13\nComparing this expression with the correlator in Eq. (29)\nwe ﬁnd\nDh\nTeiJ(0)ih\nTe\u0000iJ(\u001c)iE\n= exp(1X\nn=12nX\nl=0(\u00001)n+l\n(2n\u0000l)!l!\n\u0002\u0014\u0010\nJl(0);J(2n\u0000l)(\u001c)\u0011o\n;(B3)\nwherewetookintoaccountthatthecumulantsoftheodd\nordersvanishintheabsenceofthespinpolarizationalong\nthezaxis (hm(t)i= 0). The cumulants of the operators\nshould be calculated using the normal time ordering for\nthe powers ofJ(\u001c), reverse time ordering for J(0)and\nputtingJ(0)always to the left of J(\u001c).\nTo simplify the following, we assume, that the Mn2+\nspins, Iiin Eq. (8), are independent. In this case J(t),\nas deﬁned in Eq. (26) also consists of Nindependent\ncontributionsJi(t)\u00181=N. Then a cumulant of the sum\nof independent variables takes the form [30]\n\u0014\u0010\nJl(0);J(2n\u0000l)(\u001c)\u0011\n=NX\ni=1\u0014\u0010\nJl\ni(0);J(2n\u0000l)\ni (\u001c)\u0011\n:\n(B4)\nFrom this relation one can see the scaling law for the\ncumulants\n\u0014\u0010\nJl(0);J(2n\u0000l)(\u001c)\u0011\n/1=N2n\u00001:(B5)\nThe larger is Nthe less important are the cumulants of\nthe high orders.\nIn the limit of many independent Mn2+spins,N\u001d1,\none can neglect all the cumulants except for n= 1(the\nsecond order one). This corresponds to the normal or\nGaussian spin noise. In this case Eq. (B3) reduces to\nD\nTeiJ(0)Te\u0000iJ(\u001c)E\n= exp\u0000\nhJ(0)J(\u001c)i\u0000\nJ2(0)\u000b\ns\u0001\n;\n(B6)and from Eq. (29) we obtain\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1Z\n0e\u00002k+hJ(0)J(\u001c)i\u0000hJ2(0)isdk:(B7)\nThesecondordercorrelatorof J(t)canbepresentedas\nadoubleintegralusingitsdeﬁnition(26). Thecorrelation\nfunctionhm(t1)m(t2)idepends on t1\u0000t2only, so the\ndouble integral can be reduced to a single integral as\nfollows:\nhJ(0)J(\u001c)i=kZ\n\u0000k(k\u0000jk0j)hm(0)m(\u001c+k0=\r)idk0;\n(B8a)\n\nJ2(0)\u000b\ns= 2kZ\n0(k\u0000k0)hm(0)m(k0=\r)isdk0:(B8b)\nSubstitution of these expressions in Eq. (B7) yields the\npolarization correlation function, which deﬁnes the Ra-\nman spin ﬂip spectrum, see Eq. (48).\nAppendix C: Areas of the peaks\nIn the realistic limit \u001cs\r\u001d1using Eq. (45) we obtain\nfrom Eq. (B8)\nhJ(0)J(\u001c)i=\u0000\nJ+e\u0000i\nL\u001c+J+ei\nL\u001c\u0001\ne\u0000j\u001cj=\u001cs(C1)\nand\nJ2(0)\u000b\ns=\nJ2(0)\u000b\nwith\nJ\u0006=!2\nex\n\n2\nL\u0012\nI2\nz\u000b\n\u0007hIxi\n2\u0013\u0014\n1\u0000cos\u0012\nL\n\rk\u0013\u0015\n:(C2)\nSubstituting these expressions in Eq. (B7) and decom-\nposing the exponent into series like in Eq. (59) we ﬁnd\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1X\nn=0nX\nl=0ei(2l\u0000n)\nL\u001c\u0000nj\u001cj=\u001cs\nl!(n+l)!1Z\n0Jl\n+Jn\u0000l\n\u0000e\u00002k\u0000J+\u0000J\u0000dk: (C3)\nIn the adiabatic approximation ( \nL\u001c\r) Eq. (C2) reduces to J\u0006=k2\u0016\u0006, so from Eq. (C3) we obtain\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1X\nn=0nX\nl=0\u0016l\n+\u0016n\u0000l\n\u0000\nl!(n+l)!ei(2l\u0000n)\nL\u001c\u0000nj\u001cj=\u001cs1Z\n0e\u00002k\u0000k2(\u0016++\u0016\u0000)k2ndk: (C4)\nIn the next step we introduce n0=n\u00002landl0=lforn0\u00150andl0=n\u0000lotherwise. Then we rewrite this\nexpression as\n\np\u0003\n+(0)p+(\u001c)\u000b\n=1X\nn0=\u000011X\nl0=0\u0016l0\nsign(n0)\u0016jn0j+l0\n\u0000sign(n0)\nl0!(jn0j+l0)!e\u0000in0\nL\u001c\u0000(jn0j+2l0)j\u001cj=\u001cs1Z\n0e\u00002k\u0000k2(\u0016++\u0016\u0000)k2(jn0j+2l0)dk;(C5)14\nwhere we assume sign(0)\u00111to be speciﬁc. Finally, the Fourier transform of this expression [see Eq. 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Ubachs, “A Dense Grid of Reference Iodine Lines forOpticalFrequencyCalibrationintheRange595-655nm,”\nJ. Mol. Spec. 201, 256 (2000)."    },    {        "title": "0807.0470v2.First_principles_study_of_the_spin_lattice_coulpling_in_spin_frustrated_DyMn__2_O__5_.pdf",        "content": "arXiv:0807.0470v2  [cond-mat.mtrl-sci]  12 Oct 2008First-principles study of the spin-lattice coulpling in sp in frustrated DyMn 2O5\nTianqi Shen, Kun Cao, Guang-Can Guo, and Lixin He∗\nKey Laboratory of Quantum Information, University of Scien ce and\nTechnology of China, Hefei, 230026, People’s Republic of Ch ina\n(Dated: November 29, 2018)\nThe lattice dynamic properties and spin-phonon coupling in DyMn 2O5are studied by using the\ndensity-functional theory. The calculated phonon frequen cies are in very good agreement with\nexperiments. We then compare the phonon modes calculated fr om diﬀerent spin conﬁgurations.\nThe results show that the phonon frequencies change substan tially in diﬀerent spin conﬁgurations,\nsuggesting that the spin-phonon coupling in this material i s very strong. Especially, the short\nrange spin ordering has drastic eﬀects on the highest Raman a nd IR phonon modes that might be\nresponsible for the observed phonon anomalies near and abov e the magnetic phase transitions.\nPACS numbers: 75.25.+z, 77.80.-e, 63.20.-e\nI. INTRODUCTION\nManganese oxides including RMnO 3,1,2and RMn 2O5\n(R=Tb, Dy, Ho, etc.)3,4,5are a very special class of im-\nproper ferroelectrics that display strong magnetoelectric\n(ME) eﬀects, such as the “colossal magnetodielectric”\n(CMD) eﬀects and magneto-polarization-ﬂop eﬀects.1,2,6\nThesematerialshaveattractedgreatattention,7,8,9,10be-\ncause the strong ME eﬀects can be utilized to tune the\nelectric properties of the materials via applied magnetic\nﬁled, or vice versa, and therefore have great potential for\nfuture multifunctional device applications. The macro-\nscopicME eﬀects have their microscopic origin from the\ninterplay between the lattice degree of freedom and spin\ndegree freedom. Neutron scattering,11as well as ﬁrst-\nprinciples calculations12,13shows that the improper fer-\nroelectricity in TbMn 2O5is driven by the non-central\nsymmetric magnetic ordering, which, by coupling to the\nlattice, lower the crystal symmetry.11,12The spin-lattice\ncoupling can also modify the lattice dynamitic prop-\nerties. Indeed, recent temperature dependent Raman\nmeasurements14show anomalous phonon shift at T∗∼\n1.5 TNand near the magnetic phase transition tempera-\nture T Nin DyMn 2O5. Similar anomalies are also ob-\nserved for the IR modes.15Further B-ﬁeld dependent\nmeasurements show that the IR modes are very sensi-\ntive to local magnetic structure,16which suggest that the\nspin-phonon coupling in this compound is very strong.\nHowever, the experiments give only the overall eﬀects,\nand the detailed mechanism of the spin-phonon coupling\nis still lack.\nFirst principles method has been applied successfully\nin studying spin-phonon coupling in geometrically frus-\ntrated spinel ZnCr 2O4.17It is a challenge problem to\nstudythespin-phononcouplinginthespinfrustratedsys-\ntems, such as RMn 2O5, because in these materials, the\nmagnetic interactions are of diﬀerent magnitudes (e.g.,\n|J4|,|J5| ≫ |J3|),11,13and the spin frustration leads to\n∗corresponding author, Email address: helx@ustc.edu.cncomplicated spin-spin correlations as functions of tem-\nperature and magnetic ﬁeld. Therefore, the spin will\ndevelop diﬀerent short range ordering at diﬀerent tem-\nperature above the magnetic phase transition, and make\nit hard to identify the origin of the phonon anomalies.14\nIn this work, we study the spin-phonon coupling in\nDyMn 2O5, to shed some light on the observed phonon\nanomalies. We show that the short range spin correla-\ntion and local spin structure have signiﬁcant eﬀects to\nthe phonons frequencies in this material.\nII. METHODOLOGY\nThe ﬁrst-principles density-functional (DF) calcula-\ntions are done by using the Vienna ab initio Simula-\ntions Package (VASP).18,19We use a spin-polarized gen-\neralized gradient approximation (GGA), with Perdew-\nBurke-Ernzerhof functional.20A plane-wave basis and\nprojectoraugmented-wave(PAW)pseudopotentials21are\nused, with Mn 3 p3d4s, and Dy 5 p5d6selectrons treated\nself-consistently. A 500 eV plane-wave cutoﬀ results in\ngood convergence of the total energies. We relax the\nstructure until the changes of total energy in the self-\nconsistent calculations are less than 10−7eV, and the\nremaining forces are less than 1 meV/ ˚A. DyMn 2O5un-\ndergo several magnetic phase transitions5at low tem-\nperature. We are interested in the phonon properties in\nthe paramagnetic (PM) and anti-ferromagnetic (AFM)\nphase. To accommodate the magnetic structure, we use\na2×1×1supercell.12Forthe supercell weused, a 1 ×2×4\nMonkhorst-Pack k-points mesh converges very well the\nresults.\nIII. RESULTS AND DISCUSSION\nThe fully relaxed DyMn 2O5is an AFM insulator, and\nhas same lattice structure to that of TbMn 2O5,12,22both\nareofPb21msymmetry, asTb and Dy areisovalent. The\nlattice structure is distorted from a Pbamhigh (H) sym-\nmetry structure, due to the spin-lattice coupling.12The2\nTABLE I: Calculated phonon frequencies of ground state stru ctureL. The experiment vaules for Raman phonons are extracted\nfrom Ref. 14, whereas the IR phonons frequencies are extract ed from Ref. 16.\nB2u Ag B2g Au B3u B1g B1u B3g\nGGA Exp. GGA Exp. GGA Exp. GGA Exp. GGA Exp. GGA Exp. GGA Exp. GG A Exp.\n671.8 680.0 679.2 695.0 566.0 614.1 677.2 713.0 683.7 675.0 6 07.2 557.2 585.0\n628.4 617.7 625.0 480.8 510.0 531.2 620.1 647.0 530.0 507.9 4 95.0\n553.4 615.0 470.8 499.5 597.1 588.3 497.9 471.7\n535.8 531.5 545.0 452.3 460.0 429.1 512.7 519.0 547.5 540.0 4 37.2 450.4 440.0\n473.8 491.7 500.0 444.7 402.3 475.6 521.2 410.6 403.0 430.5\n438.0 444.2 460.0 336.5 287.5 465.5 470.1 485.0 296.7 290.0 3 24.7 320.0\n412.1 407.2 420.0 293.0 305.0 225.7 378.3 411.0 420.0 249.3 2 89.8\n351.0 336.9 350.0 273.6 132.5 351.5 371.3 330.0 142.5 264.5\n314.4 311.1 230.7 220.0 110.8 317.0 299.6 -2.0 241.5\n267.8 267.0 235.4 209.6 265.4 242.6 235.0 189.8\n223.0 224.2 215.0 96.6 216.3 217.0 220.2 205.0 103.6\n163.2 136.4 158.2 171.9\n157.5 108.4 151.5 146.1 145.0\n99.6 100.0 111.6\ncalculated lattice constants are a=7.270˚A,b=8.518˚A\nandc=5.600˚A, respectively, and are in very good agree-\nment in experimental values5(7.285, 8.487 and 5.668\n˚A respectively). The lattice constants are somewhat\nsmaller than those of TbMn 2O5.12,22Like TbM 2O5,4,12\nDyMn 2O5has two energetically degenerate lattice (and\nmagnetic) structures, LandR, in which Mn4+form an\nAFM square lattice in the abplane, whereas Mn3+cou-\nples to Mn4+either antiferromagnetically via J4alonga\naxis or with alternating sign via J3alongbaxis. Mn3+\nions in two connected pyramidsalsocouple antiferromag-\nnetically through J5. Here, we adopt the notations J3,\nJ4andJ5from Ref. 4, and deﬁne the J3to be the Mn4+-\nMn3+superexchangeinteraction through pyramidalbase\ncorners, and J4the superexchange interaction through\nthe pyramidal apex.\nWe ﬁrst calculate all zone-center optical phonon fre-\nquencies of the fully relaxed structure ( LorR) via a\nfrozen-phonon technique,12and the results are listed in\nTable I. Symmetry analysis13shows that for the high\nsymmetry ( Pbam) structure H, phonons belong to 8 ir-\nreducible representations (irreps), among them B1u,B2u\nandB3uare IR active, where B1g,B2g,B3gandAgare\nRaman active and Auis silent. However, in the low sym-\nmetry (Pb21m) structure ( LorR), the Raman and IR\nmodes couple to each other, and regroup to 4 irreps, i.e.,A1=B2u⊕Ag;A2=B2g⊕Au;B2=B3u⊕B1g;\nB1=B1u⊕B3g. The couplings between irreps. are\nvery small, therefore the results are given by their ma-\njor symmetry characters. As we see from Table I, the\nresults are in excellent agreement with experiments.14,16\nThe calculated phononfrequencies areveryclose to those\nof TbMn 2O5, because Tb and Dy are isovalent and the\ntwomaterialshaveverysimilarlatticestructures. Thein-\nner products between the correspondingphonon modes23\nof the two compounds are close to unity, suggesting that\nthey have very similar mode patterns.\nExperimentally, there are several anomalies found for\nboth Raman modes and IR modes, near the magnetic\nphase transitions. For example, anomalous phonon shift\nfor the Raman modes had been observed at T∗∼60-65\nK, and near the N´ eel temperature T N.14The highest A g\nmode show a steep hardening upon cooling between T∗\nand TN.14For the IR modes, it has been found near60K,\nthe infraredactivemodessoften andseveralmodessoften\nsubstantially.15The phonon modes are also found to be\nvery sensitive to the applied magnetic ﬁeld.16The ob-\nserved phonon anomalous phonon shift strongly suggest\nthat the spin-phonon coupling in this material is very\nstrong.\nTheoretically, the spin-lattice coupling in this com-\npound can be described via an eﬀect Hamiltonian,13\nE({uλ}) = (E0−/summationdisplay\nijJijSi·Sj)−/summationdisplay\nij/summationdisplay\nλ∂Jij\n∂uλuλSi·Sj+(1\n2/summationdisplay\nλmλω2\nλu2\nλ−/summationdisplay\nij/summationdisplay\nλρ∂2Jij\n∂uλ∂uρuλuρSi·Sj),(1)\nwhereuλis theλ-th zone-center phonon, and Siis the\nmagnetic moment of the i-th atom. Here, we consideronly the magnetic moments of the Mn3+and Mn4+ions.\nJijis the exchange interaction between two adjacent Mn3\nFIG. 1: (Color online) The spin conﬁgurations used in calcu-\nlating phonon frequencies of diﬀerent spin states.\nions.E0is the total energy of high symmetry structure\nHwithout spin-spin interactions, where the ground state\nspin conﬁguration is determined by the ﬁrst term. ωλ\nis the frequencies of the λ-th phonon, calculated in the\nhigh symmetry PM state. The spin-lattice coupling have\ntwo eﬀects: (i) The second term, which is linear in u,\nbreaks the inversion symmetry of the system, and drive\nthe structure to a low symmetry polarized state leading\nto coexistence of AFM and ferroelectricity. The coupling\nstrength is proportionalto ∂Jij/∂uλ. However, this term\nwould not change the phonon frequencies (at least to the\nﬁrst order approximation). (ii) The phonon modes ωλ\n(including frequencies and eigenvectors) are modiﬁed in\nthe presence of spin-phonon interaction, due to the third\ntermofEq. (1), wherethespin-phononcouplingstrength\nis∂2Jij/∂uλ∂uρ. Diﬀerent spin conﬁguration (SC) {Si}\nwould therefore have diﬀerent phonon frequencies.\nTo investigate how the SCs change the phonons in\nDyMn 2O5, we calculate the phonons frequencies of dif-\nferent SCs including the high temperature PM state, the\nAFM state, and the ferromagnetic (FM) state. The spin-\nlattice coupling strength J′′can be extracted by com-\nparing the force-constant matrices calculated from these\ndiﬀerent spin states.17[see Eq. (1)]. In this work, we fo-\ncus on the B 2uand A gmodes, whereas other modes can\nbe studied in similar way. To simplify the discussion,all following calculations are done in the high symmetry\nstructure H, constructed via symmetrizing structures L\nandRaccording to the Pbamsymmetry.12In reality, the\nlattice constants would be somewhat diﬀerent at diﬀer-\nent SCs. In the present calculations, this eﬀect is ignored\nand the lattice constants are ﬁxed in the calculations.\nThe phonon frequencies of PM, AFM (SC G in Fig. 1)\nand FM (SC H in Fig. 1) states are compared in Table II\nfor the A gmodes and in Table III for the B 2umodes.\nOne may attempt to calculate the phonon frequencies of\nthe PM state using spin unpolarized GGA, which is listed\nin the column under u-GGA. A quick look reveals that\nthe phonon frequencies in this column are signiﬁcantly\nlower than those of AFM and FM state. Especially, as\nshown in Table II, the A girrep has a soft mode, caus-\ning a Jahn-Teller distortion.24The phonon frequencies\ncalculated from u-GGA is not a good approximation for\nthe PM state, because the Mn ions have local magnetic\nmoments even in the PM phase, although their direc-\ntions are distributed randomly. Alternatively, we calcu-\nlate/an}bracketle{tω/an}bracketri}htdis, which are the averaged phonon frequencies\nof several fully disordered SCs ( A,B,Cin Fig. 1 ) so\nall exchange interactions are canceled out. The averaged\nphonon frequencies /an}bracketle{tω/an}bracketri}htdisare close to those of the AFM\nand FM states without soft phonons.\nHowever, the spins are fully disordered only at very\nhigh temperature. At lower temperature, especially close\nto the magnetic phase transition, the spins are somehow\ncorrelated, and develop short range ordering. Since J4\nis the largest among all the exchange interactions,13to\ncompare with experiments, we consider the ordering of\nJ4interactions. We calculate /an}bracketle{tω/an}bracketri}htJ4, which are the aver-\naged phonon frequencies of three SCs (D, E, F in Fig. 1)\nin which J4are ﬁxed in the AFM conﬁguration, whereas\nall other exchange interactions are canceled out. As we\nsee from Table II and Table III, the frequencies of most\nmodes in diﬀerent SCs diﬀer by about 3 - 5 cm−1, which\nare of similar magnitude to the phonon shifts in the mag-\nneticﬁeldexperiment.16However,thehighfrequencyB 2u\n(671 cm−1) and A g(679 cm−1) modes change dramat-\nically (∼20 cm−1) for diﬀerent SCs. Especially, /an}bracketle{tω/an}bracketri}htdis\nand/an}bracketle{tω/an}bracketri}htJ4are diﬀerent by about 10 and 13 cm−1for the\nAgandB2umodes, respectively, though both belong to\nthe PM phase. Interestingly, the two modes frequencies\nof/an}bracketle{tω/an}bracketri}htJ4are close to the those in AFM state, and /an}bracketle{tω/an}bracketri}htdis\nhave the mean values of those of AFM and FM states. It\nis usually believed that GGA overestimates the exchange\ninteractions, whereas including the on-site Coulomb cor-\nrelation may improve this problem. To see how includ-\ning the on-site Coulomb correlation will change the spin-\nphonon coupling in DyMn 2O5, we have carried out GGA\n+ U calculations of the phonons in diﬀerent SCs. The\non-siteCoulomb UofMn ions hasbeen taken to be a rea-\nsonablevalue4.0eV,whereastheexchangeparameter j=\n0.8 eV is used. The results are also listed in Table II and\nTable III for the A gand B 2umodes respectively. As we\nsee, GGA + U signiﬁcantly overestimatesthe frequencies\nof the high frequency modes when compared to experi-4\nTABLE II: The Agmodes calculated in high symmetry structure Hof diﬀerent spin conﬁgurations using GGA and GGA + U\nmethods.\nu-GGA GGA GGA + U\n/angbracketleftω/angbracketrightdis/angbracketleftω/angbracketrightj4AFM FM /angbracketleftω/angbracketrightdis/angbracketleftω/angbracketrightj4AFM FM\n571.1 667.7 677.6 679.0 662.8 746.1 759.9 759.0 723.2\n563.2 621.5 625.8 618.6 629.7 658.8 661.3 659.4 653.4\n530.7 617.7 618.4 613.5 617.3 634.2 635.3 631.0 638.3\n463.3 537.2 540.9 531.1 547.0 559.3 561.1 556.6 562.2\n432.9 490.6 492.2 491.3 494.1 505.4 506.3 506.0 506.4\n409.2 441.0 442.8 442.6 441.4 451.4 452.7 452.6 450.1\n369.6 414.3 407.9 407.0 429.8 438.2 436.6 436.0 440.3\n249.6 340.0 337.6 337.6 340.3 342.6 342.3 341.6 343.0\n178.8 321.9 329.2 311.0 331.7 328.9 331.9 323.2 334.8\n152.7 238.7 237.2 236.2 243.9 248.3 248.0 247.5 249.2\n102.9 222.3 220.2 219.6 225.9 217.7 216.9 216.6 220.1\n66.1 136.8 137.5 136.0 137.6 141.1 141.6 140.9 141.0\n-174.2 110.0 110.0 109.3 111.2 113.2 113.4 113.1 113.5\nTABLE III: The B2umodes calculated in high symmetry structure Hof diﬀerent spin conﬁgurations using GGA and GGA +\nU methods.\nu-GGA GGA GGA + U\n/angbracketleftω/angbracketrightdis/angbracketleftω/angbracketrightj4AFM FM /angbracketleftω/angbracketrightdis/angbracketleftω/angbracketrightj4AFM FM\n586.4 660.2 673.3 671.7 649.6 723.7 736.6 735.9 705.8\n512.8 627.2 625.5 626.1 623.8 659.1 659.4 659.1 657.9\n507.2 551.6 552.2 553.4 552.3 578.5 578.7 579.4 577.9\n488.7 534.0 530.9 535.3 534.1 566.6 567.0 568.3 564.6\n431.2 474.5 473.5 475.7 476.4 495.5 494.5 495.6 498.1\n391.0 439.5 437.2 439.2 438.1 461.8 462.0 462.2 459.4\n381.3 409.7 410.0 411.7 406.0 419.9 420.2 420.9 418.0\n310.0 353.6 350.2 350.8 357.6 365.7 364.9 365.2 366.1\n274.3 315.0 314.3 315.2 314.6 316.2 315.8 316.3 316.6\n257.7 272.7 266.0 267.9 280.1 283.8 282.5 282.5 285.5\n211.1 222.2 221.8 223.4 220.8 211.7 212.0 212.5 212.7\n142.6 162.0 162.2 162.0 161.4 165.6 165.7 165.5 165.7\n139.2 156.8 156.9 157.0 157.3 163.3 163.5 163.7 162.7\n93.9 98.4 98.6 99.2 96.8 94.0 94.3 94.6 93.4\nments. However, the frequency diﬀerence between diﬀer-\nent SCs are almost remain the same to those without U,\nsuggesting that the spin-phonon coupling constants do\nnot change much by including the on-site Coulomb cor-\nrelation. In the following discussion, we therefore focus\nonly the GGA results.\nTo understand the results, we analyze the vibrational\npattern of the two modes, shown in Fig. 2. For the\nhigh-frequency A gmode, about 90% of the vibration is\nassociated with O 3(the oxygens that connect Mn3+and\nMn4+along the aaxis, and convey J4interaction). The\nO3atoms vibrate in the abplane, with an angle of 21.4◦\ntoaaxis. This mode also has small components of Mn3+\nvibrating in the a-axis, and Mn4+motion in the c-axis.\nTherefore J5may also aﬀect the A gmode (but consider-\nably smaller than J4). In the high-frequency B 2umode,\nmore than 70% of the contribution comes from O 3, which\nalso vibrate in the abplane, with an angle of 19.9◦toaaxis. The rest contribution includes the motion of Mn4+\nalongaaxis. If the spin exchange interaction is local, the\nmotionthe O 3atomswouldonlytune the J4interactions.\nTherefore the J4interaction plays an essential role of the\nspin-phonon coupling in these two modes. Now it is easy\nto understand the phonon shifts of the two modes in dif-\nferent SCs. According to Eq. (1), the frequency shifts\nof the highest A gand B 2umodes are /an}bracketle{tS3·S4/an}bracketri}ht∂2J4/∂u2\nλ,\nwhereS3,S4are the spin vectors of Mn3+and Mn4+\nions associated with J4, respectively. For the fully disor-\ndered SC, /an}bracketle{tS3·S4/an}bracketri}ht=0. In the J4short range orderd state\nand in AFM state /an}bracketle{tS3·S4/an}bracketri}ht=-S3S4, whereas in FM state,\n/an}bracketle{tS3·S4/an}bracketri}ht=S3S4.S3=2.3µBandS4=1.64µB, are the lo-\ncal magnetic moments. By comparing the force-constant\nmatrices of diﬀerent SCs, we estimate ∂2J4/∂u2\nλ∼0.120\nmeV/(˚A·µB)2for the two high frequency modes. As dis-\ncussed in previous section, this value remain unchanged\nwhen including the on-site Coulomb interactions.5\nFIG. 2: (Color online) The mode pattern of the highest (a)\nAgmode and (b) B 2umode.\nExperimentally,suchlargephononhardeningdueto J4\nordering was not directly observed during the tempera-\nture dependent measurement14,15for two reasons. First,\ntheJ4ordering develop gradually with decreasing of the\ntemperature. A Monte Carlo simulation25show that the\naverage of /an}bracketle{tS3·S4/an}bracketri}htincreases from 80% to 90% of its\nmaximum value when temperature lower from 1.5 T Nto\nTN. Second, the phonon frequencies hardening due to J4\nordering is accompanied by the anharmonic eﬀects,14,15\nand it is hard to isolate the anharmonic eﬀects and spin-\nphonon eﬀects in experiments. The lower frequencies\nmodes involve collective motion of many atoms. There-\nfore, thephononfrequencyshiftsduetospin-phononcou-pling are a consequence of the competition of many J′′s,\nand are much smaller than the high frequency modes.\nBelow N´ eel temperature, T N, the material is in the\nlong range ordered AFM state. We then compare the\nphonon frequencies of short range ordered state /an}bracketle{tω/an}bracketri}htJ4\nto those of AFM state. As we see, while the highest A g\nmode hardensby 2 cm−1, and the highest B 2umode softs\nby 2 cm−1, consistent with experiments.14,15The fre-\nquencies diﬀerence between /an}bracketle{tω/an}bracketri}htJ4and AFM might come\nfrom the J5interactions.\nIV. SUMMARY\nTo summarize, we have investigated the spin-phonon\ncoupling in a spin frustrated DyMn 2O5system via ﬁrst-\nprinciples calculations. We compare the phonon modes\ncalculated from diﬀerent spin conﬁgurations. 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He, (unpublished)."    },    {        "title": "2106.15187v2.Spin_pumping_in_noncollinear_antiferromagnets.pdf",        "content": "Spin pumping in noncollinear antiferromagnets\nMike A. Lund, Akshaykumar Salimath, and Kjetil M. D. Hals\nDepartment of Engineering Sciences, University of Agder, 4879 Grimstad, Norway\nThe ac spin pumping of noncollinear antiferromagnets is theoretically investigated. Starting from\nan e\u000bective action description of the spin system, we derive the Onsager coe\u000ecients connecting the\nspin pumping and spin-transfer torque associated with the dynamics of the SO(3)-valued antifer-\nromagnetic order parameter. Our theory is applied to a kagome antiferromagnet resonantly driven\nby a uniform external magnetic \feld. We demonstrate that the reactive (dissipative) spin-transfer\ntorque parameter can be extracted from the pumped ac spin current in-phase (in quadrature) with\nthe driving \feld. Furthermore, we \fnd that the three spin-wave bands of the kagome AF gener-\nate spin currents with mutually orthogonal polarization directions. This o\u000bers a unique way of\ncontrolling the spin orientation of the pumped spin current by exciting di\u000berent spin-wave modes.\nI. INTRODUCTION\nOver the last years, there has been rapidly growing\ninterest in implementing antiferromagnetic elements in\nspin-based electronics. This has led to the development\nof antiferromagnetic spintronics1{6, in which the infor-\nmation is coded into the magnetic moments of antiferro-\nmagnets (AFs)7. Unlike ferromagnets, which have been\nthe traditional building blocks of spintronics, the AFs\nare remarkably stable against magnetic \feld noise due to\ntheir vanishing magnetization. Furthermore, the AFs are\ncharacterized by terahertz (THz) spin dynamics, which\nis a thousand times faster than the characteristic fre-\nquency of ferromagnets. The ultrahigh-frequency of AFs\nis desirable for use in future spin electronics because it\nallows for signi\fcantly higher operational speeds of the\ndevices. Ultrafast switching of AFs has been experimen-\ntally demonstrated8, and recent works have shown that\nthe antiferromagnetic order is e\u000eciently manipulatable\nvia electric currents8{21as well as optical pulses22{24.\nA cornerstone of spintronics is the ability to manipu-\nlate the order parameter of magnetic materials via spin-\ntransfer torques (STTs) { the process where spin currents\nproduce magnetic torques via direct transfer of spin an-\ngular momentum from the itinerant electrons to the or-\ndered spin system25. The reciprocal process of the STT\nis spin pumping and refers to the phenomenon where the\ncollective spin excitations of the magnet pump a spin cur-\nrent into adjacent metallic leads26. Notably, the linear re-\nsponse coe\u000ecients describing the STT and spin pumping\nare connected via the Onsager reciprocal relations27{30.\nConsequently, one can obtain signi\fcant insight into the\nstrength, symmetry, and governing mechanisms of the\nSTT by probing the reciprocal spin pumping process.\nIn AFs, the spin pumping20,31,32and STT10,15,17have\nbeen theoretically investigated in several works, and two\nexperiments recently observed sub-THz spin pumping in\nthe uniaxial insulating AFs MnF 233and Cr 2O334. Most\nof these works have concentrated on collinear AFs, which\nare antiferromagnetic systems characterized by a single\norder parameter (commonly known as the staggered \feld\nor N\u0013 eel vector). However, several AFs require two or\nthree mutually orthogonal staggered \felds to describe the\nfmfrFIG. 1: (color online). Illustration of two reciprocal spin pro-\ncesses in a bilayer consisting of an NCAF interfaced with a\nnormal metal (NM). a. A spin accumulation \u0016sin the NM\ninduces a spin current Isthat \rows into the NCAF, produc-\ning an STT on the NCAF. b. A uniformly precessing NCAF,\nwhich is driven by the e\u000bective \felds frandfm, pumps a\nspin currentIsinto the NM layer. For a uniformly precessing\nNCAF with strong exchange interaction, the antiferromag-\nnetic spin state can be parametrized by the macroscopic stag-\ngered vectors L1,L2,L3, and the dynamics can be described\nas a rotation of the coordinate system spanned by these vec-\ntors. The STT and spin pumping in aandb, respectively,\nare connected via the Onsager reciprocal relations.\nspin order correctly (see Fig. 1)36. In this case, the sys-\ntem is referred to as a noncollinear AF (NCAF). The\nspin order of NCAFs is parameterized by a rotation ma-\ntrix, which de\fnes the orientation of the reference frame\nspanned by the orthogonal staggered \felds35,36. To date,\nlittle knowledge exists on how spin currents couple to the\nSO(3) order parameter of NCAFs. In Ref. 17, the STT\nwas phenomenologically investigated, whereas the dissi-\npative coupling mechanism was derived in Ref. 37 from\na scattering matrix formalism and applied to amorphousarXiv:2106.15187v2  [cond-mat.mes-hall]  22 Nov 20212\nmagnets and kagome AFs in Refs. 38,39. However, so\nfar, there have been no thorough investigations of the\nspin pumping process in these nontrivial spin systems.\nIn this work, we derive a general theory of the reac-\ntive and dissipative ac spin pumping in NCAFs. The\ngeneral formalism is applied to NCAFs with kagome lat-\ntice structure. Importantly, we \fnd that both the reac-\ntive and dissipative STT parameters can be mapped out\nfrom the spin pumping signal measured via the inverse\nspin Hall e\u000bect (ISHE). Additionally, we show that the\nthree spin-wave bands of the kagome AF produce spin\ncurrents with orthogonal spin polarizations, which en-\nables manipulation of the spin current's orientation by\nonly tuning the frequency of the external driving \feld.\nWhen the driving \feld hits the resonance frequency of a\nspin-wave band, a current with a \fxed spin polarization\nis created. This phenomenon di\u000bers markedly from spin\npumping of ferromagnets and collinear AFs, where a re-\norientation of the magnetic state is required for changing\nthe polarization direction. Thus, our work demonstrates\nthat spin pumping could represent an e\u000bective technique\nfor exploring novel spin torque mechanisms in NCAFs.\nThis paper is organized as follows. Sec. II presents\na general e\u000bective action description of NCAFs and de-\nrives the Onsager coe\u000ecients representing the coupling\nbetween the NCAF and spin currents. From the On-\nsager coe\u000ecients, a general theory of spin pumping is de-\nrived. Then, in Sec. III, the general theory is applied to\nkagome AFs. A summary is provided in Sec. IV, whereas\nthe action and dissipation functionals of kagome AFs are\nmicroscopically derived in appendixes A and B.\nII. GENERAL THEORY\nWe consider the reciprocal processes spin pumping and\nSTT in a bilayer consisting of an NCAF of volume Vin-\nterfaced with a normal metal (NM) (see Fig. 1).20,28,41\nOur main aim is to derive a general theory for the spin\npumping of a uniformly precessing NCAF. To this end,\nwe \frst consider the STT, derive the Onsager coe\u000ecients\ngoverning the STT-driven uniformly precessional motion\nof the NCAF, and then use the Onsager reciprocal rela-\ntions to \fnd an expression for the spin pumping.\nOur model is based on the assumption that the ex-\nchange interaction of the NCAF is much stronger than\nany other interaction energies in the microscopic spin\nHamiltonian such that the mutual orientation of the sub-\nlattice spins only is weakly a\u000bected by the dynamics of\nthe NCAF.35,36The STT is produced by a spin accumu-\nlation\u0016sin the NM layer at the NM/NCAF interface.41\nThe vector\u0016shas a direction parallel to the out-of-\nequilibrium spin density in the NM and a norm equal to\nthe di\u000berence between the chemical potentials of the spin\nup and down electrons. The spin accumulation yields\na spin current that \rows into the NCAF, transferring\nits spin angular momentum to the antiferromagnetic sys-\ntem. The source of this spin accumulation does not playa role in the theory we develop and could, in principle,\noriginates from any microscopic mechanism generating\nan out-of-equilibrium spin density (e.g., spin Hall e\u000bect).\nFurthermore, we disregard the e\u000bects of the SOC that\nbreak the spin rotational symmetry of the STT. In this\ncase, the e\u000bective action Sof the NCAF can to second\norder in the space-time gradients and external forces (i.e.,\nSTTs and magnetic \felds) be written as35\nS=Z\ndVdtL: (1)\nThe Lagrangian density L=T\u0000U\u0000Usof the spin system\nconsists of a kinetic term T\nT=a1\n2V\u0001m; (2)\nthe energyUproduced by the exchange interaction, the\nspin-orbit coupling (SOC), and the magnetic \feld \u0018h\nU= \u0003\u000b\f\nij\u0002\n@\u000bRT@\fR\u0003\nij+\u0017kl\nijRijRkl+ ~\u0014ijmimj\u0000h\u0001m;\n(3)\nand a termUsrepresenting the coupling to the spin accu-\nmulation\u0016sof the itinerant quasi-particles that di\u000buse\ninto the NCAF from the adjacent NM layer\nUs=\u0015m\u0001fs: (4)\nHere,a1is a constant that depends on the lattice struc-\nture,Ris a rotation matrix that describes the orientation\nof the reference frame spanned by the staggered \felds\nof the NCAF,Vi=\u0000(1=2)\u000fijk[RT_R]jk(where\u000fijkis\nthe Leivi-Civita tensor and _R\u0011@tR),fs=\u0016s=~, and\nmis proportional to the out-of-equilibrium magnetiza-\ntion produced by a relative tilting of the magnetic sub-\nlattices.@\u000bRrepresents the partial derivatives of the ro-\ntation matrix with respect to the spatial coordinates, i.e.,\n\u000b2fx;y;zg. For further analysis, it is convenient to split\n~\u0014ijinto isotropic and anisotropic terms: ~ \u0014ij=a2\u000eij+\u0011ij\n(\u000eijis the Kronecker delta). The coe\u000ecients \u0003\u000b\f\nijanda2\n(\u0017kl\nijand\u0011ij) are proportional to the isotropic exchange\ninteraction (the anisotropy energy), whereas the constant\n\u0015parameterizes the strength of the reactive STT induced\nby the spin accumulation. Throughout, Einstein's sum-\nmation convention is implied for repeated indices.\nThe dissipative processes in the NCAF is determined\nby the dissipation functional17,40\nG=Z\ndVdt\"\n~\u000b\n8Tr\u0010\n_RT_R\u0011\n+~\u0015\n2V\u0001fs#\n; (5)\nwhere ~\u000band ~\u0015parameterize the damping of the spin\nsystem and the dissipative STT, respectively.\nIn Sec. A-B, we microscopically derive the above action\nand dissipation functional for a NCAF with a kagome lat-\ntice. For the sake of completeness, we have in Eq. (3) also\nincluded the energy contribution from the spatial varia-\ntions of the order parameter. However, when deriving3\nthe Onsager coe\u000ecients, we will restrict ourselves to the\nuniform case and disregard the gradient terms.\nWe consider small deviations from the uniform equilib-\nrium state of the NCAF. In this case, it is convenient to\nuse a Gibbs vector representation of the SO(3) rotation\nmatrixR. The Gibbs vector rcorresponding to a rota-\ntion by an angle of \u0012about the axis ^nisr= tan(\u0012=2)^n,\nand the action of Ron a general vector vis36,42\nRv=v+2\n1 +jrj2[r\u0002v+r\u0002(r\u0002v)]:(6)\nFurthermore, it is possible to represent the partial deriva-\ntives@\u0016Rof the rotation matrix in terms of the Gibbs\nvector via the relationship (here, \u00162ft;x;y;zg)42\nh\n(@\u0016R)RTi\nij=\u000fikj2\n1 +jrj2[@\u0016r+r\u0002@\u0016r]k:(7)\nNote that the identity matrix corresponds to r=0.\nThus,jrj\u001c 1 since we consider small deviations from\nthe equilibrium state. Using Eqs. (6)-(7) and keeping\nterms in Eqs. (2)-(4) up to second order in the out-of-\nequilibrium quantities fr;mgand external force \felds\nffs;hg, we \fnd the Lagrange density\nL=a1m\u0001_r\u0000U(r;m)\u0000Us(m;fs): (8)\nHere,Us(m;fs) is given by Eq. (4), and the potential\nU(r;m) takes the form of\nU= \u0000\u000b\f\nij@\u000bri@\frj+\u0014ijrirj+ ~\u0014ijmimj\u0000h\u0001m;(9)\nwhere we have introduced the anisotropy tensor \u0014ij=\n2\u000fkil[2\u000fmjn\u0017mn\nkl+\u000fljm(\u0017km\nnn+\u0017nn\nkm)] and the exchange ten-\nsor \u0000\u000b\f\nij= 4[\u0003\u000b\f\nkk\u000eij\u0000\u0003\u000b\f\nij]. The dissipation (5) becomes\nG=Z\ndVdth\n~\u000b_r2+~\u0015_r\u0001fsi\n: (10)\nEqs. (8) and (10) provide a general e\u000bective descrip-\ntion of the NCAFs' dynamics. In absence of external\nforce \felds, the Lagrangian (8) is equivalent to the phe-\nnomenological theory derived from symmetry arguments\nin Ref. 36. This can be seen by minimizing the ac-\ntion with respect to m, which yields m= (a1=2a2)[I+\n\u0011=a2]\u00001_r(Iis the identity matrix). The norm of the\nmatrix\u0011=a2is small, because \u0011is proportional to the\nSOC whereas a2is linear in the strong antiferromagnetic\nexchange interaction. This implies that [ I+\u0011=a2]\u00001\u0019\n[I\u0000\u0011=a2]. Substituting the expression for mback into\nthe Lagrange density leads to\nL=\u001fij_ri_rj\u0000\u0000\u000b\f\nij@\u000bri@\frj\u0000\u0014ijrirj; (11)\nwhere\u001fij= (a2\n1=4a2)[\u000eij\u0000\u0011ij=a2]. To second order in\nr, Eq. (11) is identical to the phenomenology developed\nin Ref. 36. The STT-induced coupling terms in Eqs. (4)\nand (5) was phenomenologically derived in Ref. 17 based\non the spin conservation principle.Next, we will use Eqs. (8)-(10) to derive a general ex-\npression for the spin pumping in NCAFs. We consider\na spatial uniform driving \feld h(t) such that the spatial\nvariations of randmcan be disregarded. The NM is\nassumed to act as a perfect spin sink, implying that the\nback\row of spin from the NM to the NCAF is negligible.\nGenerally, the state of a system can be described by\na set of thermodynamic variables fqiji= 1;2;3:::g. Let\nfidenote the thermodynamic force that induces a \rux\nJiin the quantity qi. In linear response, the \ruxes are\ngiven by the equation Ji=Lijfj, where the o\u000b-diagonal\nelements of the response matrix [ Lij] are related via the\nOnsager reciprocal relations Lij=\u000fi\u000fjLji. Here,\u000fi= 1\n(\u000fi=\u00001) if the thermodynamic variable qiis even (odd)\nunder time reversal27. At constant temperature T, the\n\ruxes and forces are chosen such that the entropy genera-\ntionSis given by T_S=P\niJifi.27NCAFs are described\nby the variables randm, which under time reversal\ntransform as r7!randm7!\u0000m. The \ruxes of the\nNCAF are _rand _m, whereas the associated forces are\nfr=\u0000V@rUandfm=\u0000V@mU, respectively.28{30Note\nthat we consider a uniformly precessing NCAF. Thus,\nrandmparametrize the uniform spin state of the en-\ntire NCAF. During precessional motion of the isolated\nNCAF, the heat generation of the antiferromagnetic sys-\ntem isfr\u0001_r+fm\u0001_m\u0018T_S. In the NM,fsis proportional\nto the out-of-equilibrium spin accumulation, which leads\nto a \row of spin from the NM into the NCAF. The heat\ngeneration of this process is fs\u0001Is\u0018T_S, which im-\nplies thatfsis the thermodynamic force producing the\npure spin current Is.28,41Thus,Isandfsare the \rux\nand force of the out-of-equilibrium spin density \u001asat the\nNM/NCAF interface, which under time reversal trans-\nforms as\u001as7!\u0000\u001as. The relationship between the \ruxes\nand thermodynamic forces are given by the equation\n0\n@_r\n_m\nIs1\nA=0\n@LrrLrmLrs\nLmrLmmLms\nLsrLsmLss1\nA0\n@fr\nfm\nfs1\nA: (12)\nBased on Eqs. (8)-(10), the Onsager coe\u000ecients govern-\ning the dynamics of randmcan be derived from the\nNCAF's equations of motion. From the variational equa-\ntion\u000eS=\u000eq=\u000eG=\u000e_q(q2fr;mg), we \fndLrirj= 0,\nLrimj=\u0000\u000eij=a1V,Lrisj=\u0015\u000eij=a1,Lmirj=\u000eij=a1V,\nLmimj= 2~\u000b\u000eij=a2\n1V, andLmisj=\u0000\u001c\u000eij=a1. Here, we\nhave de\fned \u001c= (~\u0015a1+ 2~\u000b\u0015)=a1. We see that the o\u000b-\ndiagonal elements describing the dynamics of the iso-\nlated spin system satisfy the expected reciprocity rela-\ntionsLrimj=\u0000Lmjri. The coe\u000ecients LrisjandLmisj\nde\fne the STT produced by the spin accumulation. The\nOnsager reciprocal relations implies that Lsirj=\u0000Lrjsi\nandLsimj=Lmjsi, which yields the spin pumping\nIs=\u0000\u0015\na1fr\u0000\u001c\na1fm: (13)\nEq. (13) is the \frst central results of this paper and repre-\nsents a general theory for the ac spin pumping of NCAFs.4\nIn Eq. (13),fs=0because the NM acts as a perfect spin\nsink for the spin current pumped into the metallic layer.\nyy\nx\nxy\nz\nxy\nz\nxy\nzIs\nIs\nIsh\nh\nxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\n3 2 1 -1 -2 -3 23ba\ny\nxtNM z\nNM \nNCAF VUnit cell\nFIG. 2: (color online). a. A kagome AF at equilibrium.\nThe spin order is characterized by the staggered \felds L1=\n(S3\u0000S2)=p\n3S,L2= (S2+S3\u00002S1)=3S.43The thin-\flm\nkagome AF is interfaced with a NM along z.b. The spin-wave\ndispersion relation ~ !(i)= (~!(i)2\n0+c(i)~k2\nx)1=2withKz=K= 10.\nWe have introduced the dimensionless quantities c(x)= 0,\nc(y)=c(z)= 1, ~!(i)=!(i)=!(z)\n0,~kx=kx(ap\n3JS=~!(z)\n0). For\na kagome AF resonantly driven at frequencies ~ !(i)\n0byh(yel-\nlow arrows), the three spin-wave bands pump spin currents\nwith mutually orthogonal spin polarizations (red arrows).\nIII. SPIN PUMPING IN KAGOME LATTICES\nWe now apply the general theory to a thin-\flm (mono-\nlayer) NCAF with kagome structure44interfaced with a\nNM alongz(see Fig. 2a). Important examples of kagome\nAFs include Weyl semimetals and iron jarosites45,46. The\nHamiltonian of the spin system is40,47\nH=JX\nhijiSi\u0001Sj+X\ni\u0014\nKz\u0010\nSi\u0001^e(z)\u00112\n\u0000K(Si\u0001^ni)2\u0015\n:\n(14)\nHere, the \frst term represents the nearest-neighbor ex-\nchange interaction J > 0, whereas the last term is the\nanisotropy energy with Kz>0 andK > 0.^e(z)is the\nunit vector along z. Note that the three magnetic sub-\nlattices of the kagome AF experience di\u000berent in-plane\neasy axes de\fned by ^n1= [0;1;0],^n2= [p\n3=2;\u00001=2;0],\nand ^n3= [\u0000p\n3=2;\u00001=2;0]. Consequently, the groundstate of the kagome AF is given by a 120\u000eordering of the\nsublattice spins such that Si=S^ni(orSi=\u0000S^ni).\nFollowing Ref. 35, the Lagrange density (8) of the\nabove spin Hamiltonian can be microscopically derived\n(see Sec. A). For the constants in Eq. (8), we \fnd that\na1= 24~S=p\n3a2anda2= 36S2J=p\n3a, whereais the\nlattice constant. The exchange energy tensor has the\nnon-vanishing tensor elements \u0000yy\nxx= \u0000xx\nyy= \u0000xx\nzz= \u0000yy\nzz=\n4p\n3JS2=aand \u0000xy\nxy= \u0000yx\nxy=\u00004p\n3JS2=a, whereas\nfor the second rank tensors we \fnd \u0014xx=\u0014yy=K1,\n\u0014zz=K2, ~\u0014xx= ~\u0014yy=a2, and ~\u0014zz=a2+ 4p\n3KzS2=a.\nHere, we have introduced the anisotropy constants K1=\n8p\n3(Kz+K)S2=a3andK2= 16p\n3KS2=a3. Eq. (11)\nimplies that the anisotropic part of ~ \u0014ijyields a cor-\nrection on the order of \u0018Kz=Jto the spin dynam-\nics. In AFs, the exchange energy is typically much\nlarger than the anisotropy energy. In what follows,\nwe therefore disregard the anisotropic part and assume\n~\u0014ij=a2\u000eij. The thermodynamic forces in the spin\npumping expression (13) are fr(t) =\u00002VK\u0001r(t) and\nfm=\u00002a2Vm(t) +Vh(t). Here, Kis a diagonal ma-\ntrix withKxx=Kyy=K1andKzz=K2. To \fnd the\ntime-dependence of frandfm, we solve the equations\nof motion for randmin the linear response regime.\nWithfs=0, Eq. (12) yields 2 a2m=a1_r+hand\na2\n1r+ 4a2K\u0001r=\u0000a1_h\u00004a2~\u000b_r. Substitution of the\nansatzh(t) =Re[h0ei!t] andr(t) =Re[r0ei!t] into the\nequation forr, produces the stationary solutions\nri=\u0000h0;i\u0000[L(i)\ns(!) cos (!t) +L(i)\na(!) sin (!t)];(15)\nmi=h0;iA(i)\n2a2\u0014\nL(i)\ns(!) sin (!t)\u0000L(i)\na(!) cos (!t)\u0015\n;(16)\nwhereA(i)= 4\u0000( ^e(i)\u0001K\u0001^e(i))a2=a1!(i)\n0(^e(i)are the\nthree unit vectors along the x,y, andzaxes, re-\nspectively), \u0000 = 1 =2a1\u0001!with \u0001!= 2a2~\u000b=a2\n1, and\n!(x)\n0=!(y)\n0=p\n4K1a2=a2\n1,!(z)\n0=p\n4K2a2=a2\n1are\nthe resonance frequencies for the three spin wave bands\nof the kagome AF (i.e., the frequencies of the k=0\nspin waves). L(i)\ns(!) = (\u0001!)2=((\u000e!(i))2+ (\u0001!)2) and\nL(i)\na(!) =\u000e!(i)\u0001!=((\u000e!(i))2+ (\u0001!)2) are symmetric\nand antisymmetric functions of \u000e!(i)=!\u0000!(i)\n0, respec-\ntively. In arriving at the above expressions, we have ex-\npanded around the resonance frequencies and assumed\n\u0001!=!(i)\u001c1. Substituting the above stationary solu-\ntions into the thermodynamic forces in Eq. (13), we ar-\nrive at an expression for the pumped spin current48\nIs;i=h0;i[g(i)\n1(!) sin (!t)\u0000g(i)\n2(!) cos (!t)]: (17)\nHere, the frequency-dependent functions are g(i)\n1(!) =\nVA(i)[\u001cL(i)\ns(!)=a1\u0000\u0015!(i)\n0L(i)\na(!)=2a2] andg(i)\n2(!) =\nVA(i)[\u0015!(i)\n0L(i)\ns(!)=2a2+\u001cL(i)\na(!)=a1]. Eq. (17) is the\nsecond central results of this Letter and provides a theory\nfor the spin pumping of NCAFs with kagome structure.\nImportantly, we notice that the spin current reduces to5\nIs;i=VA(i)h0;i[\u001csin (!(i)\n0t)=a1\u0000\u0015!(i)\n0cos (!(i)\n0t)=2a2]\nat the resonance frequencies !(i)\n0whereL(i)\ns= 1 and\nL(i)\na= 0. This implies that the reactive (dissipative) STT\nparameter can be extracted from the in-phase (quadra-\nture) component of Iswith respect to the driving \feld.\nBy applying the driving \feld along ^e(i), the spin cur-\nrent peaks at the resonance frequency !(i)\n0. The induced\ncurrent then only contains contributions from one of the\nthree spin wave bands { the k=0spin wave with a\nfrequency of !(i)\n0. Interestingly, this makes it very easy\nto change the polarization direction of the pumped spin\ncurrent since the di\u000berent spin wave bands lead to dif-\nferent polarizations (Fig. 2b). Speci\fcally, a driving \feld\nh=h0^e(i)cos\u0010\n!(i)\n0t\u0011\ngenerates a spin current with a\nspin polarization along ^e(i). This is very di\u000berent from\nthe situation in ferromagnets and collinear AFs, where\nthe entire magnetic state must be rotated for changing\nthe polarization direction of the pumped spin current.\nA common way to detect spin pumping is to interface\nthe magnet with a NM having a large spin Hall angle.\nThe spin current injected into the normal NM-layer gen-\nerates a transverse charge current due to the ISHE, which\nproduces a measurable Hall voltage49{51. We assume the\nNM-layer to be a heavy metal with strong SOC such\nthat the back\row of spin into the magnetic layer can be\nneglected. Further, we consider the thickness tNM of\nthe NM to be much larger than the spin di\u000busion length\n\u0015sdso that the spin current vanishes completely at the\nouter edge where z=tNM. An external magnetic \feld\nh=h0cos (!t) is used to excite the magnet. A spin\ncurrentIs(given by Eq. (17)) is then pumped into the\nnormal metal. The spin current density jsthrough the\nNM layer is found by solving the spin-di\u000busion equation\n@t\u001as=D@2\nz\u001as\u0000\u001as=\u001csfwith the boundary condition\n\u0000D@z\u001as(0;t) =Is=Aat the NCAF/NM interface and\n@z\u001as(tNM;t) =0at the outer edge. Here, \u001asis the\nspin density in the normal metal, Dis the electron dif-\nfusion constant, \u001csfis the spin-\rip time, and Ais the\ncross section area of the NCAF/NM interface. The spin\ncurrent density is found from the solution of the spin den-\nsity via the relationship js=\u0000D@z\u001as, which yields50\njs(z;t) =\u0000(Is=A) sinh [(z\u0000tNM)=\u0015sd]=sinh (tNM=\u0015sd)\n(here,\u0015sd=p\nD\u001csf). The charge current density gen-\nerated by the ISHE is50,51jISHE\nc =\rH(2e=~)[^e(z)\u0002js],\nwhere\rHis the spin Hall angle. The system constitutes\nan open circuit. Thus, the de\rected charges accumulate\nat the interfaces, which induce an electric \feld that ex-\nactly canceljISHE\nc . Integrating the net current density\nover the metallic layer, one \fnds the electric \feld50,\nEISHE =\u00002e\rH\u0015sd\n~A\u001bNMtNMtanh\u0012tNM\n2\u0015sd\u0013\n[^e(z)\u0002Is];(18)\nwhere\u001bNMis the conductance of the NM-layer. We have\nhere disregarded any electric currents in the kagome AF\nlayer because it is a thin \flm. EISHE is proportional to\nthe spin current. Thus, the reactive (dissipative) STTcan be determined from the in-phase (quadrature) com-\nponent of the electric signal with respect to h(t).\nIV. CONCLUSION\nIn conclusion, we have derived a general theory for spin\npumping in NCAFs and applied the formalism to NCAFs\nwith kagome lattice. Our \fndings reveal that spin pump-\ning represents a powerful mechanism for exploring both\nthe reactive and dissipative STTs of NCAFs. We show\nthat the reactive (dissipative) part of the STT is pro-\nportional to the in-phase (quadrature) component of the\npumped spin current at resonance. Additionally, we \fnd\nthat the three spin-wave bands of the kagome AF lead to\ncurrents with spin polarizations along the x,y, andzaxis,\nrespectively. This makes it possible to orient the spin cur-\nrent along any axis by exciting di\u000berent spin-wave modes.\nThus, our work demonstrates that the spin pumping of\nNCAFs is richer and more complex than in ferromagnets\nand collinear AFs, and opens the door for tuning the spin\ncurrent's orientation through the frequency of h.\nV. ACKNOWLEDGEMENTS\nThis work received funding from the Research Council\nof Norway via the Young Research Talents Grant No.\n286889 \"Antiferromagnetic Spinmechatronics\".\nAppendix A: E\u000bective action of kagome AFs\nThe kagome AF is modeled by the spin Hamilto-\nnian (14), where the three magnetic sublattices are\nconnected by the vectors ^e1= [1=2;p\n3=2;0],^e2=\n[1=2;\u0000p\n3=2;0] and ^e3= [\u00001;0;0] (see Fig. 2a). The\norder parameter of the spin system is a rotation matrix\nR2SO(3), which de\fnes the local orientation of the ref-\nerence frame spanned by the mutually orthogonal stag-\ngered \felds.43Additionally, we introduce a vector \feld\nLrepresenting a tilting of the spins S1,S2, andS3.R\nandLboth have three degrees of freedom each. Thus, R\nandLcan parametrize all possible con\fgurations of the\nthree sublattice spins in a unit cell. The spin on the ith\nsublattice can therefore be expressed as\nSi=SR(^ni+aL)p\n1 + 2a^ni\u0001L+a2L2: (A1)\nHere,ais the lattice constant and the denominator is\nintroduced to ensure that the spin vectors are correctly\nnormalized. We consider an AF with large exchange en-\nergy, which implies that jaLj\u001c1.\nThe action governing the spin dynamics is given by\nS=X\niZ\ndtLi; (A2)6\nwhereLi=Ti\u0000Ui\u0000Us;iis the Lagrangian density of the\nspin at lattice site i. The \frst term in Lidescribes the\nkinetic energy and is given by Ti=~A(Si)\u0001_Si, where\nAis a vector potential satisfying r\u0002A(Si) =Si=S.\nUi=Hi\u0000gB\u0001Sirepresents the interaction energy,\nwhere we have included the coupling to an external mag-\nnetic \feldB.Hiis the contribution of lattice site ito\nthe Hamiltonian (14). The coupling to the spin accumu-\nlation isUs;i=~\u0015rfs\u0001Si, where\u0015rparametrizes the\nreactive STT.41Because we consider a thin-\flm NCAF,\nfsis constant and determined by the spin accumulation\nat the NM/NCAF interface. Here, and below in the case\nof the dissipative STT, we disregard e\u000bects of the SOC\nthat breaks the spin rotational symmetry of the STT.\nIn the following, we will derive an e\u000bective action\ndescribing the low-frequency, long-wavelength dynam-\nics of the kagome AF. To this end, we follow Ref. 35\nand expand the action to second order in the exter-\nnal force \feldsfB;fsgand out-of-equilibrium quantities\n(i.e.,@\u0016R(\u0016=ft;x;y;zg) andaL). Consequently, we\nonly require terms up to \frst order in aLin the spin\nvector (A1) and use the approximation:\nSi\u0019SR(^ni+\u0001i); (\u0001i\u0001^ni= 0): (A3)\nHere, \u0001i=a(L\u0000(^ni\u0001L)^ni). The total spin polariza-\ntion of a unit cell is then Stot=P3\nk=1Sk= 3aSR(TL),\nwhere we have introduced the operator T\u000b\f=\u000e\u000b\f\u0000\n(1=3)P3\nk=1nk;\u000bnk;\f. In our case, the operator is diago-\nnal with elements 2 Txx= 2Tyy=Tzz= 1.\nTo formulate a continuum model of the action in\nEq. (A2), it is convenient to \frst consider the energy\ncontribution from one unit cell (by grouping the three\nspins in each unit cell) and then sum over all unit cells.\nWe follow this approach for all the calculations below.\nFirst, we consider the kinetic energy term. The\nkinetic energy of one unit cell is given by T(\u0001)=P\nk~A\u000b[Sk]_Sk;\u000b, wherek2 f1;2;3glabels the three\nspins in the unit cell and Einstein's summation conven-\ntion is implied for the \u000b-index. Using Eq. (A3) and ex-\npanding the vector potential A(Sk) to \frst order in the\nout-of-equilibrium quantities, one \fnds\nX\nk~A\u000b[Sk]_Sk;\u000b\u0019X\nk~S[A\u000b(R^nk)\u0001(_R^nk)\u000b+\na\u000f\u000b\f\rL\u000bnk;\f(RT_R^nk)\r]:(A4)\nHere, we have (in the last term) utilized the relation-\nship between the vector potential and spin vector, and\nthe property \u000f\u000b\f\rR\u000b\u000b0R\f\f0R\r\r0=\u000f\u000b0\f0\r0of the rota-\ntion matrix. The \frst term in Eq. (A4) is a topological\nterm35that does not a\u000bect the dynamics. Therefore, we\ndisregard this term in what follows. Because the rota-\ntion matrix is orthogonal (i.e., RTR=I), the quantity\nRT_Ris antisymmetric. It can therefore be written as\n(RT_R)ij=\u0000\u000fij\u000bV\u000b, whereVx,Vy, andVzparameterize\nthe three independent matrix elements. Using this ex-\npression and summing over the three spins, the kinetic\nenergy of one unit cell becomes T(\u0001)= 3a~S(TL)\u0001V.Second, we consider interaction energy U. We start\nwith the Heisenberg exchange term H=JP\nhijiSi\u0001Sj,\nwhere the energy contribution from one unit cell is\nH(\u0001)\nex=J[Sl\n1\u0001(Sl+^e1\n3+Sl\u0000^e1\n3) +Sl\n2\u0001(Sl+^e2\n1+\nSl\u0000^e2\n1) +Sl\n3\u0001(Sl+^e3\n2+Sl\u0000^e3\n2)]: (A5)\nHere,ldenotes the position of the spin in the unit cell,\nwhereasl\u0006^e1is the neighboring lattice site connected to\nlvia the lattice vector \u0006a^ei(see Fig. 2a). Substituting\nthe gradient expansion Sl\u0006^ei\nj\u0019Sl\nj\u0006a(^ei\u0001r)Sl\nj+a2\n2(^ei\u0001\nr)2Sl\njalong with the expression (A3) into Eq. (A5),\nyield to second order in aLand the spatial gradients of\nRthe following exchange energy of a unit cell\nH(\u0001)\nex= 9a2S2J(TL)2+Vc\u0003\u000b\f\nij[@\u000bRT@\fR]ij:(A6)\nThe tensor in the last term is de\fned as\n\u0003\u000b\f\nij=\u0000(4S2J=ap\n3)[n1;in3;je1;\u000be1;\f+n2;in1;je2;\u000be2;\f+\nn3;in2;je3;\u000be3;\f], whereVc=a3p\n3=4.\nThe in-plane anisotropy energy per unit cell is given\nbyH(\u0001)\nin=\u0000P\nkK(Sk\u0001^nk)2. Using Eq. (A3), we \fnd\nto second order in the out-of-equilibrium quantities\nH(\u0001)\nin=\u0000KS2X\n\r(n\r;in\r;jn\r;kn\r;l)(RijRkl):(A7)\nHere, we have made the substitution k!\rfor the sum-\nmation index to get agreement with the indices used in\nEq. (3). Similarly, we \fnd for the out-of-plane anisotropy\nenergyH(\u0001)\nout=P\nkKz(^z\u0001Sk)2the expression\nH(\u0001)\nout=KzS2X\n\rn\r;in\r;jRziRzj+ 3a2S2Kz(^z\u0001TL)2:\n(A8)\nThe potential energy associated with the coupling of the\nspins to an external magnetic \feld is given by the Zeeman\nenergyH(\u0001)\nB=\u0000P\nkgB\u0001Sk. Considering the \feld to be\nspatially uniform, H(\u0001)\nBto second order in the external\nforce \feld and out-of-equilibrium quantities is\nH(\u0001)\nB=\u00003gaSB\u0001(TL); (A9)\nwhere we have used thatP\nkSk= 3aSR(TL). Thus,\nthe total interaction energy of a unit cell becomes U(\u0001)=\nH(\u0001)\nex+H(\u0001)\nin+H(\u0001)\nout+H(\u0001)\nB.\nLastly, we consider the interaction energy U(\u0001)\ns=P\nk~\u0015rfs\u0001Sk(per unit cell) produced by a spin ac-\ncumulation. We disregard the spatial variations in the\nspin accumulation. To second order in fsand the out-\nof-equilibrium quantities, one then \fnds\nU(\u0001)\ns= 3~aS\u0015rfs\u0001(TL): (A10)\nCombining the interaction terms and summing over\nall unit cell, the action functional becomes S=R\ndtP\n\u0001L(\u0001). Here,L(\u0001)=T(\u0001)\u0000U(\u0001)\u0000U(\u0001)\nsis7\nthe Lagrangian density of a unit cell. In order to ob-\ntain the continuous action, we take the continuum limitP\n\u0001!Rdxdydz\nVc. Since we consider a monolayer with\na thickness of Lz\u0018a, the constant Vc=aacwhere\nac=a2p\n3=4 denotes the area of the unit cell of the\n2D kagome lattice. Thus, we obtain the action S=R\ndVdtL=R\ndVdt (T\u0000U\u0000U s), where the kinetic, inter-\naction, and STT energies are given by\nT=a1\n2V\u0001m; (A11)\nU= \u0003\u000b\f\nij[@\u000bRT@\fR]ij+\u0017kl\nijRijRkl+ ~\u0014ijmimj\u0000h\u0001m;\n(A12)\nUs=\u0015m\u0001fs: (A13)\nThe kinetic term is parametrized by the constant\na1= 24~S=a2p\n3, and we have introduced the vec-\ntor \feldm=TL. Furthermore, we have de\fned\nthe tensors \u0017kl\nij= (4S2=a3p\n3)P3\n\r=1(Kzn\r;jn\r;l\u000ez;i\u000ez;k\u0000\nKn\r;in\r;jn\r;kn\r;l) and ~\u0014ij=a2\u000eij+\u0011ij, witha2=\n36S2J=ap\n3 and\u0011ij= (12KzS2=ap\n3)\u000ezi\u000ezj. The vector\nhis related to the external \feld Bbyh= 12gSB=a2p\n3,\nand the STT coupling parameter is \u0015= 12~S\u0015r=a2p\n3.\nAppendix B: Dissipation functional\nThe dissipative processes of the spin system is captured\nby the Rayleigh dissipation functional41\nG=X\niZ\ndt\u0012~\u000bG\n2(_Si)2+~\u0015d_Si\u0001(fs\u0002Si)\u0013\n;(B1)\nwhere\u000bGis the Gilbert damping and \u0015ddetermines\nthe dissipative STT. Note that works on collinear AFshave shown that additional cross-sublattice dissipative\nprocesses could play a role in cases where the AF/NM\ninterface breaks the sublattice symmetry.31,32,52\nWe \frst consider the damping term in the dissipa-\ntion functional. Substituting Eq. 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(A1) de\fnes the orientation of the reference frame\nspanned byfL1;L2;L1\u0002L2g.\n44We assume the kagome AF to be a monolayer with a thick-\nnesses of Lz\u0018a.\n45K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. A.\nNugroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama,\nS. Akebi, R. Noguchi, R. Ishii, N. Inami, K. Ono, H. Kumi-\ngashira, A. Varykhalov, T. Muro, T. Koretsune, R. Arita,\nS. Shin, T. Kondo, and S. Nakatsuji, Nature Materials 16,\n1090 (2017).\n46K. Matan, D. Grohol, D. G. Nocera, T. Yildirim, A. B.\nHarris, S. H. Lee, S. E. Nagler, and Y. S. Lee, Phys. Rev.\nLett. 96, 247201 (2006); K. Matan, B. M. Bartlett, J. S.\nHelton, V. Sikolenko, S. Mat'a\u0014 s, K. Proke\u0014 s, Y. Chen, J. W.\nLynn, D. Grohol, T. J. Sato, M. Tokunaga, D. G. Nocera,\nand Y. S. Lee, Phys. Rev. B 83, 214406 (2011).\n47C. Ulloa and A. S. Nunez, Phys. Rev. B 93, 134429 (2016).\n48Note that Eqs. (16) and (17) also contain a term \u0018\nh0cos(!t), which does not depend on the functions L(i)\ns\nandL(i)\na. However, close to the resonance frequencies this\nterm is negligible.\n49E. Saitoh, M. Ueda, and H. Miyajima, Appl. Phys. Lett.\n88, 182509 (2006)\n50O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G.\nE. W. Bauer, S. D. Bader, and A. Ho\u000bmann, Phys. Rev.\nB82, 214403 (2010).\n51H. J. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110,\n217602 (2013).\n52A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas,\nPhys. Rev. B 98, 184402 (2018)."    },    {        "title": "1903.11122v1.Nuclear_spin_assisted_quantum_tunnelling_of_magnetic_monopoles_in_spin_ice.pdf",        "content": "Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice\nC. Paulsen,1,\u0003S. R. Giblin,2E. Lhotel,1D. Prabhakaran,3K. Matsuhira,4G. Balakrishnan,5and S. T. Bramwell6\n1Institut N\u0013 eel, CNRS and Universit\u0013 e Grenoble Alpes, 38042 Grenoble, France\n2School of Physics and Astronomy, Cardi\u000b University, Cardi\u000b, CF24 3AA, United Kingdom\n3Clarendon Laboratory, Physics Department, Oxford University, Oxford, OX1 3PU, United Kingdom\n4Kyushu Institute of Technology, Kitakyushu 804-8550, Japan\n5Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom\n6London Centre for Nanotechnology and Department of Physics and Astronomy,\nUniversity College London, 17-19 Gordon Street, London, WC1H 0AJ, United Kingdom\nExtensive work on single molecule magnets has identi\fed a fundamental mode of relaxation arising\nfrom the nuclear-spin assisted quantum tunnelling of nearly independent and quasi-classical magnetic\ndipoles. Here we show that nuclear-spin assisted quantum tunnelling can also control the dynamics of\npurely emergent excitations: magnetic monopoles in spin ice. Our low temperature experiments were\nconducted on canonical spin ice materials with a broad range of nuclear spin values. By measuring\nthe magnetic relaxation, or monopole current, we demonstrate strong evidence that dynamical\ncoupling with the hyper\fne \felds bring the electronic spins associated with magnetic monopoles to\nresonance, allowing the monopoles to hop and transport magnetic charge. Our result shows how\nthe coupling of electronic spins with nuclear spins may be used to control the monopole current. It\nbroadens the relevance of the assisted quantum tunnelling mechanism from single molecular spins\nto emergent excitations in a strongly correlated system.\nINTRODUCTION\nIn the canonical dipolar spin ice materials (Dy 2Ti2O7,\nHo2Ti2O7) [1{4], rare earth ions with total angular mo-\nmentumJ= 15=2 (Dy3+) andJ= 8 (Ho3+) are densely\npacked on a cubic pyrochlore lattice of corner-linked\ntetrahedra. The ions experience a very strong h111icrys-\ntal \feld, resulting in two e\u000bective spin states ( MJ=\u0006J)\nthat de\fne a local Ising-like anisotropy. At the millikelvin\ntemperatures discussed here (0 :08 K< T < 0:2 K), a\nlattice array of such large and closely-spaced spins would\nnormally be ordered by the dipole-dipole interaction [5],\nbut the pyrochlore geometry of spin ice frustrates the\ndipole interaction and suppresses long-range order. In-\nstead, the system is controlled by an ice-rule, that maps\nto the Pauling model of water ice [1{4]. In the e\u000bec-\ntive ground state, the spins describe a \rux with closed-\nloop topology and critical correlations, that may be de-\nscribed by a local gauge symmetry rather than by a tra-\nditional broken symmetry [6]. This strongly correlated\nspin ice state is stabilised by a remarkable self-screening\nof the dipole interaction [7, 8]. Excitations out of the\nspin ice state fractionalise to form e\u000bective magnetic\nmonopoles [6, 9], but the excited states are no longer\nself-screened and this manifests as an e\u000bective Coulomb\ninteraction between monopoles. The static properties\nof spin ice are accurately described by the monopole\nmodel [1]. The dynamic properties can also be described\nby assuming an e\u000bective monopole mobility [11{13], but\nthere have been few studies of the microscopic origin of\nthe monopole motion [14].\nThe \feld and energy scales involved in monopole mo-\ntion are illustrated in Fig. 1, a-e. When a monopole\n\u0003carley.paulsen@neel.cnrs.frhops to a neighbouring site a spin is \ripped (Fig. 1a).\nFor an isolated monopole (far from any others) this\nspin \rip takes place at nominally zero energy cost (Fig.\n1b) because contributions from near-neighbour antiferro-\nmagnetic superexchange and ferromagnetic dipole{dipole\ncoupling individually cancel. The cancellation of the \feld\ncontribution relies on the dipolar self-screening [8] that\nmaps the long-range interacting system [7] to the degen-\nerate Pauling manifold of the near neighbour spin ice\nmodel [2]. This surprising cancellation is a key result\nof the many{body physics of spin ice. In practice, a\nmonopole hop may also involve a \fnite energy change\narising from longitudinal \felds at the spin site: the main\nsource of \felds is nearby monopoles [6] (Fig. 1b), while\nfurther contributions arise from corrections to the map-\nping, which give a \fnite energy spread to the Pauling\nmanifold [15] (here of order \u00180.1 K [16]). The mech-\nanism of the hop is believed to be quantum tunnelling\nand several key signatures of this have been observed in\nthe high temperature regime between 2 K and 10 K [12{\n14, 17, 18].\nAt lower temperatures ( T < 0:6 K), spin ice starts to\nfreeze [18]. This is due in part to the rarefaction of the\nmonopole gas whose density n(T) varies as\u0018e\u0000j\u0016j=T\nwhere the chemical potential j\u0016j= 4:35 and 5.7 K\nfor Dy 2Ti2O7and Ho 2Ti2O7respectively [6], and also\nin part to geometrical constraints that create noncon-\ntractable, monopole-antimonopole pairs that cannot eas-\nily annihilate [19]. These factors, which are independent\nof the monopole hopping mechanism, suggest that the re-\nlaxation rate \u0017(T)/n(T) will fall to exponentially small\nvalues at low temperature ( T < 0:35 K).\nPrevious thermal quenching experiments have demon-\nstrated monopole populations well below the nominal\nfreezing temperature that are both long lived and able\nto mediate magnetic relaxation [20]. This paradoxicalarXiv:1903.11122v1  [cond-mat.str-el]  26 Mar 20192\n(a)E (B111 = 0)△E=10-5Κ-J+J(c)\n(b)E (B111 ≠ 0 + bhf )bhf\n-J+J(e)E (B111 ≠ 0)XX-J+J(d)B111=0.81 TB111=0\n00.510P (B)\n Longitudinal Field (T)\n00.050.10.150.2\n00.511.5\n0246810\nE (K)\nSeparation (# of spins)B (T)\nFIG. 1. How magnetic monopoles tunnel in spin ice. A magnetic monopole is a many-body state that moves via the dynamics\nof local \rippable spins. (a) A qualitative schematic of the longitudinal \feld distribution ( P(B)) around a central \rippable spin\n(red), showing how the distribution is centred around zero \feld ( B= 0 T) when there is local monopole (red sphere), and\ncentred around 0.81 T when there is no monopole, which is the case for the vast majority of spins at the millikelvin temperatures\ndiscussed here. (Note that (i) the 0 T peak is greatly exaggerated to show on the same scale as the 0.81 T peak; (ii) 0.81 T\nrepresents the true \feld for Dy 2Ti2O7{ including antiferromagnetic exchange reduces the molecular \feld to about 0.43 T).\nThe broadening of the distribution arises in part from the presence of monopoles and in part from the \fnite energy spread of\nPauling states. (b) The longitudinal \feld and energy cost of a monopole hop for an isolated monopole-antimonopole pair as a\nfunction of the distance between them. (c) The resonant tunnelling process of a \rippable spin associated with a monopole; a\nlongitudinal \feld less than the tunnel splitting for an isolated spin, \u0001 E\u001910\u00005K [14], will allow tunnelling transitions between\nthe plus and minus spin states. (d) A schematic showing how monopole \felds can take the \rippable spins o\u000b-resonance, such\nthat tunnelling is suppressed. (e) If the spins are not to far o\u000b the resonance condition, then rapidly varying hyper\fne \felds\nbhffrom the precession of nuclear moments can bring otherwise blocked spins into resonance and thus relaxation continues by\ntunnelling.\nfrozen but dynamical character of the system suggests the\nrelevance of resonant magnetic tunnelling, where mag-\nnetisation reversal can only occur when the longitudinal\n\feld is smaller than the tunnelling matrix element \u0001 E.\nThe monopolar \felds may add a longitudinal component\nthat takes the spin o\u000b the resonance condition (Fig. 1c,d)\nbut in addition may add a transverse component that\nampli\fes \u0001 E: together these lead to a suppression and\ndispersion of the monopole mobility.\nIn the following, we will demonstrate experimentally\nthat hyper\fne interactions (Fig. 1e) play a signi\fcant\nrole in bringing monopoles back to their resonance con-\ndition, enabling dynamics at very low temperatures\n(T < 0:35 K).RESULTS\nSamples\nTo investigate the e\u000bect of nuclear spins on the mag-\nnetic relaxation in spin ice, we studied four spin ice sam-\nples: Ho 2Ti2O7, withI= 7=2 and three Dy 2Ti2O7sam-\nples spanning a range of nuclear spin composition from\nI= 0 toI= 5=2. Details of nuclear spins and hyper-\n\fne parameters are given in Table 1. Ho3+is a non-\nKramers ion with intrinsically fast dynamics owing to\nthe possibility of transverse terms in the single-ion spin\nHamiltonian, while Dy3+, being a Kramers ion, has in-\ntrinsically much slower dynamics. However, it should be\nnoted that, at low temperature, bulk relaxation is slower\nin Ho 2Ti2O7than in Dy 2Ti2O7, owing to its larger j\u0016j\nand hence much smaller monopole density (See Supple-\nmentary Fig. 1).3\nCompound I\u0016NA(K)BN(T)j\u0016j(K)\u0001E(K)\nHo2Ti2O7 7/24.17 0.3 0.034 5.7 1\u000210\u00005\nnatDy2Ti2O7\u001919 %,161Dy5/2-0.48\n0.0216-0.0039\n4.35 6\u000210\u00006\u001925 %,163Dy5/20.67 0.0054\n\u001966 %,otherDy 0 0 0\n162Dy2Ti2O7 0 0 0 0\n163Dy2Ti2O7 5/20.67 0.0599 0.0054\nTABLE I. Hyper\fne properties of the used materials R2Ti2O7(R= Ho, Dy). Here, Iis the nuclear spin, \u0016Nthe nuclear\nmoment,Athe hyper\fne coupling constant, BNthe e\u000bective \feld due to the nuclear moment at r= 0:5\u0017A,\u0016is the monopole\nchemical potential (conventionally negative, hence we quote j\u0016j) and \u0001Eis the tunnel splitting for a typical transverse \feld of\n0.5 T [14]. Note that about 13 % of Ti ions have a nuclear moment giving approximately 20 \u0016T \feld at the site of a Ho or Dy\nion.\nThermal protocol\nIn previous experiments we have accurately manipu-\nlated the monopole density in Dy 2Ti2O7by rapid mag-\nnetothermal cooling (Avalanche Quench Protocol, AQP)\nthe sample through the freezing transition, allowing the\ncontrolled creation of a non-equilibrium population of\nmonopoles in the frozen regime [20]. However it is more\nproblematic to cool samples containing Ho, due to the\nlarge Ho nuclear spin which results in a Schottky heat ca-\npacity of 7 J mol\u00001K\u00001at 300 mK. Indeed this anomaly\nhas been exploited by the Planck telescope where the\nbolometers are attached to the cold plate by yttrium-\nholmium feet thus allowing passive \fltering with a sev-\neral hour time constant that was crucial to the operation\nof the system [21]. For Ho 2Ti2O7this means di\u000eculty\nin cooling. Therefore during some of the runs the sam-\nple temperature was recorded via a thermometer directly\nmounted on the sample face. Fig. 2a shows the monitor-\ning of the sample temperature as it approaches equilib-\nrium for Ho 2Ti2O7and Dy 2Ti2O7during and after the\nAQP. The inset of Fig. 2a shows that only a few sec-\nonds are required to cool the samples from 0.9 K to 0.2\nK, which is well below the freezing transition. Whereas\nDy2Ti2O7continues to cool, reaching 80 mK after only\n10 s, Ho 2Ti2O7takes nearly 2000 s to reach the same\ntemperature. Hence the data shown here were taken at\n80 mK for Dy 2Ti2O7and 200 mK (and 80 mK when\npossible) for Ho 2Ti2O7.\nMonopole density\nWe have phenomenologically estimated how the\nmonopole density depends upon the rate of sample cool-\ning,dT=dt and the spin relaxation time \u001c(T) = 1=\u0017(T),\nwhich is derived from the peaks in the imaginary com-\nponent of the ac susceptibility. Di\u000berentiation of \u001c(T)\nto gived\u001c=dT and hencedT=d\u001c , allows de\fnition of an\nequilibrium cooling rate dT=d\u001c , that gives the maximum\ncooling rate that may still maintain equilibrium. Fig.\n2b compares dT=dt anddT=d\u001c for both Dy 2Ti2O7and\nHo2Ti2O7. It can be seen that after the AQP, dT=dt\nfor Ho 2Ti2O7crosses the equilibrium curve and goes outof equilibrium at \u00190:9 K, and for Dy 2Ti2O7at\u00190:72\nK. The upper limit of the monopole density at low tem-\nperature can be estimated by equating it to the theo-\nretical value at the crossing temperature: thus we \fnd\none monopole on approximately every 103tetrahedra for\nboth Ho 2Ti2O7and Dy 2Ti2O7.\nSpontaneous relaxation\nWe studied the e\u000bect of wait time twbetween the end\nof the avalanche quench and the application of the \feld\nwith the aim to determine the e\u000bect of nuclear spins on\nthe monopole dynamics. Varying the wait time deep in\nthe frozen regime allowed us to gauge the spontaneous\nevolution of the zero-\feld monopole density as a function\nof time: that is, if monopoles recombine in a time tw,\nthen the observed monopole current will be smaller, the\nlonger the wait time. Two separate experiments were\ndesigned to study these e\u000bects. In the \frst experiment\n(Fig. 3) after waiting we applied a constant \feld and\nmeasured the magnetization Mas a function of time. In\nthe second experiment (Fig. 4) we investigated the e\u000bect\nof wait time on the magnetothermal avalanches [22{24]\nthat occur on ramping the \feld to high values. Both\nof these allowed access to the magnetic current density\nJm=dM=dt . Full details of the experimental conditions\nare given in Supplementary Note 1 and Supplementary\nFig. 2.\nThe monopole current is controlled by multiple factors.\nIn the simplest model [9] there are three of these: the\nmonopole density n, the monopole mobility u(related\nto the spin tunnelling rate) and the bulk susceptibility\n\u001f. ThusJm=dM=dt =\u0017(Meq\u0000M) whereMeq=\u001fH\nis the equilibrium magnetisation and \u0017/un. In gen-\neral it is di\u000ecult to deconvolve these various factors. In\nRef. 13 it was achieved by independent measurement\nofn(T) and\u001f(T) to reveal u(T). In the present time-\ndependent experiments we cannot perform such a direct\nseparation, but by studying Dy 2Ti2O7samples with dif-\nferent isotopes, it seems reasonable to assume that the\nsusceptibility and starting density are roughly the same,\nso the variation in mobility (hop rate) will dominate dif-4\n00.511.5\n1101001000104T (K)\nt (s)wait timeµ0H (T)-0.30.3measureAQp(a)\n00.20.40.60.8\n012345T (K)tw(s)HTODTO\n00.20.40.60.81\n00.20.40.60.81dT/dt (K s-1)\nT (K)0.9 K0.72 KDTOHTOHTODTOdT/dτ (K s-1)00.20.40.81\n0.6Cu holdersamplethermometer(b)\nFIG. 2. Controlled cooling of spin ice below its freezing tem-\nperature. How the temperature of the samples varies during\nand after the AQP: (a) The applied \feld (black) during an\nAQP, and the temperatures measured by a small thermometer\nglued directly on top of the samples (schematically shown in\n(b)) vs log time for Ho 2Ti2O7(HTO, red) andnatDy2Ti2O7\n(DTO, blue). The inset shows a zoom of the \frst 6 seconds\nvs time. (b) Comparison of the sample cooling rates dT=dt as\na function of temperature after the AQP for Ho 2Ti2O7and\nnatDy2Ti2O7from the data in (a) to the equilibrium cool-\ning ratedT=d\u001c extracted from ac susceptibility data for the\ntwo samples (see Supplementary Fig. 6). The cooling rate\nfor Ho 2Ti2O7crosses the equilibrium rate at \u00180:9 K, and\nnatDy2Ti2O7at 0.72 K.\nferences between the samples. Inclusion of Ho 2Ti2O7in\nthe comparison gives a further point of reference: the\nstarting monopole densities (see above) and susceptibili-\nties for Ho 2Ti2O7are expected to be comparable to those\nof Dy 2Ti2O7, while the the tunnel splitting (which con-\ntrols the intrinsic mobility) is also estimated to be of the\nsame order [14] in the appropriate range of internal \felds\n(see Fig. 1 and Ref. 14, Fig. 5).\nFig. 3 summarizes results for the relaxation of the\nmagnetization M(t) for the di\u000berent samples, as wellas the value of M(t= 400 s) and the monopole cur-\nrentJm(t= 0) as a function of wait time, for a con-\nstant applied \feld of 0.08 T. The Dy 2Ti2O7samples\nshow a clear progression in wait time e\u000bect that cor-\nrelates strongly with their relative densities of nuclear\nspin states. Thus the monopoles recombine during the\nwait period much more e\u000bectively the larger the nuclear\nspin: that is, the larger the nuclear spin the higher the\nmonopole mobility, the faster the recombination, and the\nfewer the monopoles at the start of the measurement. In\nFig. 3e, f, higher mobility means the relaxation curves\n(M(t= 400 s) and Jm(t= 0)) shift both up and to the\nleft, so a crossover in curves is expected { and this is in-\ndeed observed at the longer times. Near to equilibrium a\nsecond crossover would be expected (i.e. the equilibrium\ncurrent density is higher for the highest mobility), but\nthis crossover is clearly very far outside our time win-\ndow. Hence our Dy 2Ti2O7samples are always far from\nequilibrium.\nThe e\u000bects observed for Dy 2Ti2O7are yet more dra-\nmatic in Ho 2Ti2O7, consistent with the Ho3+non-\nKramers character, large nuclear spin, and large hyper-\n\fne coupling. Relaxation at 200 mK covers more than\ntwo orders of magnitude but is practically extinguished\nfor long wait times, showing that excess monopoles spon-\ntaneously recombine to eliminate themselves from the\nsample. The plots indicate that the half life for monopole\nrecombination in Ho 2Ti2O7would be approximately 150\nseconds (much shorter than the equilibrium relaxation\ntime) and suggests that equilibrium in the monopole den-\nsity is reached at long times. Using the above estimate\nfor the initial monopole density n(t= 0)\u001810\u00003, we re-\ncover a nominal equilibrium density of neq= 10\u00005(per\nrare earth atom). Although this estimate is an upper\nlimit it is nevertheless far from the expected equilibrium\ndensity,n200 mK\neq\u001810\u000013(calculated by the method of\nRef. 1, see Supplementary Fig. 1). It continues to evolve\nwith temperature, being lower by a further order of mag-\nnitude atT= 80 mK (Fig. 3f). Most likely, the actual\nequilibrium monopole density is ampli\fed by defects and\ndisorder in the sample.\nMagnetothermal avalanches\nFig. 4 illustrates the e\u000bect of twon the magnetother-\nmal avalanches. These occur when the injected power\n(\u00160H\u0001Jm) overwhelms the extraction of thermal en-\nergy from the sample to the heat bath [23] such that\nmonopoles are excited in great excess as the tempera-\nture steeply rises. The faster and more abundant the\nmonopoles, the lower the avalanche \feld. To obtain the\ndata in Fig. 4, after the AQP and tw, the applied \feld\nwas swept at a constant rate, 0.02 T.s\u00001up to 0.4 T. If\nthe avalanche \feld Hava(tw), is de\fned as the \feld where\nthe magnetization crosses 1 \u0016Bper rare earth ion, (0.5\n\u0016Bfor the163Dy sample) then the di\u000berence in avalanche\n\feld \u0001Hava=Hava(tw)\u0000Hava(tw= minimum) allows us\nto compare the spread of \felds for all samples.\nFig. 4a and b show the experimental results for the iso-5\n00.020.040.060.080.10.120.14 M (µB per Dy)wait=100 swait =160 swait=553 swait= 2070 swait = 8072 s163 Dy  (µN=0.67)(b)\nT=80 mK\n0.0010.01\n101001000104 dM/dt (µB s-1 per Dy or Ho)\ntw (s)T=80 mKT=200 mKHTO162DyDTO163DyT=80 mK(f)-0.4-0.3-0.2-0.100.10.20.30.4\n0510051015µ0 H (T)\nt (s) AQpwait period(5-50,000 s)apply H and measure(d)\n00.050.10.150.2\n050100150200250300350400 M (µB per Ho)\nt (s)wait=9 swait =38 swait=99 swait= 248 swait = 7688 sHTO  (µN=4.17)T=200 mK(c)00.050.10.150.20.25 M (µB per Dy or Ho)\nHTO162DyDTO163Dy(e)\nT=80 mKT=200 mK00.020.040.060.080.1 M (µB per Dy)wait=18 swait =47 swait=168 swait= 1207 swait = 18006 s162 Dy   (µN=0)\nT=80 mK(a)\nFIG. 3. Spontaneous evolution of the monopole density during a wait time in zero applied \feld. This is gauged by the growth\nof magnetization ( M) and monopole current density ( Jm=dM=dt ) after a \feld is applied; comparison of the di\u000berent isotopic\nsamples reveals the e\u000bect of nuclear spins on the monopole mobility. (a)162Dy2Ti2O7(162Dy) and (b)163Dy2Ti2O7(163Dy),\nboth measured at T= 80 mK, and (c) Ho 2Ti2O7(HTO,T= 200 mK).natDy2Ti2O7(DTO) can be seen in Supplementary\nFig. 4. The samples were \frst prepared using the AQP protocol outlined in (d) and discussed further in Methods. After the\nspeci\fed wait periods, a \feld of 0.08 T was applied and the magnetization as a function of time was recorded. All measurements\nshown in the \fgure were made with the \feld along the [111] axis; examples for other directions are given in the Supplementary\nFigs. 10 and 11. (e) Plot of the value of the magnetization Mobtained after the \frst 400 seconds for the three samples shown\nto the left, and fornatDy2Ti2O7vs log wait time. Note that the magnetization values at 400 s remain far from the expected\nequilibrium value. (f) The monopole current Jm=dM=dt att= 0 for the three samples shown to the left vs log wait time.\nAlso shown is the monopole current fornatDy2Ti2O7, and the monopole current for Ho 2Ti2O7measured at 80 mK.\ntopically enriched Dy 2Ti2O7samples at 80 mK, demon-\nstrating a very clear pattern. In general the spread of\nHava(tw) becomes larger, the larger the nuclear spin,\nshowing again that the nuclear spins strongly enhance\nthe monopole mobility. Thus, the162Dy sample (no nu-\nclear spin) shows negligible evolution of the position of\nthe avalanche \feld. FornatDy2Ti2O7(shown in Supple-\nmentary Fig. 4) the e\u000bect is small, while for the163Dysample (maximum nuclear spin) the e\u000bect of twcan be\nclearly seen as a steady progression of Hava(tw) to higher\n\felds for increasing twdue to the smaller initial monopole\ndensity at the start of the \feld ramp. Also shown in the\n\fgure are the curves that result from slow conventional\nzero \feld cooling (CC) from 900 mK to 80 mK (at 1\nmK.s\u00001) followed by a 1000 s wait period. For the162Dy\nandnatDy2Ti2O7samples the CC avalanche \feld is o\u000bset6\n00.040.080.120.16\n101001000104Δ µ0 H (T)\ntw (s)HTO\nDTO163Dy162Dy(e)-0.4-0.200.20.4\n051005101520µ0 H (T)\nt (s) AQpwait period(10-50,000 s)ramp H and measure(d)\n00.511.522.533.5 M (µB per Dy)CCwait=20swait=56,000s162Dy (µN=0)80 mK(a)\n00.511.52\nwait=16,000swait=30sCC M (µB per Dy)163Dy (µN=0.67)80 mK(b)\n00.511.522.533.5\n00.050.10.150.20.250.30.350.4 M (µB per Ho)µ0 H (T)CCwait=5.2swait=30,000sHTO  (µN=4.17)200 mK(c)\nFIG. 4. Wait time and isotope dependences of magnetothermal avalanches. This gives further evidence of the e\u000bect of nuclear\nspins on the monopole mobility. Avalanches of the magnetization were recorded while the \feld was ramped at 20 mT s\u00001for\n(a)162Dy2Ti2O7(162Dy) and (b)163Dy2Ti2O7(163Dy), both measured at T= 80 mK, and (c) Ho 2Ti2O7(HTO, measured at\nT= 200 mK). Magnetothermal avalanches fornatDy2Ti2O7(DTO) can be seen in Supplementary Fig. 4. The samples were\n\frst prepared using the AQP and then followed by various wait times (as outlined in (d) and discussed in methods) except\nfor the curves marked CC, where the sample was \frst prepared using the conventional zero \feld cooled protocol (red squares).\nAlso shown for each of the samples is the equilibrium Mvs\u00160Htaken at 900 mK (solid black dots). All measurements shown\nin the \fgure were made with the \feld along the [111] axis; examples for other directions are given in the Supplementary Figs.\n9, 10 and 11. (e) Plot of di\u000berence in avalanche \feld \u0001 Hava=Hava(tw)\u0000Hava(tw= minimum) against log wait time for the\ndata shown in the left as well asnatDy2Ti2O7(DTO, see Supplementary Fig. 4).\nto higher \felds, well outside the distribution of Hava(tw).\nFor the163Dy sample the CC curves falls within the dis-\ntribution but near the long wait time curves. Also, we\nnote for163Dy, which happens to have better thermal\ncontact, and thus faster cooling during the AQP, the CC\ncurve again falls outside the distribution (shown in Sup-\nplementary Fig. 10 b). Thus slow cooling is more ef-\n\fcient at approaching equilibrium in Dy 2Ti2O7than is\nthe AQP cooling followed by a long tw, especially for the\nlow nuclear moment samples. This is typical behaviour\nfor frustrated or disordered systems because slow cool-\ning allows the system time to explore all available phase\nspace.\nFig. 4c shows a much greater e\u000bect of twfor\nHo2Ti2O7with a larger spread of \felds, saturating near\n0.32 T for the the longest tw. This is again consistentwith the conclusion that that the larger the nuclear spin\nmoment, the more e\u000bective the spontaneous monopole\nrecombination. The measurements were performed pri-\nmarily at 200 mK, but the same conclusion follows from\nmeasurements at 80 mK. Ho 2Ti2O7also exhibits some\nunusual behaviour suggesting that the monopole density\nand magnetization do not approach equilibrium in a sim-\nple way. First, the magnetization jumps fall short of the\nMvsHequilibrium curve taken at 900 mK, even though\nthermometers placed on the sample indicate that the\nsample does indeed heat above 900 mK (see Supplemen-\ntary Note 2 and Supplementary Fig. 3 for more details).\nSecondly, in contrast to the behaviour of Dy 2Ti2O7dis-\ncussed above, the CC curve of Ho 2Ti2O7falls in the mid-\ndle of the distribution of Hava(tw) indicating, unusually,\nthat waiting long enough at low temperature is an equally7\ne\u000ecient way of approaching equilibrium as slow cooling.\nDISCUSSION\nThe experimental result demonstrated here is that\nmagnetic monopole dynamics in the frozen regime of spin\nice are greatly enhanced by the hyper\fne coupling of the\nelectronic and nuclear moments. We now argue that this\nobservation \fnds a natural { albeit surprising { expla-\nnation by analogy with the properties of single-molecule\nmagnets [25]. These are metal-organic clusters with large\ncomposite spins: some of the most studied include the\nso called Mn 12and Fe 8systems, both of which can be\nthought of as an ensemble of identical, weakly interact-\ning nanomagnets of net spin S = 10 with an Ising-like\nanisotropy. The degenerate Ms=\u0006Sstates are split\nby the ligand electric \feld into a series of doublets. At\ntemperatures smaller than the level separation, the spins\n\rip by resonant tunnelling through a quasi-classical bar-\nrier. The signature of a resonant tunnelling e\u000bect in Fe 8\nis a peak in the low temperature relaxation rate around\nH= 0 [26]. It quickly became clear that to understand\nthe resonant tunnelling both dipolar and dynamic nu-\nclear spin contributions to the interactions need to be\naccounted for. The typical dipolar \feld in such a sys-\ntem is\u00190:5 K, and the relevant tunnel splitting \u0001 E\nof the order 10\u00008K, meaning that a broad distribution\nof dipolar \feld and a static hyper\fne contribution would\nforce all the spins o\u000b resonance. Prokof'ev and Stamp [4]\nproposed that dynamic nuclear \ructuations can drive the\nsystem to resonance, and the gradual adjustment of the\ndipole \felds in the sample caused by tunnelling, brings\nother clusters into resonance and allows a continuous re-\nlaxation. Hence the observation of relaxation in single\nmolecule magnets is fundamentally dependent on the hy-\nper\fne coupling with the \felds of nuclear spins [28].\nThe Prokof'ev and Stamp model [4] certainly does not\napply in detail to spin ice at low temperatures. First, in\nsingle molecule magnets the spin of any particular com-\nplex in the system is available to be brought to reso-\nnance, whereas in spin ice, only those spins that are in-\nstantaneously associated with a di\u000busing monopole are\navailable to tunnel (and this presumes that more ex-\ntended excitations can be neglected). The remaining\nspins { the vast majority { are, in contrast, static and\ninstantaneously ordered by the ice rules. The rate of\n\ripping of these quasi-ordered spins, which corresponds\nto monopole pair creation, is negligible at the tempera-\ntures studied and the process is not relevant to our ex-\nperiments. Thus, even at equilibrium, spin ice has an\ne\u000bective number of \rippable spins that depends on tem-\nperature (see Supplementary Fig. 1). Away from equilib-\nrium, where our experiments are performed, the number\nof \rippable spins in spin ice further depends on time, with\nmonopole recombination depleting their number. In ad-\ndition, it seems reasonable to assume that the reduction\nof the density of monopoles is even more important dur-ing the relaxation process; as monopoles move through\nthe matrix magnetizing the sample they will annihilate\nwhen they encounter a monopole of opposite charge, or\nbecome trapped on a defect or on the sample surface.\nThis feature of spin ice is a second important di\u000berence\nwith single molecule magnets, as modelled in Ref. 27.\nA third di\u000berence relates to the distribution of internal\n\felds in the system. In spin ice only, the actual \feld asso-\nciated with a \rippable spin, both before and after a \rip,\nis a monopolar \feld. Flipping a spin transfers a monopole\nfrom site to site (Fig. 1a), dragging the monopolar \feld\nwith it: a \feld that is much stronger and of longer range\nthan any conventional dipole \feld. However, the change\nin \feld on a spin \rip is dipolar, as in single molecule\nmagnets.\nIn short, the \rippable spins in spin ice are really an\naspect of the emergent monopole excitation rather than\na perturbed version of an isolated (composite) spin as as-\nsumed for the single molecule magnets in Ref. 4. Yet de-\nspite this di\u000berence, it seems reasonable to suggest that\nthe basic idea of Ref. 4 does apply to spin ice. The\nlongitudinal monopolar \felds will take \rippable spins o\u000b\nresonance (Fig. 1c-e), while the transverse ones will tend\nto broaden the resonance well beyond the tunnel split-\nting calculated for an isolated spin. i.e. \u0001 E= 10\u00005K\n[14]. An applied \feld can also take \rippable spins on\nor o\u000b resonance or broaden the resonance, depending on\nits direction. Nevertheless, in zero applied \feld, at very\nlow temperatures we would expect all \rippable spins as-\nsociated with isolated monopoles to be o\u000b resonance and\nhence unable to relax, unless they are brought back to\nresonance by a combination of the monopole \felds and\nthe \ructuating nuclear spins: nuclear assisted \ripping of\nspins will then bring further spins to resonance via the\nchange in dipolar \felds, as in the Prokof'ev-Stamp pic-\nture [4]. Our experimental results for the wait time de-\npendence of various properties clearly support this propo-\nsition: in zero \feld (during tw) the sample with no nu-\nclear spin is scarcely able to relax its monopole density,\nwhile the larger the nuclear spin, the quicker the relax-\nation. For \rippable spins associated with closely{spaced\nmonopole-antimonopole pairs the situation is slightly dif-\nferent. Although they are strongly o\u000b-resonance (Fig.\n1b), the decreasing transition matrix elements will be\ncompensated by the increasing Boltzmann factors re-\nquired for detailed balance. Also, for the \fnal recom-\nbination, a favourable change in exchange energy will\nreduce the \feld required to bring spins to resonance (see\ncaption, Fig. 1).\nWe note in passing that the di\u000berences between sin-\ngle molecule magnets and spin ice are also evident in our\ndata. Speci\fcally, a t1=2initial relaxation of the mag-\nnetisation is a property of single molecule magnets, with\nthet1=2form arising from the dipole interactions [4, 29].\nGiven the very unusual \feld distribution in spin ice, and\nthe complicating factor of monopole recombination, as\ndescribed above, it is hardly likely that this functional\nform will apply. We test for a t1=2decay in the Supple-8\nmentary Fig. 5 and con\frm that it can only be \ftted\nover a narrow time range: to calculate the true time de-\npendence in spin ice poses a theoretical problem.\nOur main result has implications for both the the-\nory of spin ice and the theory of nuclear spin assisted\nquantum tunnelling. First, in previous work [11] we\nhave shown how the low-temperature quenched monopole\npopulations of Dy 2Ti2O7obey the nonlinear and non-\nequlibrium response of monopole theory [30] that was\ndeveloped assuming a single hop rate. In view of our\n\fndings, the theory should apply most accurately to the\nDy2Ti2O7sample with no nuclear spins and least accu-\nrately to Ho 2Ti2O7where the hyper\fne splitting ener-\ngies are of a similar order to the Coulomb energies. In\nother measurements, presented in Supplementary Figs.\n7 and 8, we con\frm that this is the case; hence a gen-\neralisation of the theory of Ref. 30 to include the e\u000bect\nof nuclear spins seems an attainable goal. We also note\nthat Ho 2Ti2O7o\u000bers the unusual situation that, at low\ntemperatures ( <0.35 K) and su\u000ecient wait times, the\nnuclear spins are ice-rule ordering antiparallel to their\nelectronic counterparts; hence spin ice o\u000bers a rare chance\nto investigate the e\u000bect of correlation on nuclear spin as-\nsisted quantum tunnelling in a controlled environment.\nPerhaps this will shed light on some of the unusual prop-\nerties particular to Ho 2Ti2O7, as noted above.\nSpin ice thus exempli\fes a remarkable extension of the\nconcept of nuclear spin assisted quantum tunnelling [4]\nto the motion of fractionalised topological excitations [6].\nThis is made possible by the fact that the emergent ex-\ncitations of the system { the monopoles { are objects lo-\ncalised in direct space that move through \ripping spins.\nAs well as illustrating this generic point, our result may\nalso have practical consequences. We have established\nhow coupling with nuclear spins controls the magnetic\nmonopole current and the spectacular magnetothermal\navalanches: hence any experimental handle on the nu-\nclear spins of the system would also be a rare experimen-\ntal handle on the monopole current. Any future applica-\ntion of magnetic monopoles in spin ice will surely rely on\nthe existence of such experimental handles.\nMETHODS\nSamples. Single crystals were grown by the \roating\nzone method for all samples, the naturalnatDy2Ti2O7and\nHo2Ti2O7samples (DTO, HTO) were prepared at the Insti-\ntute of Solid State Physics, University of Tokyo, Japan, and\n162Dy2Ti2O7,163Dy2Ti2O7at Warwick University and Ox-\nford University respectively.\nMeasurements. Measurements were made using a low tem-\nperature SQUID magnetometer developed at the Institut N\u0013 eel\nin Grenoble. The magnetometer is equipped with a miniature\ndilution refrigerator with a base temperature of 65 mK. The\nfast dynamics after a \feld change were measured in a relative\nmode, the slower measurements were made by the extraction\nmethod, and the initial relative measurements were adjusted\nto the absolute value extraction points. The \feld could berapidly changed at a rate up to 2.2 T s\u00001.\nFor all the data shown here the \feld was applied along the\n[111] crystallographic direction. Measurements were also per-\nformed perpendicular to the [111] direction, as well as along\nthe [001] and [011] directions and on a polycrystalline sam-\nple, examples of which are discussed in Supplementary Note\n3. In total ten di\u000berent samples were studied. The direction\nof the applied \feld as well as di\u000berences in the sample shapes\nand thermal contact with the sample holder can e\u000bect some\nof the details of the measurements. However this does not\nchange the main conclusion of the paper: the demonstration\nof the importance of nuclear assisted quantum tunnelling to\nthe relaxation.\nThe measurements of temperature vs time shown in Fig.\n2, a bare-chip Cernox 1010-BC resistance thermometer from\nLakeShore Cryogenics was wrapped in Cu foil and glued on\ntop of the sample as shown in the inset of Fig. 2b.\nCooling Ho 2Ti2O7was di\u000ecult and warming was also\ntricky using the AQP, depending on the initial temperatures\nand wait times. Therefore to ensure the sample was heated\nabove 900 mK, two AQP were used, separated by 300 seconds,\nwhich explains why the starting temperature for Ho 2Ti2O7\nwas higher in Fig. 2 (see Supplementary Note 2 and Supple-\nmentary Fig. 3 for further discussions).\nA schematic of the AQP used for the preparation of the\nsamples is shown in Fig. 3d. First a \feld of \u00000:3 T was\napplied and the sample was allowed to cool to base tempera-\nture for 20 minutes. The \feld was then reversed at 2.2 T s\u00001\nto +0:3 T for 4 s then reduced to zero. After a wait period\nranging from 10 to 50,000 s, a \feld of 0.08 T was applied and\nthe relaxation of the magnetization was recorded. The \feld\nB= 0:08 T was chosen because it is large enough to get size-\nable relaxation, but small compared to the avalanche \felds\nshown in Fig. 4. In this way, when applying the magnetic\n\felds, the relaxation is well behaved and the sample does not\nheat.\nThe AQP used for the data of Fig. 4 was similar to the\nabove, except the avalanche \feld was \u00060:4 T. After the wait\nperiod the \feld was ramped at 0.02 T s\u00001while the mag-\nnetization and temperature of the sample were continuously\nrecorded. For the slow CC protocol measurements shown in\nFig. 3, the samples were \frst heated to 900 mK for 10 s, then\ncooled at a rate of approximately 0.01 K s\u00001, followed by a\nwaiting period of 1000 s.\nDATA AVAILABILITY\nInformation on the data underpinning the results\npresented here, including how to access them, can\nbe found in the Cardi\u000b University data catalogue at\nhttp://doi.org/10.17035/d.2019.0069144874.\nThe datasets obtained and/or analyzed in this study are also\navailable from the corresponding author on reasonable re-\nquest.9\n[1] Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske\nT. & Godfrey, K. W. Geometrical frustration in the ferro-\nmagnetic pyrochlore Ho 2Ti2O7.Phys. Rev. Lett. 79, 2554\n{ 2557 (1997).\n[2] Bramwell, S. T. & Harris, M. J. Frustration in Ising-type\nspin models on the pyrochlore lattice. J. Phys.: Condens.\nMatter 10, L215 - L220 (1998).\n[3] Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan,\nR. B. & Shastry, S. Zero-point entropy in spin ice. Nature\n399, 333 { 335 (1999).\n[4] Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frus-\ntrated magnetic pyrochlore materials. Science 294, 1495\n{ 1501 (2001).\n[5] Luttinger, J. M & Tisza, L. Theory of dipole interactions\nin crystals. Phys. Rev. 70, 954 (1946).\n[6] Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic\nmonopoles in spin ice. Nature 451, 42{45 (2008).\n[7] den Hertog, B. C. & Gingras, M. J. P. Dipolar interactions\nand origin of spin ice in Ising pyrochlore magnets. Phys.\nRev. Lett. 84, 3430 (2000).\n[8] Isakov, S. V., Moessner, R. & Sondhi, S. L. Why spin ice\nobeys the ice rules Phys. Rev. Lett. 95, 217201 (2005).\n[9] Ryzhkin, I. A. Magnetic relaxation in rare-earth py-\nrochlores. J. Exp. and Theor. Phys. 101, 481{486 (2005).\n[10] Kaiser, V., Bloxsom, J. A., Bovo, L., Bramwell, S. T.,\nHoldsworth, P. C. W. & Moessner, R. Emergent Electro-\nchemistry in Spin Ice: Debye{H uckel Theory and Beyond.\nPhys. Rev. 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G., den Hertog, B. C. & Gingras, M. J. P. Long\nrange order at low temperatures in dipolar spin ice. Phys.\nRev. Lett. 87, 067203 (2001).\n[17] Ehlers, G., Cornelius, A. L., Fennell, T. Koza, M.,\nBramwell, S. T. & Gardner, J. S. Evidence for two distinct\nspin relaxation mechanisms in `hot' spin ice Ho 2Ti2O7.J.\nPhys.: Condens. Matter 16, S635{S642 (2004).\n[18] Snyder, J., Ueland, B. G., Slusky, J. S., Karunadasa, H.\nCava, R. J. & Schi\u000ber, P. Low-temperature spin freezing\nin the Dy 2Ti2O7spin ice. Phys. Rev. B 69, 064414 (2004).\n[19] Castelnovo, C., Moessner, R. & Sondhi, S. L. Thermal\nquenches in spin ice. Phys. Rev. Lett. 104, 107201 (2010).\n[20] Paulsen, C., Jackson, M. J., Lhotel, E., Canals, B., Prab-\nhakaran, D., Matsuhira, K., Giblin, S. R. & Bramwell, S.T. Far-from-equilibrium monopole dynamics in spin ice.\nNature Physics 10, 135{139 (2014).\n[21] Planck Collaboration. Planck Early Results. II. The ther-\nmal performance of Planck, Astronomy & Astrophysics\n536(2011). doi: 10.1051/0004-6361/201116486.\n[22] Slobinsky, D., Castelnovo, C., Borzi, R. A., Gibbs, A.\nS., Mackenzie, A. P., Moessner, R. & Grigera, S. A. Un-\nconventional Magnetization Processes and Thermal Run-\naway in Spin-Ice Dy 2Ti2O7.Phys. Rev. Lett. 105, 267205\n(2010).\n[23] Jackson, M. J., Lhotel, E., Giblin, S. R., Bramwell, S.\nT., Prabhakaran, D., Matsuhira, K., Hiroi, Z., Yu, Q.\n& Paulsen, C. Dynamic behavior of magnetic avalanches\nin the spin-ice compound Dy 2Ti2O7.Phys. Rev. B 90,\n064427 (2014).\n[24] Krey, C., Legl, S., Dunsiger, S. R., Meven, M., Gardner,\nJ. S., Roper, S. R. & P\reiderer, C. First Order Metamag-\nnetic Transition in Ho 2Ti2O7Observed by Vibrating Coil\nMagnetometry at Milli-Kelvin Temperatures. Phys. Rev.\nLett.108257204.\n[25] Gatteschi, D. & Sessoli, R. Quantum Tunneling of Mag-\nnetization and Related Phenomena in Molecular Materi-\nals.Angewandte Chemie 42, 268 { 297 (2003).\n[26] Sangregorio, C., Ohm, T., Paulsen, C., Sessoli, R. and\nGatteschi, D. Quantum tunneling of the magnetization in\nan Iron cluster nanomagnet. Phys. Rev. Lett. 78, 4645 {\n4648 (1997).\n[27] Prokof'ev, N. V. & Stamp, P. C. E. Low-temperature\nquantum relaxation in a system of magnetic nano\nmolecules. Phys. Rev. Lett. 80, 5794 (1998).\n[28] Wernsdorfer, W., Caneschi, A., Sessoli, R., Gatteschi, D.,\nCornia, A., Villar, V., & Paulsen, C. E\u000bects of Nuclear\nSpins on the Quantum Relaxation of the Magnetization\nfor the Molecular Nanomagnet Fe 8.Phys. Rev. Lett. 84,\n2965 { 2968 (2000).\n[29] Ohm, T., Sangregorio, C. & Paulsen, C. J. Low Temp.\nPhysics 113, 1141 { 1146 (1998).\n[30] Kaiser, V., Bramwell, S. T., Holdsworth, P. C. W. &\nMoessner, R. ac Wien E\u000bect in Spin Ice, Manifest in\nNonlinear, Nonequilibrium Susceptibility. Phys. Rev. Lett.\n115, 037201 (2015).\nAcknowledgements: S.R.G. thanks Cardi\u000b University\nfor `seedcorn' funding, and acknowledges the EPSRC\nfor EP/L019760/1. S.T.B. thanks Patrik Henelius for\ncommunicating his independent ideas on nuclear assisted\nquantum tunnelling in spin ice, and acknowledges the\nEPSRC for EP/S016465/1. E.L. and C.P. acknowledge\n\fnancial support from ANR, France, Grant No. ANR-\n15-CE30-0004. G.B. wishes to thank \fnancial support\nfrom EPSRC, UK, through grant EP/M028771/1.\nAuthor Contributions: The experiments were de-\nsigned and performed by C.P. with inputs and discussions\nfrom E.L. and S.R.G. The data were analyzed by C.P.,\nE.L., S.R.G., and S.T.B. Contributed materials were fab-\nricated by K.M., D.P. and G.B. The paper was written\nby C.P., E.L., S.R.G., and S.T.B.10\nNuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice\nSupplementary Information\n10-1310-1110-910-710-50.0010.1\n00.20.40.60.81n\nT (K)DTOHTO\nSupplementary Figure 1.Density of single-charge monopoles (equilibrium number per diamond lattice site in\nzero \feld) versus temperature , calculated by the Debye-H uckel theory of Kaiser et al. [S1]. The analytic calculation is very\naccurate for the monopole model of spin ice: it includes both single and double-charge monopoles, but only the single-charge\nmonopoles are relevant at the temperatures we study. The density of `\rippable' spins per spin site is 3 =2 times the monopole\ndensity.\nSupplementary Note 1. Experimental details\nCreating a large density of monopoles using the avalanche quench protocol in Ho 2Ti2O7.We have\npreviously described the avalanche quench protocol (AQP) in detail for Dy 2Ti2O7(see Ref. S2 and its Supplementary\nInformation). From magnetisation measurements recorded during and after the AQP, we inferred that samples of\nDy2Ti2O7heat systematically to temperatures above 900 mK, even though the reference thermometer on the sample\nholder only registered a small jump [S3].\nHowever from the onset, it was clear that Ho 2Ti2O7was di\u000berent. For example, when performing measurements\nwhere the \feld is ramped at a steady rate, the magnetic avalanches of Ho 2Ti2O7never reach the 900 mK equilibrium\nvalue as seen in Fig. 4c. Sometimes depending on previous measurements, the AQP worked very poorly, or did not\nseem to work at all.\nThis was the motivation for measuring the sample temperature directly by mounting a thermometer on the samples\nduring some of the runs.\nTemperature measurements during the AQP. We attempted direct temperature measurements with 4 di\u000berent\nthermometers; 2 homemade RuO 2resistance thermometers (\fled down to reduce mass with wires attached with silver\nepoxy), a Cernox 1010-SD thermometer with platinum leads, and a bare-chip Cernox 1010-BC, both from LakeShore\nCryogenics. All had short comings but in the end most of the measurements shown here were made using the bare\nchip thermometer. This was the lightest of the four, but had a small but noticeable magnetoresistance that we have\ncorrected for. A constant current source delivered 10 nA, and the voltage was measured with a Stanford Instruments\nmodel 830 lock-in ampli\fer running at 1100 Hz. This setup was a compromise, the measurements of the temperature\nwere fast, but prone to some drift and noise.\nDuring normal measurements samples are sandwiched between two long narrow pieces of Cu that are anchored to\nthe mixing chamber of a miniature dilution refrigerator. The samples are glued in place, then te\ron tape is tightly\nwrapped around the Cu strips, clamping the samples to the Cu. For measurements with the thermometer glued\nto the sample, only one Cu strip was used, the second was suspended away from the thermometer, as shown in\nSupplementary Figure 2.11\nSupplementary Figure 2.Pictures of the sample mounting . Upper left: Sample of Dy 2Ti2O7and Cernox bare chip\nresistor with leads protected by kapton tape. Upper right: Ho 2Ti2O7glued onto the bare chip resistor with thin Cu foil\nprotruding. Lower left: the sample + thermometer have been turned over, and the sample surface has been glued onto the\nCu sample holder using GE varnish. The Cu foil has been folded back to cover the upper half of the sample and bare chip\nresistor. Lower right: stando\u000bs hold upper part of sample holder away, te\ron tape has been wrapped around the sample and\nthermometer. The thermometer is isolated from the Cu sample holder by the sample.\nSupplementary Note 2. Results\nAs already discussed, cooling samples that contain Ho can be problematic, and this is also true for heating samples\nwith the AQP when the starting temperature was well below 200 mK. This is shown in Supplementary Figure 3 for\na series of AQP taken on Ho 2Ti2O7when the sample was \frst cooled to 65 mK after waiting 4 hours. Point (a) is\nthe beginning of sequence when we applied a \feld of \u00000:3 T on the sample followed by a wait period of 180 s. For\nDy2Ti2O7, already a large >1 K spike in temperature would be seen at (a) as the sample rapidly magnetizes in the\n\feld, but for Ho 2Ti2O7only a small jump of about 0.3 K was recorded. The jump in temperature at the \frst AQP is\nalso small, only reaching about 0.6 K, compared to >1:4 K for Dy 2Ti2O7under similar conditions. At (c) we begin a\nsecond AQP, again setting \u00160H=\u00000:3 T, and waiting 180 s. But this time the jump in temperature reaches nearly\n0.8 K, and the second AQP at (d) shows that now, the sample has warmed >1:3 K.\nDuring the AQP, heat from the \ripping of the electronic spins is absorbed by the sample. But because of the\nlarge heat capacity of the Ho nuclei, much of the energy is absorbed by the nuclear spin bath, raising its temperature\nbut resulting in a small overall jump in sample temperature. However for the second AQP, the starting sample\ntemperature is now greater, nearly 0.16 K, and this is enough to heat the spins above one Kelvin.\nThus Ho 2Ti2O7measurements were systematically made with 2 or 3 AQP in succession in order to ensure the\nsample is warmed above 1 K.\nNote that the need for several AQP also suggests that at low temperature, well below 300 mK, the nuclear spins\nbegin to freeze out, and anti-align with their respective electronic spin, thus 2 in { 2 out for the electronic spin becomes\n2 out { 2 in for its nuclear counterpart.\nAnother nagging problem was that the samples of Ho 2Ti2O7did not reach M= 0 after the AQP. The origin of\nthis is not clear, but our data suggests that Ho 2Ti2O7cools too fast. When the \feld is switched o\u000b, the applied\n\feldHgoes to zero before the sample even starts to change its magnetisation. The sample then feels the internal\n\feldHinternal =\u0000D\u0001Mwhere D is the demagnetisation factor and avalanches against this. As the magnetisation\ndecreases,Hinternal also decreases, the sample heats, but then cools so rapidly that the magnetisation gets `stuck' at\na small positive value of the order 1 emu/g. For convenience, the solution was to add a small overshoot for the \feld\nof about\u00000:004 T for 1 s, then switch back to H= 0. This resulted in a starting Mclosest to zero. Note that\nmeasurements without the overshoot gave the same results, but with an o\u000bset. This ultra rapid cooling may also\nexplain why the avalanches shown in Fig. 4c for Ho 2Ti2O7(while ramping of the \feld) fall below the equilibrium\nvalue expected for 900 mK, in contrast to Dy 2Ti2O7.12\nSupplementary Figure 3.The applied \feld and the sample temperature for Ho 2Ti2O7as a function of running\ntime for a double AQP. The sample was cooled for 4 hours to base temperature of approximately 65 mK. (a) At t= 0, the\n\feld was changed from 0 to \u00000:3 T. A relatively weak jump in the temperature can be seen. (b) At t= 180 s, the \frst AQP\nis performed: the \feld goes from \u00000:3 to +0:3 T, then after 4 s from +0 :3 T to zero. The temperature on the sample reaches\nabout 0.55 K, not su\u000ecient to randomize the spins. (c) At t= 200 s the \feld is again put at \u00000:3 T in preparation for the\nnext AQP. This time the jump in sample temperature is larger, but still less than required. (d) At t= 380 s the second AQP\ntakes place, warming the sample above 1 K.\n00.511.522.533.5\n00.050.10.150.20.250.30.350.4M (µB / Dy)\nµ0H  (T)DTO80 mK\nwait=22000swait=10sCC(a)\n00.511.522.53\n0.150.160.170.180.19M (µB /Dy)\nµ0H  (T)DTO80 mK\nwait=22000swait=10sCC(b)\n00.020.040.060.080.10.120.14\n050100150200250300350400M (µB / Dy)\ntime (s)wait=17 swait=68 swait=488 swait=3847 swait=30727 sDTO80 mK(c)\nSupplementary Figure 4.Plots for natural Dy 2Ti2O7corresponding to Figs. 3 and 4 of the main manuscript.\n(a) Avalanches of the magnetisation recorded while the \feld was ramped at 0.02 T/s fornatDy2Ti2O7(DTO). The samples\nwere \frst prepared using the AQP and then followed by various wait times except for the curve marked `ZFC', where the\nsample was \frst prepared using the conventional zero \feld cooled (CC) protocol (red circles). Also shown is the equilibrium\nMvs\u00160Htaken at 900 mK (solid black dots). (b) Magni\fcation, showing the spread in avalanche \felds as a function of the\nwait time, and the ZFC far outside the pack. (c) The e\u000bect of wait time on the relaxation of the magnetisation Mvs time for\nnatDy2Ti2O7measured at 80 mK. The samples were again \frst prepared using the AQP. After the speci\fed wait periods, a\n\feld of 0.08 T was applied and the magnetisation as a function of time was recorded.13\nSupplementary Figure 5.Time dependence of the magnetisation at short times . The same data of Fig. 3 in the\nmain text as well as data for thenatDy2Ti2O7sample plotted againstp\ntime. An important prediction from Prokofev and\nStamp [S4] was that the initial relaxation of the magnetisation should follow a square root time dependence. This worked well\nfor the SMM Fe 8up to 1000 s or more. The situation for spin ice is quite di\u000berent, the right hand side of the \fgure shows the\ndata can at best be \ftted over a very restricted range in time only up to 1 s.14\n0.00010.011100104\n0.40.50.60.70.80.911.11.2τ (s)\nT (K)DTOHTO\nSupplementary Figure 6.Experimental values of relaxation time \u001cfrom susceptibility and magnetisation\nmeasurements. Relaxation time \u001cvs temperature for Ho 2Ti2O7andnatDy2Ti2O7. The green and blue data points ( \u001cless\nthan 100 s) were taken from the peaks in the imaginary susceptibility. The red points for Ho 2Ti2O7come from analyzing dc\nrelaxation (all raw data was \frst corrected for demagnetisation e\u000bects). The slope dT=d\u001c de\fning the equilibrium cooling rate\nshown in Fig. 2 are taken from \fts to these curves.\nSupplementary Figure 7.E\u000bect of monopole current Jm=dM=dt on wait time for Ho 2Ti2O7left: Monopole\ncurrentJm=dM=dt obtained from the relaxation of the magnetisation of Ho 2Ti2O7measured after di\u000berent waiting times,\nand measured at 800 Oe. Jmis obtained by extrapolating the derivative of the magnetisation with respect to time at t= 0\n(See Ref. S2 for the detailed procedure). right: Jmvs 1=Tobtained from the saturation value of the left \fgure, i.e. when the\ncurrent value does not depend anymore on the waiting time.15\n\u0011\u000f\u0011\u0011\u0011\u0012\u0011\u000f\u0011\u0011\u0012\u0011\u000f\u0011\u0012\u0011\u000f\u0012\n\u0011\u0016\u0012\u0011\u0012\u0016\u0013\u0011\u0013\u0016\u0014\u0011\u0014\u0016\u0015\u0011E.\u0010E5\u0001\u0001\tµ#\u0010T\nTRSU\t)\n\u0001\u0001\u0001\t0F\u0012\u0010\u0013\n\u0012\u0017\u0013%Z%50\n\u0011\u000f\u0011\u0011\u0011\u0012\u0011\u000f\u0011\u0011\u0012\u0011\u000f\u0011\u0012\u0011\u000f\u0012\n\u0016\u0012\u0011\u0012\u0016\u0013\u0011\u0013\u0016\u0014\u0011\u0014\u0016\u0015\u0011TRSU\t)\n\u0001\u0001\u0001\t0F\u0012\u0010\u0013\nE.\u0010E5\u0001\u0001\tµ#\u0010T\n\u0012\u0017\u0014%Z)50\n\u0011\u000f\u0011\u0011\u0011\u0012\u0011\u000f\u0011\u0011\u0012\u0011\u000f\u0011\u0012\u0011\u000f\u0012\n\u0016\u0012\u0011\u0012\u0016\u0013\u0011\u0013\u0016\u0014\u0011\u0014\u0016\u0015\u0011\u0015\u0016E.\u0010E5\u0001\u0001\tµ#\u0010T\nTRSU\t)\n\u0001\u0001\u0001\t0F\u0012\u0010\u0013\n)50\u0001\u0001\u0001\"2Q\u0001\f\u0001XBJU\u001e\u0014\u0017\u0011T;'$\u0001\f\u0001XBJU\u0001\u001e\u0015\u0014\u0011T\"2Q\u0001\f\u0001XBJU\u001e\u0012\u0013\u0011\u0011T\nSupplementary Figure 8.E\u000bect of the nuclear spins on the monopole current . Monopole current Jm=dM=dt vsp\nHdetermined fornatDy and162Dy2Ti2O7samples (left),163Dy2Ti2O7and Ho 2Ti2O7samples (middle), and in di\u000berent\ncooling conditions for the Ho 2Ti2O7sample.Jmis obtained by extrapolating the derivative of the magnetisation with respect\nto time at t= 0 (See Ref. S2 for the detailed procedure). Natural and162Dy (no nuclear spin) Dy 2Ti2O7follows thep\nH\nbehavior expected for magnetic monopoles interacting through the Coulomb force,163Dy2Ti2O7and Ho 2Ti2O7samples do not,\nwhatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the\nidealised emergent chemical kinetics of monopole theory does not apply in163Dy2Ti2O7and Ho 2Ti2O7, where the dynamics\nis strongly a\u000bected by the nuclear spin e\u000bects, because the hyper\fne splitting energies are of a similar order to the Coulomb\nenergies.\n\u0011\u000f\u0012\u0017\u0011\u000f\u0012\u0017\u0016\u0011\u000f\u0012\u0018\u0011\u000f\u0012\u0018\u0016\u0011\u000f\u0012\u0019\u0011\u000f\u0012\u0019\u0016\u0011\u000f\u0012\u001a\n\u0011\u0016\u0011\u0011\u0011\u0012\u0001\u0012\u0011\u0015\u0012\u000f\u0016\u0001\u0012\u0011\u0015\u0013\u0001\u0012\u0011\u0015\u0013\u000f\u0016\u0001\u0012\u0011\u0015! µ\u0011\u0001)\u0001\t5\nwait time (s)%50\u0001\u0004\u0012%50\u0001\u0004\u0013\u0001SVO\u0001\u0012%50\u0001\u0004\u0013\u0001SVO\u0001\u0013\tB\n\u0011\u0011\u000f\u0011\u0011\u0012\u0011\u000f\u0011\u0011\u0013\u0011\u000f\u0011\u0011\u0014\u0011\u000f\u0011\u0011\u0015\u0011\u000f\u0011\u0011\u0016\u0011\u000f\u0011\u0011\u0017\n\u0011\u0016\u0011\u0011\u0011\u0012\u0001\u0012\u0011\u0015\u0012\u000f\u0016\u0001\u0012\u0011\u0015\u0013\u0001\u0012\u0011\u0015\u0013\u000f\u0016\u0001\u0012\u0011\u0015! µ\u0011\u0001)\u0001\t5\nwait time (s)%50\u0001\u0004\u0012%50\u0001\u0004\u0013\u0001SVO\u0001\u0012%50\u0001\u0004\u0013\u0001SVO\u0001\u0013\tC\nSupplementary Figure 9.The e\u000bect of wait time twon the magneto-thermal avalanches for two di\u000berent\nsamples of natural DTO, and for two di\u000berent runs (a) shows the value of the avalanche \feld Hava(tw), de\fned as\nthe \feld where the magnetization crosses 1 \u0016Bper rare earth ion. (b) is a plot of the di\u000berence in avalanche \feld \u0001 Hava=\nHava(tw)\u0000Hava(tw=minimum ).\nSupplementary Note 3. Di\u000berent Samples and Measuring Directions\nDuring the course of this study, 10 di\u000berent samples were measured with some samples measured along multiple axis.\nThe mass of the samples ranged between 3 to 40mg, and they had various shapes. No corrections for demagnetization\ne\u000bects have been taken into account for the results presented in the main text. This is because for most of the\nmeasurements shown in the main text, the sample was far from equilibrium, and the magnetization was very small,\nand thus the demagnetizing \feld -NM was small. Nevertheless the sample shape and \feld direction do e\u000bect the\nobserved relaxation curves and the avalanche \felds.\nIn this section we show that although there are some variations between samples, between cooling runs, and for\ndi\u000berent directions, these di\u000berences do not change the main conclusion of the paper; the demonstration that nuclear\nassisted quantum tunneling is operative regardless of \feld direction.16\nSupplementary Figure 10. Measurements made on a polycrystalline sample of163Dy2Ti2O7(sample 2) The\nsample was \frst prepared using the same avalanche quench protocols (AQP) outlined in the main text and methods section,\nfollowed by various waiting times. (a) shows the e\u000bects of wait time on the relaxation of the magnetization in a \feld of 0.8 T\nat 80mK. (b) shows the e\u000bects of wait time on the position of the avalanche \feld when the \feld is ramped from 0 to 0.4 T at a\nconstant rate of 0.02 T/s at 80mK. Also shown in the \fgure is the conventional zero \feld cooling (CC) curve were the sample\nwas slowly cooled from 900 mK to 80 mK (at 1 mK/s) followed by a 1000 s wait period. For this sample the CC avalanche\n\feld is o\u000bset to higher \felds, and is well outside the distribution of Hava(tw).\nSupplementary Figure 9(a) shows the e\u000bect of twon the magneto-thermal avalanches for two di\u000berent samples of\nnatural DTO, and for two di\u000berent runs. Sample 1 (also shown in the main text) was rectangular shaped parallelepiped\nand sample 2 was a square thin platelet shaped sample. The measurements shown in the \fgures were taken with the\n\feld along the [111] axis for both samples. The left panel shows the value of the avalanche \feld Hava(tw), de\fned\nas the \feld where the magnetization crosses 1 \u0016Bper rare earth ion. The curves are clearly o\u000bset from one another,\neven the two curves taken on the same sample, but during di\u000berent runs. The initial position of the avalanche \feld\nis very sensitive to thermal contact with the sample holder. For sample 2 run 1, the thermal contact was made using\ntwo Cu bands with the sample sandwiched between the two. For sample 2 run 2 only one Cu band was used, thus\nthe thermal contact was worse. The better thermalized sample has a higher avalanche \feld, because leading up to\nthe avalanche heat could be more e\u000eciently evacuated from the sample. Supplementary Figure 9(b) is a plot of the\ndi\u000berence in avalanche \feld \u0001 Hava=Hava(tw)\u0000Hava(tw=minimum ). As can be seen, for sample 2, the two runs\ncollapse onto one another, but the shape of the curve for sample 1 is slightly di\u000berent. A more systematic study needs\nto be made to understand if this is a shape dependent e\u000bect, or sample dependent.\nFor163Dy2Ti2O7, two samples were studied. Sample 1 was an odd shaped disk. The [111] direction was perpen-\ndicular to the surface of the disk, and resulted in a very large demagnetization factor for this direction. Importantly\nthis resulted in di\u000eculty thermalizing the sample along this direction to our Cu sample holder. This resulted in a\nmuch less e\u000ecient AQP cooling. We estimate that the sample took about 5 seconds to cool below 500mK, and about\n40 seconds to cool below 100mK. This is much slower than the usual AQP as described in Fig. 2 of the main text,\nbut still faster than the CC method. Sample 1 was measured along the [111] direction (shown in the main text)\nand perpendicular to the [111] direction. Sample 2 was a polycrystalline sample and the e\u000bects of wait time on the\nrelaxation of the magnetization and position of the avalanche \feld are shown in Supplementary Figure 10 (a) and (b).\nThese data sets are very similar to those presented in the main text in terms of the strength of the e\u000bect of wait time\nfor163Dy2Ti2O7. (see Figures 3 and 4)\nHowever there are two interesting di\u000berences.\nFirstly, Supplementary Figure 11 (a) shows the monopole current Jm=dM=dt att= 0 vs log wait time for sample\n1 [111] and perpendicular to [111] as well as for polycrystalline sample 2. As can be seen in the \fgure, although\nthe slopes of the three curves are roughly the same, the [111] data fall signi\fcantly below the two perpendicular\ncurves. Most likely this is not an intrinsic e\u000bect, but comes from the poor thermalization for the [111] sample run:\nas mentioned above, for this direction after the AQP the sample cooled much slower, therefore the initial monopole17\n\u0011\u000f\u0011\u0011\u0012\u0011\u000f\u0011\u0012\u0011\u000f\u0012\n\u0012\u0011\u0012\u0011\u0011\u0012\u0011\u0011\u0011\u0012\u0011\u0015\u0001E.\u0010EU\u0001\tµ#\u0010T\u0010%Z\u0001PS\u0001)P\n wait time (s)5\u001e\u0013\u0011\u0011N,)50\u0012\u0017\u0014%Z\tB\n5\u001e\u0019\u0011N,\n<\u0012\u0012\u0012>\u0001\t\u0004\u0012\n<\u0012\u0012\u0011>\u0001\t\u0004\u0013\n<\u0012\u0012\u0012>\u0001\t\u0004\u0012\n<QFSQ\u0001\u0012\u0012\u0012>\u0001\t\u0004\u0012\nQPMZ\u000eDSZTUBM\u0001\t\u0004\u0013\n\u0011\u0011\u000f\u0011\u0016\u0011\u000f\u0012\u0011\u000f\u0012\u0016\u0011\u000f\u0013\n\u0011\u0016\u0011\u0011\u0011\u0012\u0001\u0012\u0011\u0015\u0012\u000f\u0016\u0001\u0012\u0011\u0015\u0013\u0001\u0012\u0011\u0015)50\n\u0012\u0017\u0014%Z%50<\u0012\u0012\u0012>\u0001\t\u0004\u0012\n<\u0012\u0012\u0011\u0001>\u0001\t\u0004\u0012\n<\u0011\u0011\u0012>\u0001\t\u0004\u0014\n<\u0011\u0011\u0012>\u0001\t\u0004\u0013\n<\u0012\u0012\u0012>\u0001\t\u0004\u0012\nQPMZ\u000eDSZTUBM\u0001\t\u0004\u0013\n<QFSQ\u0001\u0012\u0012\u0012>\u0001\t\u0004\u0012\n! µ\u0011\u0001)\u0001\t5\nwait time (s)\tC\nSupplementary Figure 11. comparison of di\u000berent samples and di\u000berent measuring directions for HTO,\n163Dy2Ti2O7andnatDy2Ti2O7(DTO) (a) the monopole current Jm=dM=dt att= 0 vs log wait time for two samples of\n163Dy2Ti2O7and three samples of HTO. (b) Plot of di\u000berence in avalanche \feld \u0001 Hava=Hava(tw)\u0000Hava(tw=minimum )\nagainst wait time for various directions and various samples of HTO,163Dy2Ti2O7andnatDy2Ti2O7(DTO). The top 4 curves\nin the \fgure are measurements for 3 di\u000berent samples of HTO (squares). Sample 1 was a needle shaped sample measured along\nthe [111] (long) direction. Sample 2 was also needle shaped and measured along the [001] direction. Sample 3 was a square\nplatelet, and was measure along the [001] and [110] axis. The middle 3 curves are for two di\u000berent samples of163Dy2Ti2O7(solid\ndots) measured at 80mK. Sample 1 was measured along the [111] direction and perpendicular to the [111] direction, and sample\n2 was a poly-crystal. The bottom two curves are for natural DTO (triangles) taken on two di\u000berent samples along the [111]\nand [001] directions. (same data as shown in Supplementary Figure 9 (b))\ndensity at the beginning of the wait period was much reduced, so the initial monopole current was less, shifting\nthe [111] curve down in the plot. A second di\u000berence that can be seen in Supplementary Figure 10 (b) is that the\navalanche \feld for the CC method occurs at much higher \felds and is well outside the distribution of curves obtained\nby the AQP method. The same result was found for sample 1 perpendicular to [111]. This can be contrast to the\ndata shown for the [111] sample in the main text, and again can be explain by the slower cooling for the [111] sample\nrun.\nThe results for measurements on 3 di\u000berent samples of HTO are also shown in Supplementary Figure 11. Sample\n1 was a needle shaped sample measured along the [111] (long) direction. Sample 2 was also needle shaped and\nmeasured along the [001] direction. Sample 3 was a square platelet, and was measure along the [001] and [110] axis.\nSupplementary Figure 11 (a) shows the monopole current Jm=dM=dt att= 0 vs log wait time for sample 1 [111]\n(also in the main text) compared to sample 2 [110]. The e\u000bect of wait time on the currents for these two samples\nare very similar; the rate at which monopoles recombine is seen to be much faster than that of163Dy2Ti2O7, and\nboth seem to saturate at very long wait times. Supplementary Figure 11 (b) are plots of di\u000berence in avalanche \feld\n\u0001Hava=Hava(tw)\u0000Hava(tw=minimum ) against log wait time for the three samples of HTO, as well as two samples\nof163Dy2Ti2O7and for two samples ofnatDy2Ti2O7(DTO) (same data as shown in Supplementary Figure 9 (b)).\nS1. Kaiser, V., Bloxsom, J. A., Bovo, L., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. Emergent Electrochemistry\nin Spin Ice: Debye{H uckel Theory and Beyond. Phys. Rev. B 98, 144413 (2018).\nS2. Paulsen, C., Giblin, S. R., Lhotel, E., Prabhakaran, D., Balakrishnan, G., Matsuhira, K., and Bramwell S. T. Experi-\nmental signature of the attractive Coulomb force between positive and negative magnetic monopoles in spin ice. Nature\nPhysics 12, 661 (2016).\nS3. Jackson, M. J., Lhotel, E., Giblin, S. R., Bramwell, S. T., Prabhakaran, D., Matsuhira, K., Hiroi, Z., Yu, Q., and\nPaulsen, C. Dynamic behavior of magnetic avalanches in the spin-ice compound Dy 2Ti2O7.Phys. Rev. B 90, 064427\n(2014).\nS4. Prokof'ev, N. V. & Stamp, P. C. E. Low-temperature quantum relaxation in a system of magnetic nano molecules.\nPhys. Rev. Lett. 80, 5794 (1998)."    },    {        "title": "2403.08621v1.Spin_resolved_counting_statistics_as_a_sensitive_probe_of_spin_correlation_in_transport_through_a_quantum_dot_spin_valve.pdf",        "content": "arXiv:2403.08621v1  [cond-mat.mes-hall]  13 Mar 2024Spin-resolved counting statistics as a sensitive probe of s pin correlation in\ntransport through a quantum dot spin valve\nGuanjian Hu,1Shikuan Wang,2Jing Hu,1RuiQiang Li,1Yiying Yan,1and JunYan Luo1,∗\n1Department of Physics, Zhejiang University of Science and Te chnology, Hangzhou 310023, China\n2Department of Physics, Hangzhou Dianzi University, Hangzh ou 310018, China\nWe investigate the noise in spin transport through a single q uantum dot (QD) tunnel coupled\nto ferromagnetic electrodes with noncollinear magnetizat ions. Based on a spin-resolved quantum\nmaster equation, auto- and cross-correlations of spin-res olved currents are analyzed to reveal the un-\nderlying spin transport dynamics and characteristics for v arious polarizations. We ﬁnd the currents\nof majority and minority spins could be strongly autocorrel ated despite uncorrelated charge trans-\nfer. The interplay between tunnel coupling and the Coulomb i nteraction gives rise to an exchange\nmagnetic ﬁeld, leading to the precession of the accumulated spin in the QD. It strongly suppresses\nthe bunching of spin tunneling events and results in a unique double-peak structure in the noise of\nthe net spin current. The spin autocorrelation is found to be susceptible to magnetization align-\nments, which may serve as a sensitive tool to measure the magn etization directions between the\nferromagnetic electrodes.\nPACS numbers:\nI. INTRODUCTION\nThe manipulation of the spin degrees of freedom lies\nat the heart of spintronics and spin-based information\nprocessing [1–10]. Signiﬁcant eﬀorts have been devoted\nto explore new spin-based functional devices with high\nperformance and eﬃciency [11–14]. Among them, a\nnanoscale spin valve is widely considered as an essential\ncandidate building block in spintronic devices [15–25]. It\nis of great importance to investigate its transport prop-\nerties, where the spin-resolved correlations have a vital\nrole to play.\nSpin valves could be constructed in layered het-\nerostructures, which are typically characterized by the\npresence of a strong tunnel magnetoresistance eﬀect\n[26, 27]. In recent years, these systems have especially\nbeneﬁted from the progress of discovering suitable 2D\nmagnetic materials [28–30]. A quantum dot (QD) sys-\ntem is another ideal platform for spin valve devices, with\nthe uniqueadvantageofprecisemanipulationand control\nof individual spins. This is attributed to the rapid devel-\nopment of nanofabrication, which enables accurate con-\nﬁnement of single electrons and their accumulated spins\nin QDs [31–37] due to Coulomb blockade and spin block-\nade, respectively.\nIn the pioneering experiments, spin blockade takes\nplacedueto ablockingtriplet state[38]in adoublequan-\ntum dot [39–43]. Instead, we investigate the intriguing\nnonequilibrium spin on a single QD spin valve, where\nthe spin blockade mechanism is ascribed to the spin se-\nlection and ﬁltering between spins in the QD and the\nnoncollinearly polarized ferromagnetic (FM) electrodes.\nThe magnitude and direction of the accumulated spin on\nQD is determined by two processes. First, the tunnel\n∗Electronic address: jyluo@zust.edu.cncoupling to the FM electrodes gives rise to a competi-\ntion mechanism between injection and decay of spin on\nthe QD. Second, the interplay between the Coulomb in-\nteraction inside the QD and the tunnel coupling to the\nFM electrodes gives rise to an exchange (eﬀective) mag-\nnetic ﬁeld [44–49], leading to spin precessional dynamics\ninside the QD. This results in a prominent negative dif-\nferential conductance in the charge current [50–52] and\nremarkable suppression of the low frequency charge cur-\nrentnoise[53]. However,theexchangeﬁeldisreminiscent\nof spin torque in magnetic structures, which translatesto\na net spin angular momentum transfer or a spin current\nbetween the QD and FM electrodes. It is thus appeal-\ning to investigate the spin current noise characteristics,\ninstead of its charge counterparts.\nSpin current noise, due to the discreteness of the spin\ncarrier, is a measure of the correlations between spin\ntransfer [54]. It is able to provide additional information\nabout the spin dynamics and spin transfer processes, dif-\nferent from their charge behaviors. Recent studies have\nshown that the spin current noise could be generated\neven in the absence of a net charge current [55, 56]. Fur-\nthermore, spin current ﬂuctuations are predicted to be\nable to sensitively explore the eﬀect of spin-ﬂip scatter-\ning [57–61], and intriguinglyprobe repulsive orattractive\ninteractions in QD transport systems [62, 63].\nInthiswork,weinvestigateexclusivelythespincurrent\nand its noise correlations in transport through a spin-\nvalve, where the setup is composed of a single QD sand-\nwiched between two noncollinearly polarized FM elec-\ntrodes, asschematicallyshownin Fig.1. Basedonaspin-\nresolved quantum master equation, we obtain individual\nspin-resolved currents, as well as their autocorrelations\nand cross-correlations. Our analysis takes fully into ac-\ncount the interplay between spin injection and spin pre-\ncessional dynamics. In case of perpendicular alignment,\nit is found that even for low bias with almost an empty\nQD, the autocorrelations of the majority and minority2\nFIG. 1: (a) Schematics of the QD spin valve system: A QD\nis tunnel coupled to the left and right FM electrodes, whose\nmagnetizations directions are mLandmR, respectively. A\nbias voltage V=VL−VRis applied accross the device, leading\nto spin and charge transport through the system. (b) The\nspin quantization axis of the QD, ez, which is chosen to be\nperpendicular to the plane spanned by mLandmRenclosing\nan angle θ∈[0,180◦]. Theexandeyare deﬁned according\ntoex≡mL+mR\n|mL+mR|andey≡mL−mR\n|mL−mR|, respectively.\nspin currents exhibit opposite dependence on polariza-\ntion, although the charge transfers independently. At a\nmedium bias, the exchange ﬁled strongly suppresses the\nbunching of spin tunneling events and results in a unique\ndouble-peak structure in the net spin current noise. Fur-\nthermore, the spin current autocorrelations show a strik-\ning diﬀerence for perpendicular and antiparallel align-\nments, which may serve as a sensitive tool to measure\nthe magnetization directions between the two FM elec-\ntrodes.\nThe rest of the paper is organized as follows. In Sec.\nII, we introduce the single QD spin valve setup with the\ncorresponding Hamiltonian. It is then followed by the\nintroduction of the spin-resolved full counting statistics\n(FCS) in Sec. III. Section IV is devoted to the discussion\nofspin-resolvedcurrentsandtheirnoiseforvariouspolar-\nizations and magnetization directions in the electrodes.\nFinally, we summarize our work in Sec. V.\nII. MODEL SYSTEM\nThe QD spin valve system is schematically shown in\nFig.1, where the single QD is tunnel coupled to the left\nand right FM eletrodes. The Hamiltonian of the entire\nsystemHTotalreads\nHTotal=HL+HR+HQD+HTL+HTR.(1)\nThe ﬁrst two terms describe the left and right FM\nelectrodes, which are modeled as reservoirs of nonin-\nteracting electrons. According to the Stoner model of\nferromagnetism[64], there is ﬁnite asymmetry in the den-\nsityofstatesformajority[ Dℓ+(ω)]andminority[ Dℓ−(ω)]\nspins in the electrode ℓ∈ {L,R}. Without loss of gener-\nality, the quantization axis for the electron spins in each\nelectrode is chosen along the direction of the majority\nspins. The magnetizations of the two FM electrodes are\nnot necessarily to stay in the same direction and nor-\nmally enclose an angle θ∢(mL,mR), see Figs.1(a) and(b). For simplicity, the densities of states are approx-\nimated to be energy independent in the following, i.e.,\nDℓ±(ω)→ Dℓ±. The degree of spin polarization thus\ncan be characterized by pℓ= (Dℓ+−Dℓ−)/(Dℓ++Dℓ−),\nwherepℓ= 0 corresponds to a nonmagnetic electrode,\nandpℓ= 1 a half-metallic electrode with majority spins\nonly. The corresponding Hamiltonian of the FM elec-\ntrodes thus read\nHB=/summationdisplay\nℓ=L,RHℓ=/summationdisplay\nℓkνεℓkνa†\nℓkνaℓkν, (2)\nwhereaℓkν(a†\nℓkν) is the annihilation (creation) operator\nfor an electron with momentum kof majority ( ν= +) or\nminority ( ν=−) spin in the FM electrode ℓ={L,R}.\nEach electrode is in thermal equilibrium and is charac-\nterized by the Fermi function fℓ(ω) ={1+eβℓ(ω−µℓ)}−1,\nwhereβℓ= (kBTℓ)−1is the inverse temperature and µℓ\nthe chemical potential of the electrode ℓ. The diﬀerence\nin the chemical potentials deﬁnes the bias voltage across\nthe two electrodes, i.e., V=µL−µR. Hereafter, we\nchoose/planckover2pi1=e= 1, unless stated otherwise.\nThe Hamiltonian HQDdescribes the single QD, whose\nexplicit form depends on the choice of the spin quanti-\nzation axis. Here we chose neither mLnormR, but the\nz-axis perpendicular to the plane spanned by mLand\nmR. The unit vectors exandeyare deﬁned according\ntoex≡mL+mR\n|mL+mR|andey≡mL−mR\n|mL−mR|, respectively, as\nshown in Fig.1(b). The Hamiltonian of the single QD\nthus reads\nHQD=/summationdisplay\nσ=↑,↓εc†\nσcσ+Uc†\n↑c↑c†\n↓c↓, (3)\nwherecσ(c†\nσ) is the annihilation (creation) operator for\nan up- (σ=↑) or a down-spin ( σ=↓) electron, εis the\nspin-degenerate energy level of the single QD, and Uis\nthe Coulomb energy cost for double occupation.\nElectron tunneling between the FM electrodes and QD\nis described by the tunneling Hamiltonians HTLand\nHTR. In the spin quantization axis as shown in Fig.1(b),\nHTLis given by\nHTL=1√\n2/summationdisplay\nk/braceleftbigg\ntLk+a†\nLk+(e+iθ\n4c↑+e−iθ\n4c↓)\n+tLk−a†\nLk−(−e+iθ\n4c↑+e−iθ\n4c↓)/bracerightbigg\n+H.c.,(4)\nwheretLkνis the tunneling amplitude between the QD\nand the left electrode. The tunneling between the right\nelectrode and the QD HTRcan be obtained simply by\nthe replacement L →R andθ→ −θ. The corresponding\ntunnel coupling strengths are characterizedby the intrin-\nsic tunneling widths Γ ℓ±(ω) = 2π/summationtext\nk|tℓk±|2δ(ω−εℓkν),\nwhich will be approximated to be energy independent in\nthe wide band limit, i.e., Γ ℓ±(ω) = Γℓ±.3\nIII. SPIN-RESOLVED FULL COUNTING\nSTATISTICS\nTo keep track of spin and charge transport between\nthe QD and the FM electrodes, we utilize the power-\nful FCS and introduce a group of counting ﬁelds χ=\n(χL+,χL−,χR+,χR−) associated with transfer of spin\n“+”or“−”throughtheleft (L)orright(R) junctions. In\nthe Fock state of the QD: |0/an}bracketri}ht,| ↑/an}bracketri}ht,| ↓/an}bracketri}ht, and|d/an}bracketri}ht, standing\nfor no extra electron, an extra spin- ↑electron, an ex-\ntra spin- ↓electron, and double occupation respectively,\nthe reduced density matrix can be expressed in a col-\numn vector ρ(χ) = (ρ00,ρ↑↑,ρ↓↓,ρdd,ρ↑↓,ρ↓↑)T, where\nthe diagonal element ραα≡ /an}bracketle{tα|ρ|α/an}bracketri}htdenotes the proba-\nbility of ﬁnding the QD in the state |α/an}bracketri}ht(α=0,↑,↓, d),\nand the oﬀ-diagonal elements ρ↑↓=ρ∗\n↓↑≡ /an}bracketle{t↑ |ρ| ↓/an}bracketri}htstand\nfor the quantum coherences. The other oﬀ-diagonal ele-\nments between states with diﬀerent electron numbers are\ndynamically decoupled and thus not included.The central quantity of the FCS is the spin-resolved\ncumulant generating function (CGF) F(χ,t), which is\ndeﬁned via [65]\neF(χ,t)=trS[ρ(χ,t)] =ρ00(χ,t)+ρ11(χ,t)+ρdd(χ,t),\n(5)\nwhere tr S[···] stands for the trace over the QD degrees\nof freedom and ρ11(χ,t) =ρ↑↑(χ,t) +ρ↓↓(χ,t) is the\nprobability to ﬁnd the QD occupied by one electron re-\ngradless of spin orientations. With the knowledge of the\nspin-resolved CGF F(χ,t), various spin-resolved cumu-\nlants can be evaluated simply by taking partial deriva-\ntive with respect to the corresponding counting ﬁelds.\nEquation(5)motivatesustoinvestigatethespin-resolved\nprobabilities ρ00(χ,t),ρ11(χ,t), andρdd(χ,t). Underthe\nusualsecond-orderBorn-Markovapproximation,theyare\nfound to satisfy\n˙ρ00=−2/summationdisplay\nℓ=L,Rγ+\nℓρ00+/summationdisplay\nℓ=L,R(K+\nℓ++K+\nℓ−)γ−\nℓρ11\n+2/summationdisplay\nℓ=L,R(K+\nℓ+−K+\nℓ−)γ−\nℓcos(θ\n2)Sx+2[(K+\nL+−K+\nL−)γ−\nL−(K+\nR+−K+\nR−)γ−\nR]sin(θ\n2)Sy, (6a)\n˙ρ11=2/summationdisplay\nℓ=L,R(K−\nℓ++K−\nℓ−)γ+\nℓρ00−/summationdisplay\nℓ=L,R(γ−\nℓ+ ˜γ+\nℓ)ρ11+2/summationdisplay\nℓ=L,R(K+\nℓ++K+\nℓ−)˜γ−\nℓρdd\n+2/summationdisplay\nℓ=L,Rpℓ(˜γ+\nℓ−γ−\nℓ)cos(θ\n2)Sx+2[pL(˜γ+\nL−γ−\nL)−pR(˜γ+\nR−γ−\nR)]sin(θ\n2)Sy, (6b)\n˙ρdd=/summationdisplay\nℓ=L,R(K−\nℓ++K−\nℓ−)˜γ+\nℓρ11−2/summationdisplay\nℓ=L,R˜γ−\nℓρdd−2/summationdisplay\nℓ=L,R(K−\nℓ+−K−\nℓ−)˜γ+\nℓcos(θ\n2)Sx\n−2[(K−\nL+−K−\nL−)˜γ+\nL−(K−\nR+−K−\nR−)˜γ+\nR]sin(θ\n2)Sy, (6c)\nwhere we have introduced K±\nℓν=κℓνe±iχℓνandκℓ±=\nDℓ±/(Dℓ++Dℓ−) for simplicity. Apparently, the proba-\nbilities are coupled to each other due to tunneling be-\ntween the QD and the FM electrode, where the tun-\nneling rates are given by γ±\nℓ= Γℓf±\nℓ(εℓ) and ˜γ±\nℓ=\nΓℓf±\nℓ(εℓ+U), withf+\nℓ(ω) =fℓ(ω) the usual Fermi func-\ntion and f−\nℓ(ω) = 1−fℓ(ω). Furthermore, it is foundthat these probabilities are also coupled to the average\nspin on the QD\nSx=ρ↑↓+ρ↓↑\n2, Sy=iρ↑↓−ρ↓↑\n2, Sz=ρ↑↑−ρ↓↓\n2,\nwhich are described by4\n˙Sx=/summationdisplay\nℓ=L,R(K−\nℓ+−K−\nℓ−)γ+\nℓcos(θ\n2)ρ00+1\n2/summationdisplay\nℓ=L,Rpℓ(˜γ+\nℓ−γ−\nℓ)cos(θ\n2)ρ11\n−/summationdisplay\nℓ=L,R(˜K+\nℓ−˜K+\nℓ)˜γ−\nℓcos(θ\n2)ρdd−/summationdisplay\nℓ=L,R(˜γ+\nℓ+γ−\nℓ)Sx−(BL−BR)sin(θ\n2)Sz, (6d)\n˙Sy=[(K+\nL+−K+\nL−)γ+\nL−(K+\nR+−K+\nR−)γ+\nR]sin(θ\n2)ρ00+1\n2[pL(˜γ+\nL−γ−\nL)−pR(˜γ+\nR−γ−\nR)]sin(θ\n2)ρ11\n−[(K+\nL+−K+\nL−)˜γ−\nL+(K+\nR+−K+\nR−)˜γ−\nR]sin(θ\n2)ρdd−/summationdisplay\nℓ=L,R(˜γ+\nℓ+γ−\nℓ)Sy+(BL+BR)cos(θ\n2)Sz,(6e)\n˙Sz=(BL−BR)sin(θ\n2)Sx−(BL+BR)cos(θ\n2)Sy−/summationdisplay\nℓ=L,R(γ−\nℓ+ ˜γ+\nℓ)Sz. (6f)\nIt is apparent that the average spin is coupled to the\nprobabilities. These terms are responsible for the build-\ningupofspinintheQD.However,thetunnelingcoupling\nalso leads to a decay of the average spin, with the decay\nrate given by γdec=/summationtext\nℓ=L,R(γ−\nℓ+ ˜γ+\nℓ). Intriguingly,\nthe interplay between Coulomb interaction and tunel-\ncoupling between the QD and the FM electrode ℓ={L,\nR}gives rise to an exchange magnetic ﬁeld [47, 66]\nBℓ=pℓΓℓ\nπ/integraldisplay′\ndω/parenleftbiggf+\nℓ(ω)\nω−εℓ−U+f−\nℓ(ω)\nω−εℓ/parenrightbigg\nmℓ,(7)\nwhere the prime at the integral stands for the Cauchy’s\nprinciple value. The total exchange magnetic ﬁeld B=\nBL+BRleads to precession of the average spin in the\nQD. According to Eqs.(6d)-(6f), it is described by\n˙S=S×B. (8)\nWe will reveal the essential roles it plays in the spin-\nresolved transport properties.\nIt should be noted that in the steady state limit ( t→\n∞), the spin-resolved CGF is reduced to [49, 54, 67]\nF(χ,t) =λ0(χ)t, (9)\nwhereλ0(χ)istheuniqueeigenvalueoftheLiouvillianas-\nsociated with the quantum master equation (6) that sat-\nisﬁesλ0(χ)|χ→0→0. For instance, the ﬁrst cumulant,\ni.e., the individual spin- ν(ν∈ {+,−}) current through\njunction ℓ(ℓ∈L,R), is given by\n/an}bracketle{t/an}bracketle{tJν\nℓ/an}bracketri}ht/an}bracketri}ht= (−i)∂\n∂χℓνλ0(χ)|χ→0. (10)\nBy making use of the quantum master equation (6), one\narrives at\n/an}bracketle{t/an}bracketle{tJ±\nℓ/an}bracketri}ht/an}bracketri}ht=−2κ±{γ+\nℓρ00+1\n2(˜γ+\nℓ−γ−\nℓ)ρ11−˜γ−\nℓρdd\n∓(γ−\nℓ+˜γ+\nℓ)S·mℓ}. (11)\nApparently, it not only depends on the electron occupa-\ntion probabilities but also the average spin . The netcharge and spin currents through the junction ℓare de-\nﬁned as\n/an}bracketle{t/an}bracketle{tJch\nℓ/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tJ+\nℓ/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tJ−\nℓ/an}bracketri}ht/an}bracketri}ht, (12a)\n/an}bracketle{t/an}bracketle{tJsp\nℓ/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tJ+\nℓ/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tJ−\nℓ/an}bracketri}ht/an}bracketri}ht. (12b)\nSpeciﬁcally, utilizing Eq.(11), one ﬁnds\n/an}bracketle{t/an}bracketle{tJch\nℓ/an}bracketri}ht/an}bracketri}ht=−2[γ+\nℓρ00+1\n2(˜γ+\nℓ−γ−\nℓ)ρ11−˜γ−\nℓρdd]\n+2pℓ(γ−\nℓ+ ˜γ+\nℓ)S·mℓ, (13a)\n/an}bracketle{t/an}bracketle{tJsp\nℓ/an}bracketri}ht/an}bracketri}ht=−2pℓ[γ+\nℓρ00+1\n2(˜γ+\nℓ−γ−\nℓ)ρ11−˜γ−\nℓρdd]\n+2(γ−\nℓ+ ˜γ+\nℓ)S·mℓ. (13b)\nAnalogously, the second cumulants, corresponding to\nthe correlations between the spin- νcurrents through\njunction ℓand spin- ν′current through junction ℓ′, can\nbe obtained via taking the second-order partial deriva-\ntives with respect to their corresponding counting ﬁelds\n/an}bracketle{t/an}bracketle{tJν\nℓJν′\nℓ′/an}bracketri}ht/an}bracketri}ht= (−i)2∂2\n∂χℓν∂χℓ′ν′λ0(χ)|χ→0.(14)\nAccording to Eq.(12), the noises of the net charge and\nspin currents are thus given by\nSch\nℓℓ′=/an}bracketle{t/an}bracketle{tJ+\nℓJ+\nℓ′/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tJ−\nℓJ−\nℓ′/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tJ+\nℓJ−\nℓ′/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tJ−\nℓJ+\nℓ′/an}bracketri}ht/an}bracketri}ht,(15a)\nSsp\nℓℓ′=/an}bracketle{t/an}bracketle{tJ+\nℓJ+\nℓ′/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tJ−\nℓJ−\nℓ′/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tJ+\nℓJ−\nℓ′/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tJ−\nℓJ+\nℓ′/an}bracketri}ht/an}bracketri}ht,(15b)\nwhere each term is evaluated according to Eq.(14).\nHigher order cumulants can be obtained in an analo-\ngous manner. Although there were investigations about\ncharge noise in spin valves, it is still of great importance\ntoinvestigatethespin-resolvednoises, whichareessential\nto reveal the underlying spin correlations in transport.\nIV. RESULTS AND DISCUSSION\nA. Current-voltage characteristics\nUtilizing Eq.(6) it is easy to check that in the station-\nary limit /an}bracketle{t/an}bracketle{tJch\nL/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht= 0, which ensures the charge5\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s112 /s61/s48/s46/s48/s48 /s32/s112 /s61/s48/s46/s48/s48/s32/s40 /s66\n/s76 /s61 /s66\n/s82/s61/s48/s41\n/s32/s112 /s61/s48/s46/s55/s53 /s32/s112 /s61/s48/s46/s55/s53/s32/s40 /s66\n/s76 /s61 /s66\n/s82/s61/s48/s41\n/s32/s112 /s61/s48/s46/s57/s57 /s32/s112 /s61/s48/s46/s57/s57/s32/s40 /s66\n/s76 /s61 /s66\n/s82/s61/s48/s41/s225/s225 /s74/s32/s43 /s82/s241/s241 /s32/s91\n/s93/s40/s97/s41\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s225/s225 /s74/s32/s45\n/s82/s241/s241 /s32/s91 /s93/s40/s99/s41 /s40/s100/s41/s40/s98/s41\n/s48/s46/s50/s48/s46/s52\n/s48/s46/s48/s225/s225 /s74/s32/s99/s104 /s82/s241/s241 /s32/s91\n/s93/s40/s101/s41 /s40/s102/s41\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s225/s225 /s74/s32/s115/s112 /s82/s241/s241 /s32/s91\n/s93\n/s86 /s32/s91/s32 /s107\n/s66/s84/s32 /s93/s40/s103/s41\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s66 /s32/s91 /s71 /s93\n/s86 /s32/s91 /s107\n/s66/s84 /s93/s32/s66\n/s76\n/s32/s66\n/s82\n/s86 /s32/s91/s32 /s107\n/s66/s84/s32 /s93/s40/s104/s41/s113/s32 /s61/s32 /s112 /s32/s47/s32/s50 /s113/s32 /s61/s32 /s112\n/s40/s105/s41\nFIG. 2: Individual spin-resolved currents ( /angbracketleft/angbracketleftJ+\nR/angbracketright/angbracketrightand/angbracketleft/angbracketleftJ−\nR/angbracketright/angbracketright),\nthe net charge and spin currents ( /angbracketleft/angbracketleftJch\nR/angbracketright/angbracketrightand/angbracketleft/angbracketleftJsp\nR/angbracketright/angbracketright) versus\nbias voltage through the right junction for perpendicularl y\n(θ=π\n2)andantiparallely ( θ=π)alignedmagnetizations with\nvarious polarizations. For comparison, the results neglec ting\nthe exchange magnetic ﬁeld are also plotted in symbols. We\nchoose symmetric tunneling couplings Γ L= ΓR= Γ/2 and\nsame polarizations pL=pR=p. The other parameter are\nε= 10kBTandU= 30kBT. Insect: The exchange ﬁeld due\nto coupling to the left and right electrodes versus bias volt age\nfor full polarization pL=pR=p= 1.\nconservation. It is thus enough to analyze either of the\ncurrents through the left or right junctions. In Fig.2,\nwe plotted the individual spin-resolved currents /an}bracketle{t/an}bracketle{tJ±\nR/an}bracketri}ht/an}bracketri}ht,\nnet charge current /an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht, and net spin current /an}bracketle{t/an}bracketle{tJsp\nR/an}bracketri}ht/an}bracketri}ht\nthrough the right junction vs bias voltage for various po-\nlarizations with perpendicular aligned ( θ=π\n2) and an-\ntiparallel ( θ=π) electrode magnetizations.\nLet us ﬁrst consider the case of perpendicular align-\nment (θ=π\n2). For nonmagnetic electrodes ( p= 0), the\nexchange ﬁeld has a vanishing contribution [cf. Eq.(7)],and thus the system can be mapped onto a resonant tun-\nneling model. At low bias, the transport is blocked due\nto vanishing occupation of the QD. As the bias increases,\nwhenever an excitation energy level ( εorε+U) falls into\nthe energy window deﬁned by the chemical potentials\nof the left and right electrodes, a new transport channel\nopens. This leads to step-like structures in the individual\nspin-resolved currents and net charge current, as shown\nby the solid lines in Figs.2(a), (c), and (e). A step oc-\ncurs at a bias twice of the excitation energy level due\nto symmetric application of the bias voltage across the\nQD (µL/R=±V/2). In this case, /an}bracketle{t/an}bracketle{tJ+\nR/an}bracketri}ht/an}bracketri}htand/an}bracketle{t/an}bracketle{tJ−\nR/an}bracketri}ht/an}bracketri}htare\nequal in magnitude and the net spin current thus is zero\naccording to Eq.(12b), cf. the solid line in Fig.2(g).\nFinite spin polarization in the electrodes gives rise to\nspin accumulation in the QD, which will have essential\nrole to play in transport. At the ﬁrst current plateau\n(2ε < V < 2(ε+U)), double occupation on the QD\nis energetically not allowed and the tunneling rates are\ngreatlyreducedatlowtemperatures,i.e., γ+\nR,˜γ+\nR→0and\nγ−\nR,˜γ−\nR→ΓR. As a result, the spin-resolved currents in\nEq.(11) are simpliﬁed to\n/an}bracketle{t/an}bracketle{tJ±\nR/an}bracketri}ht/an}bracketri}ht=κR±ΓRρ11±√\n2κR±ΓR(Sx−Sy).(16)\nThe ﬁrst term is directly associated with the probabil-\nity of an electron in the QD to tunnel out via the right\njunction, where the rates are modulated by the corre-\nsponding densities of states of majority or minority spins\nin the electrode. The second term is unambiguously re-\nlated to the spin accumulation on the QD. In the absence\nof the exchange ﬁeld ( B= 0), the occupation and aver-\nage spin on the QD in the steady state can be obtained\nby utilizing Eqs.(6):\nρ11=2ΓL\n2ΓL+(1−p2)ΓR, (17a)\nand\nSx=−p3ΓR√\n2(2ΓL+(1−p2)ΓR),(17b)\nSy=p(2−p2)ΓR√\n2(2ΓL+(1−p2)ΓR). (17c)\nAs polarization pincreases, the QD is inclined to be oc-\ncupied by one extra electron ( ρ11→1) and the average\nspinsSx→ −√\n2ΓR/(4ΓL) andSy→√\n2ΓR/(4ΓL) such\nthat the total spin tends to align along −mR, i.e., an-\ntiparallel to the magnetization of the right electrode (see\nFig.1). Due to the presence of Coulomb repulsion, no\nmore electrons can tunnel into the QD and thus trans-\nport is strongly suppressed. This explains the strong re-\nduction of the spin-resolved currents with rising p, see\nthe dotted curves in Figs.2(a) and (c).\nThe presence of exchange ﬁeld leads to a precession of\nthe averagespin. It is shown in the inset of Fig.2(h) that\nBLplaysthedominantrole, especiallywhenthe biasisin\nresonancewith the energylevels εandε+U. Theaverage6\nspin now rotates dominantly about BLand thus weakens\nthe spin valve eﬀect. This increases the probability for\nan electron to leave the dot and leads to an enhanced\nspin-resolved current J+\nR, particularly prominent at V=\n2εandV= 2(ε+U). This also explains the strong\nenhancement of the net spin current /an}bracketle{t/an}bracketle{tJsp\nR/an}bracketri}ht/an}bracketri}htatV= 2ε,\nsee for instance the dotted curve in Fig.2(g). However,\nthe suppression of /an}bracketle{t/an}bracketle{tJ−\nR/an}bracketri}ht/an}bracketri}htis not lifted due to strongly\nannihilated κR−in the limit of large polarization.\nAs the bias further increases to the regime V >2(ε+\nU), double occupation on the QD is now energetically\nallowed. This opens up an additional transport channel\nand the spin-resolved currents rise to the second plateau,\ncf. the solid curves in Figs.2(a) and (c) for p= 0. For\nbias far away from the energy level ε+U, the inﬂuence\nof the exchange ﬁeld is greatly reduced even for a strong\npolarization (see the inset of Fig.2(h)). At low temper-\natures, the tunneling rates in Eq.(11) are well approxi-\nmated by either 0 or Γ R. One obtains simple expressions\nof the spin-resolved currents for ﬁnite spin polarization:\n/an}bracketle{t/an}bracketle{tJ±\nR/an}bracketri}ht/an}bracketri}ht=κR±ΓR(ρ11+2ρdd)±√\n2κR±ΓR(Sx−Sy).(18)\nAs the polarization increases, κR−is strongly reduced,\nleading thus to a prominent suppression of /an}bracketle{t/an}bracketle{tJ−\nR/an}bracketri}ht/an}bracketri}ht, in\ncomparison with that of p= 0, cf. Fig.2(c). Remark-\nably, ﬁnite polarization gives rise to a slight increase of\n/an}bracketle{t/an}bracketle{tJ+\nR/an}bracketri}ht/an}bracketri}ht. For symmetric tunnel couplings (Γ L= ΓR) that\nweconsideredhere, utilizing Eq.(6) oneﬁnds the station-\nary probabilities and average spin\nρ11+2ρdd→1, Sx→0, Sy→ −√\n2p/4.(19)\nThus, there is a competition between the ﬁrst and second\nterm in Eq.(18). For p= 0, the ﬁrst term dominates,\nwhichgivesthesecondcurrentplateau,cf. thesolidcurve\nin Fig.2(a). An increase of pleads to a rising κR+but a\ndecreasing Sy, which explains the slight enhancement of\n/an}bracketle{t/an}bracketle{tJ+\nR/an}bracketri}ht/an}bracketri}htin comparison with that of p= 0, as shown by the\ndashed and dotted curves in Fig.2(a).\nLet us now consider the situation for the antiparallel\nconﬁguration ( θ=π). Forp= 0, the charge and spin\ncurrents increase with bias in a typical step-like man-\nner, the same as those for perpendicular conﬁguration\n(θ=π\n2). In the case of large polarization and in the\nbias region 2 ε < V < 2(ε+U), whenever a spin “+”\nelectron tunnels into the QD, it will be trapped in the\nQD for a long time. Its average spin is found to be along\nmL, almost the same direction of the exchange ﬁeld (see\nEq.(7) and the inset of Fig.2(h)), which can not weaken\nthe spin valve eﬀect. Furthermore, the Coulomb inter-\naction energetically prohibits a second electron to tunnel\ninto QD and transport is thus strikingly suppressed with\nincreasingpolarization. For a largebias V >2(ε+U), al-\nthough double occupation is allowed, the probability for\na second electron of spin “ −” tunneling through the QD\nis strongly reduced due to vanishing density of state for\n“−” spin in the left electrode. One thus observes promi-\nnent suppression of spin-resolved currents as well as net/s48/s46/s48/s49/s46/s48/s50/s46/s48/s51/s46/s48/s225/s225 /s74/s32/s43 /s82\n/s74/s32/s43 /s82\n/s241/s241 /s32/s47/s32/s50 /s225/s225 /s74/s32/s99/s104 /s82\n/s241/s241\n/s112 /s61/s48 /s112 /s61/s48/s32/s40 /s66\n/s76 /s61 /s66\n/s82/s61/s48/s41\n/s112 /s61/s48/s46/s55/s53 /s112 /s61/s48/s46/s55/s53/s32/s40 /s66\n/s76 /s61 /s66\n/s82/s61/s48/s41\n/s112 /s61/s48/s46/s57/s57 /s112 /s61/s48/s46/s57/s57/s32/s40 /s66\n/s76 /s61 /s66\n/s82/s61/s48/s41/s40/s98/s41\n/s40/s99/s41 /s40/s100/s41/s40/s97/s41\n/s40/s101/s41 /s40/s102/s41\n/s40/s103 /s41 /s40/s104/s41\n/s40/s105/s41 /s40/s106/s41/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s225/s225 /s74/s32/s45 /s82\n/s74/s32/s45 /s82\n/s241/s241 /s32/s47/s32/s50 /s225/s225 /s74/s32/s99/s104 /s82\n/s241/s241\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s225/s225 /s74/s32/s43 /s82\n/s74/s32/s45 /s82\n/s241/s241 /s32/s47/s32/s50 /s225/s225 /s74/s32/s99/s104 /s82\n/s241/s241\n/s48/s46/s48/s49/s46/s48/s50/s46/s48/s51/s46/s48/s83/s99/s104 /s82/s82\n/s32/s47/s32/s50 /s225/s225 /s74/s32/s99/s104 /s82\n/s241/s241\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s49/s46/s48/s50/s46/s48/s83/s115/s112 /s82/s82\n/s32/s47/s32/s50 /s225/s225 /s74/s32/s99/s104 /s82\n/s241/s241\n/s86 /s32/s91/s32 /s107\n/s66/s84 /s32/s93/s113 /s32/s32/s61/s32 /s112 /s32/s47/s32/s50 /s113 /s32/s32/s61/s32 /s112\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48\n/s86 /s32/s91 /s32/s107\n/s66/s84/s32 /s93\nFIG. 3: Spin autocorrelations ( /angbracketleft/angbracketleftJ+\nRJ+\nR/angbracketright/angbracketrightand/angbracketleft/angbracketleftJ−\nRJ−\nR/angbracketright/angbracketright), cross-\ncorrelation ( /angbracketleft/angbracketleftJ+\nRJ−\nR/angbracketright/angbracketright), and noises of net charge and spin cur-\nrents (Sch\nRRandSsp\nRR) versus bias voltage for perpendicular\n(θ=π\n2) and antiparallel ( θ=π) magnetization alignments\nwith various polarizations. The results neglecting the ex-\nchange magnetic ﬁeld are also plotted in symbols for compar-\nison. The other parameters are the same as those in Fig.2.\ncharge and spin currents, as shown by the dotted curves\nin Fig.2(b), (d), (f), and (h).\nB. Spin current noise characteristics\nNow we are in a position to investigate the spin-\nresolved current noises based on Eq.(14). Again, we con-\nsidertwodiﬀerentmagnetizationconﬁgurations,i.e., per-\npendicular ( θ=π\n2) and antiparallel ( θ=π) alignments.\nThe numerical results /an}bracketle{t/an}bracketle{tJν\nRJν′\nR/an}bracketri}ht/an}bracketri}htare presented in Fig.3,\nwhere noises measured in terms of the Fano factors are\nplotted as functions of the bias voltage. The noises be-\ntweendiﬀerent electrodesarequantitativelysimilar. Fur-7\nthermore, the spin cross-correlations satisfy /an}bracketle{t/an}bracketle{tJ+\nRJ−\nR/an}bracketri}ht/an}bracketri}ht=\n/an}bracketle{t/an}bracketle{tJ−\nRJ+\nR/an}bracketri}ht/an}bracketri}htsuch that one only needs to consider one of\nthem.\nLet us ﬁrst consider the situation of perpendicular\nalignment ( θ=π\n2). At low bias ( V≪kBT), the thermal\nnoise dominates, which is described by the well-known\nhyperbolic cotangent behavior. This leads to divergent\nspin auto-correlations( /an}bracketle{t/an}bracketle{tJν\nRJν\nR/an}bracketri}ht/an}bracketri}ht/2/an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht → ∞,ν= +,−)\nfor various polarizations, as shown in Figs.3(a) and (c).\nThe spin “+” and spin “ −” currents are found to be\nuncorrelated for p= 0, see the solid curve in Fig.3(e).\nYet, ﬁnite polarization gives rise to a negative cross-\ncorrelation ( /an}bracketle{t/an}bracketle{tJ+\nRJ−\nR/an}bracketri}ht/an}bracketri}ht/2/an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht), as shown by the dashed\nand dotted curves in Fig.3(e).\nAs bias increases but remains lower than the ﬁrst ex-\ncitation energy level ( V <2ε), electron transport is ex-\nponentially suppressed. Charge tunneling events are un-\ncorrelated and thus both noises of net chargecurrent and\nspin current exhibit Poissonian statistics independent of\npolarizations ( Sch\nRR/2/an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht= 1 and Ssp\nRR/2/an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht= 1),\nas shown in Figs.3(g) and (i), respectively. Remarkably,\nthe spin autocorrelations depend sensitively on the po-\nlarizations: S++\nRRincreases but S−−\nRRdecreases with rising\np. This demonstrates that the autocorrelations of spin-\nresolved currents may serve as a skeptical tool to detect\ndegree of spin polarization in FM materials.\nAs the bias further increases, the ﬁrst and then the\nsecond channels open. Both /an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htand/an}bracketle{t/an}bracketle{tJ−\nRJ−\nR/an}bracketri}ht/an}bracketri}htde-\ncreases in a step-like manner for p= 0. As polarization\nincreases,aspin “+”electrontunneled intothe QDtends\nto stay there for a long time due to the spin valve eﬀect.\nWhen it is tunneled out, a bunching of “+” spin elec-\ntrons can ﬂow during a short time window, leading thus\nto a dynamical spin blockade mechanism [68–70]. This\nexplains the strongly enhanced super-Poissonianspin au-\ntocorrelation ( /an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}ht/2/an}bracketle{t/an}bracketle{tJch\nR/an}bracketri}ht/an}bracketri}ht>1) in the bias regime\n2ε < V < 2(ε+U), cf. the dotted curve in Fig.3(a). The\nexistence of the exchange ﬁeld weakens the spin-valve ef-\nfect and consequently reduces the noise, leading thus to a\ndouble-peak structure in /an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}ht. The cross-correlation\nbetween the “+” and “ −” spin currents ( /an}bracketle{t/an}bracketle{tJ+\nRJ−\nR/an}bracketri}ht/an}bracketri}ht) is\nalso sensitive to the polarization: It changes its sign from\nnegative to positive as pincreases. The noise characteris-\ntics of the net charge current ( Sch\nRR) and net spin current\n(Ssp\nRR) thus can be understood in terms of the individual\ncomponents according to Eq.(15). We remark that the\nunique double-peak structure in the noise of the net spin\ncurrent may serve as a sensitive means to measure the\nexchange magnetic ﬁeld.\nFor the antiparallel conﬁguration ( θ=π) and in the\nlow bias regime ( V≪kBT), current is suppressed and\nthe noises are quantitatively similar to those for the per-\npendicular conﬁguration. As bias increases, the noises\nshow typical step-like structure. In comparison with the\nsituation ofperpendicular alignment, the noisesfor θ=π\nshow distinct behaviors. First, the noises for antiparallel\nconﬁguration is insensitive to the exchange ﬁled. This\nis due to the fact that in this case the accumulated spin/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s50/s51/s225/s225 /s74/s32/s43 /s82/s74/s32/s43 /s82/s241/s241 /s32/s47/s32/s50 /s225/s225 /s74/s32/s99/s104 /s82/s241/s241\n/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32 /s32/s61/s32/s48\n/s32 /s32/s61/s32 /s32/s47/s32/s50\n/s32 /s32/s61/s32/s51 /s32/s47/s32/s52\n/s32 /s32/s61/s32\nFIG. 4: Spin current autocorrelation /angbracketleft/angbracketleftJ+\nRJ+\nR/angbracketright/angbracketrightversus polar-\nization for various magnetization alignments at a particul ar\nbias voltage V= 40kBT. The other parameter are the same\nas those in Fig.2.\nis almost in the same direction of mLsuch that the ex-\nchange ﬁeld has a vanishing role to play. Second, for bias\nV <2ε, both spinautocorrelationsandcross-correlations\nare independent of the polarizations ( /an}bracketle{t/an}bracketle{tJµ\nRJν\nR/an}bracketri}ht/an}bracketri}ht/an}bracketri}ht/an}bracketri}ht →1\n2and\n/an}bracketle{t/an}bracketle{tJ+\nRJ−\nR/an}bracketri}ht/an}bracketri}ht →0), implying that in this case the spin “+”\nand “−” currents are uncorrelated for an arbitrary po-\nlarization. Third, in the bias regime 2 ε < V < 2(ε+U),\nthe autocorrelation /an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htshows opposite dependence\non polarization in comparison to the perpendicular con-\nﬁguration. For a deep analysis, we plotted in Fig.4 the\nautocorrelation /an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htversuspfor various θwith a\ngiven bias voltage V= 40kBTinside the ﬁrst current\nplateau. As p→0,/an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htfor diﬀerent alignments are\nconsistent. For θ= 0 orπ\n2,/an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htrises monotonically\nwith polarization, leading to prominent super-Possionian\nnoises at a large polarization, see the solid and dashed\ncurves. For θ=3\n4π,/an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htﬁrst decreases for a wide\nrangeofpandthenincreasesrapidlyinthelimitof p→1.\nForantiparallelconﬁguration( θ=π),/an}bracketle{t/an}bracketle{tJ+\nRJ+\nR/an}bracketri}ht/an}bracketri}htdecreases\nmonotonically with polarization and vanishes at p= 0.\nWe remark that this unique noise feature may serve as\na sensitive tool to measure the magnetization directions\nbetween the left and right FM electrodes.\nV. CONCLUSION\nWe have investigated the spin-dependent transport\nthrough a spin valve composed of a quantum dot tunnel\ncoupled to external ferromagnetic electrodes with non-\ncollinear magnetizations. The analysis is based on the\nspin-resolved full counting statistics, which allowed us\nto determine systematicallyspin-resolvedtransportchar-\nacteristics. In particular, we have analyzed individual\nspin-resolved currents, as well as their auto- and cross-\ncorrelations versus bias for diﬀerent magnetization con-\nﬁgurations and polarizations. In the case of perpendicu-8\nlar alignment, we found that the charges are transferred\nuncorrelated at a low bias; however, the autocorrelations\nof spin currents were shown to be sensitive to the polar-\nization in the electrodes. As bias increases to the regime\nwhere single occupation of electron is allowed, the inter-\nplay between tunnel coupling and the Coulomb interac-\ntion gives rise to a prominent exchange ﬁeld. It leads to\nthe precession of the accumulated spin in QD, which lifts\nthe bunching of spin tunneling events and thus results\nin a unique double-peak structure in the noise of the net\nspin current. 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Shiraishi1 \n \n1Graduate School of Engineering Science, Osaka University, Osaka, Japan    \n2Department of Electronics, Kyushu University, Fukuoka, Japan  \n \n \n*Corresponding author: E-mail:  ando@ee.es.osaka -u.ac.jp  \n \nSpincurrentronics , which involves the generation, propagation and control of spin current s, \nhas attracted a great deal of attention because  of the possibility of realizing  \ndissipation -free information propagation. Whereas electrical generation of spin current s \noriginally  made the field  of spincurrentronics possible, and significant  advance s in \nspin-current devices has been made, novel spin-current -generation approaches such as \ndynamical method s have  also been vigorously investigated. However, the low spin-current  \ngeneration efficiency associated with  dynamical method s has impeded further progress \ntowards practical spin devices. Here we show that by introducing a  Heusler -type \nferromagnet ic material , Fe 3Si, pure spin currents can be generated about twenty times \nmore efficien tly using  a dynamical method. This achievement paves the way to the \ndevelopment of novel spin-based devices.  \n \nA central issue with respect to  the practical application of spintronic  devices  is to establish a \nmechanism for highly efficient spin-current generation .1 The development of magnetic tunnel \njunction s with single -crystal MgO barriers  has addressed this  challenge ,2,3 resulting in application s in magnetic  head s and magnetoresistive random access memory. The electrical spin -injection from \nferromagnetic materials (FM)  into nonmagnetic materials  (NM) is also in a similar situation .4,5 The \nuse of a tunnel barrier in order to overcome  the conductance mismatch problem ,6 and the use of \nhalf-metal materials  as spin injector s have  also been proposed .7 While steady progress has been \nmade in the application of  electrical method s for spin -current generation in spintronic s devices, \nrecent studies have also focused on more radical approaches such as dynamical,8-16 thermal,17 and \nacoustic18 methods . These methods are expected to pave the way  for a new generation of  novel \nspintronic s devices  that involve  no charge current . Spin pumping is a dynamical  method in which a \nspin current is generated by a precession of the  magnetization . It has been the subject of considerable \ninterest because  a spin current can be produced over a large area  without  the presence of a  charge \ncurrent , which is  expected to reduce the problem of conductance mismatch.13 Whereas, spin \npumping is a promising technique for a next generation spin current devices, low efficiency of \ngeneration of pure spin current impedes further progress towards practical spin devices , \nunfortunately . For this reason, identifying a novel FM material that is capable of  highly efficient spin \ninjection  is of the utmost importance . Here, we focus on singl e-crystal Fe 3Si, which has desirable \nproperties such as a smaller damping constant and a larger resistivity than those for Ni80Fe20 (Py), \nthe most commonly used  spin source .19 Moreover,  high-quality single -crystal Fe3Si can be easily \ngrown on semiconducting substrates such as Si, Ge and GaAs with atomically flat interface s.19-21 \nThis means that  Fe3Si can be applied to  a wide variety of materials , allowing the development  of \nnovel semiconductor -based  spintronic devices in addition to metal -based  device s. In the present \nstudy , a significant enhancement of spin-injection efficiency is demonstrated  by using a \nsingle -crystal Fe 3Si layer .  \nA 25-nm-thick Fe 3Si epitaxial layer was grown on a high-resistivity FZ -Si(111)  substrate  by \nmolecular beam epitaxy  (MBE) at room temperature . 19,20 A 5-nm-thick Pd layer was then formed  by electron beam (EB) evaporation at room temperature. Two contact  wires  (separated by w=1.0 mm \ngap) for measuring the DC electromotive force were attached to the edge of the Pd film using Ag \npaste. During the measurements, microwave s with a frequency of 9.6 10.01 GHz were generated in \na TE102 cavity of an electron spin resonance (E SR) system, and an external static magnetic field, H, \nwas applied  at an angle, H, as shown in Fig. 1(a). The sample w as placed inside the cavity in a \nnodal position where the rf electric and magnetic field components were a minimum  and a maximum , \nrespectively . The DC electromotive force , VEMF, was measured using a nanovoltmeter. See the \nMethods section for details concerning device fabrication and  the measurement setup.  All \nmeasurements were carried out at room temperature.  \nFigure 1(b) shows ferromagnetic resonance ( FMR ) spectra,  i.e., dI(H)/dH as a function of \nH-HFMR, for the Pd/Fe 3Si/Si  sample recorded at H = 0, 80, 110, and 180, where I, H, and HFMR are \nthe microwave absorption intensity, external magnetic field, and FMR  field, respectively . \nUnfortunately, FMR could not be measured at H = 90 due to the limit ed external magn etic field  \nstrength , i.e.,  the maximum magn etic field of 1.3 T in the  ESR system is smaller than the anisotropy \nfield for the Fe 3Si thin film (~1.5 T). For H = 0, 80, 110, and 180 , clear FMR spectra were  \nobserved. From the  obtained  resonant magnetic field  (HFMR = 92.9 mT) at H = 0, the saturation  \nmagnetization , Ms, is estimated to be 828  emu/cc, which is consistent with previous ly measured  \nvalues  using a vibrating sample m agnetometer ,19,22 indicating that the spectra are associated with \nFMR in the Fe3Si layer . Figure 1 (c) shows VEMF as a function of H-HFMR. For H = 0, a clear signal \ncan be seen at the  FMR  condition . The EMF signal s were analyzed using a deconvoluted fitting \nfunction with independent contributions from the the inverse spin Hall effect (ISHE , symmetrical \nLorentzian curve centered on  HFMR) and the anomalous Hall effect (AHE,  asymmetrical curve ) as \nfollows:11 \n     𝑉𝐸𝑀𝐹 = 𝑉𝐼𝑆𝐻𝐸Γ2\n(𝐻−𝐻𝐹𝑀𝑅 )2+Γ2+𝑉𝐴𝐻𝐸−2Γ(𝐻−𝐻𝐹𝑀𝑅 )\n(𝐻−𝐻𝐹𝑀𝑅 )2+Γ2 ,         (1) where  is the damping constant.  As shown in Fig. 1(d), a theoretical fit using Eq. (1) nicely \nreproduces the experimental results.  VISHE and VAHE are estimated to be 67.1 and 17.5 V/mm, \nrespectively. Figure 1 (e) shows VISHE and VAHE as a function of θH. The polarity reversal observed \nfor VISHE when θH is changed from 0° to 180°  is consistent with the theoretical ly predicted symmetry \nof the ISHE,  expressed as  𝐽𝑐=𝐽𝑠×𝜎, where  σ, Js and Jc are the direction s of the spin , spin current \nand charge current ,11 respectively,  thus indicating successful dynamical spin injection into the Pd \nlayer from the Fe 3Si layer.  This is also supported by the linear relationship between VISHE and the \nmicrowave power, PMW, shown in the inset of Fig. 1(f) (see also Supplementary Information  (SI) A). \nSince the conductance s of the Pd and Fe 3Si layer s are in parallel to each other , the electromotive \nforce generated in the Fe 3Si layer is also detected. Although the  anisotropic magnetoresistance  \n(AMR)  effect can produce signals with a Loren tzian line shape in the VEMF-H curve ,23 the H \ndependence of VISHE induced by the AMR is quite d ifferent from that shown in Fig. 1(e). In addition, \nno such Loren tzian line shape was obtained for the Fe3Si layer in the absence of the Pd layer . \nConsidering these results , it can be conclude d that the contribution of the AMR  effect is negligibly  \nsmall  (see SI  B). Furthermore , although  in the FMR condition,  a temperature gradient  is induced  in \nthe sample , and this can lead to an  additional DC electromotive force d ue to the Seebeck  effect, the \nspin Seebeck effect,17 and the anomalous Nernst -Ettingshausen  effect ,24,25 these  contribution s were  \nalso found to be negligible  (see SI  B). Therefore , it can be concluded that the origin of VISHE is the \nISHE in the Pd layer due to  a pure spin current generated by spin pumping  of the Fe 3Si layer . In fact, \nwhen the NM layer was changed from Pd to Al , in which spin -orbit interaction s are weaker than in \nPd, VISHE was drastically reduced to 4.20 V/mm , which is one -sixteenth of the value for the \nPd/Fe 3Si/Si sample  (see SI  C).     \nFor comparison, the  spin injection efficiency was investigated for several  FM materials : Ni80Fe20 \n(Py), polycrystalline Fe 3Si, and single -crystal Co 6Fe4. The polycrystalline  Py and Fe 3Si layer s were formed by EB eva poration  and pulse laser deposition , respectively. T he single -crystal  Co6Fe4 was \ngrown by MBE.26 The d etailed growth procedures are described in the Methods section.  To \ndistinguish between the single -crystal Fe 3Si grown by MBE and  the polycrystalline Fe 3Si grown by \nPLD, these layers  are referred to as “single -Fe3Si” and “poly-Fe3Si”, respectively. Figure 2 shows \nthe H dependence of the (top) FMR signal, d I(H)/d H, and the (bottom) electromotive force, VEMF, \nfor θH = 0 and 180 , for  a) Pd/Py/SiO 2/Si, b) Pd/poly -Fe3Si/SiO 2/Si, and c) Pd/Co 6Fe4/Si. The \nmicrowave excitation power was 200 mW. Clear FMR signals and EMF s were obtained for all \nsamples. In order to estimate the generated spin current, 𝐽𝑠0, the following equation  was used :12  \n𝑉𝐼𝑆𝐻𝐸\n𝑤=𝜃𝑆𝐻𝐸 𝜆𝑁𝑡𝑎𝑛 ℎ(𝑑𝑁\n2𝜆𝑁)\n𝑑𝑁𝜎𝑁+𝑑𝐹𝜎𝐹(2𝑒\nℏ)𝐽𝑠0 ,           (2) \nwhere dF and F are the thickness and electric conductivity of the FM layer, and dN, and N are those \nof the Pd layer, respectively.  From  the VEMF vs. H curve s, VISHE/w for the Pd/Py, Pd/poly -Fe3Si, and \nPd/Co 6Fe4 sample s was estimated to be 2.85, 15.0, and 2.92 V/mm, respectively . This leads to the \nsurprising conclusion that  VISHE/w for the single -Fe3Si sample (67 .1 V/mm) is more than twenty \ntimes  higher than that for samples using a conventional FM material such as Py . From E q. (2), 𝐽𝑠0 \nfor the single -Fe3Si, poly-Fe3Si, Py, and Co6Fe4 sample s is calculated  to be 2.7510-8, 5.7610-9, \n1.2510-9, and 1.7610-9 J/m2, respectively  (see Table 1) . Thus, for the single -Fe3Si sample, the \ngenerated spin current is more than twenty times  higher than that for the Py samples. Since 𝐽𝑠0 is a \ngood  indicator of the s pin injection efficiency , these result s clearly indicate that highly efficient spin \ninjection is realized for the single -Fe3Si sample.  In general,  𝐽𝑠0 is expressed as12 \n𝐽𝑠0=𝑔𝑟↑↓𝑟2ℎ2ℏ[4𝜋𝑀𝑠𝛾+√(4𝜋𝑀𝑠)2𝛾2+4𝜔2]\n8𝜋𝛼2[(4𝜋𝑀𝑠)2+4𝜔2] ,        (3) \nwhere h,ℏ, 𝑔𝑟↑↓, Ms and  are the microwave magnetic field, the Dirac constant , the real part of the \nmixing conductance, the saturation magnetization  and the Gilbert damping constant, respectively.  \n(=2f) is the angular frequency of the magnetization precession, where f is the microwave frequency.  The e stimated 𝑔𝑟↑↓ values and other physical parameters for the different samples are summarized in \nTable 1.  The parameters and Ms are estimated from width of FMR spectrum and HFMR, \nrespectively. These parameters are strongly depend ent on the FM layer , and  Eq. (3)  implies that for \nhigh spin injection efficiency, should be as small as possible and Ms should be optimized to \nmaximize 𝐽𝑠0 . However, even though the values for the poly-Fe3Si sample is smaller than that for \nthe single -Fe3Si sample , and the Ms value is comparable , 𝐽𝑠0 for the poly-Fe3Si sample is \nconsiderably small er than that for the single -Fe3Si sample . This indicate s that and Ms are not the \nmain factors responsible for the large VISHE for the single -Fe3Si sample. We therefore  focus on 𝑔𝑟↑↓ , \nwhich is generally related to the conductance between non-collinear FMs.  In this study, s ince 𝑔𝑟↑↓ is \ncalculated using  Eq. (3), other extrinsic contribution s that affect the spin injection efficiency , which  \nare not considered  in the conventional theory , are also included  in 𝑔𝑟↑↓. As can be seen  from Table 1 , \n𝑔𝑟↑↓ for the single -Fe3Si sample is clearly larger than those for the other samples.  It should also be \nnoted that  𝑔𝑟↑↓ for the Co 6Fe4 sample is also relatively large despite the small 𝐽𝑠0 value . Both of \nthese results were  reproduc ible over  several samples. This unexpected behavior of 𝑔𝑟↑↓ might \nprovide an important clue  for understand ing the mechanism that gives rise to the large VISHE for the \nsingle -Fe3Si sample.  \nBased on the result s shown  in Fig. 2 and Table 1 , a possible mechanism  is now considered . Figure \n3 schematic ally illustrates the spin -current  flow generated by spin pumping  in samples consisting of \nNM and FM layers, for  different interface  conditions between the FM layer and the substrate. Figure \n3(a) shows the case for a single FM layer with an atomically flat interface  with the substrate , Fig. \n3(b) shows the case for a rough interface between a single FM layer and the substrate , and  Fig. 3( c) \nshows a situation where there are two different FM layers with different saturation magnetizations, \nand an atomically flat interface exists between the lo wer FM2 layer and the substrate . Although the \nideal spin pumping condition  is represented by Fig. 3(a), it  is possible that a  non-zero interfac e rougness exists  between FM1 layer and substrate. In this cas e, since ferromagnetic resonance \ncondition of the magnetization near the interface is changed due to shape anisotropy,27 magnetization \nnear the interface does not precess under the same FM R condition as that for the FM1  layer . Since \nthe spin diffusi on length in FM1 layer  is short, the FM 1 layer near the interface in Fig. 3(b) can act \nas an effective spin sink , resulting in  a reduction of  the spin current flow ing into the NM layer. The \nFM2 layer also induce s an additional EMF , which can cause  a significant change in  the measured \nVISHE. A similar situation should occur  for the sample  shown in Fig. 3(c) , with atomically flat FM2 \nlayer . In the present study, Fig. 3(a) corresponds to the case for the single -Fe3Si and th e Co 6Fe4 \nsample s, and Fig. 3( b) correspond s to the case for the Py and poly -Fe3Si sample s because the \nthermally oxidized Si substrate has a non -zero surface roughness. In fact, a measureable VISHE was \nfound for a Py layer  without  any NM layer, which indicates the existence of another spin sink .28 \nAlthough the Co 6Fe4 sample also has an ideal atomically flat interface , across which spins can be \nelectrically injected f rom the Co 6Fe4 into a Si channel  even at room temperature,29, 30 the large  and \nMs values lead to a significant reduction in the spin -current density , as indicated by  Eq. (3). Thus, \n𝑔𝑟↑↓ might be also reflect the crystal and magnetic quality of the ferromagnetic layer . \nTo experimentally investigate whether this was in fact the case , two additional  single -Fe3Si \nsamples were fabricated with different interfacial conditions . First, an  as-grown  single -Fe3Si/Si \nsample was annealed at 350 C for 30 min in an Ar atmosphere  because an earlier study \nrevealed that slight intermixing  occurs  between the Fe3Si layer and the Si substrate at around \n300 C.22 Thus, the  annealed sample expected to have  a rough interfacial layer, which \ncorrespond s both to the sample in Fig. 3(b) and that in Fig. 3(c). Following  annealing, a \n5-nm-thick Pd layer was form ed. FMR spectra for the as-grown and annealed samples are \nshown in Fig. 4(a). To highlight the differences in the FMR  field between these two samples , the \nmicrowave absorption intensity, I, rather than d I/dH, is plotted as a function  of H-HFMR. As can be seen, the width of the FMR spectrum for the annealed sample is slightly  larger . As indicated \nby the blue arrow,  the absorption intensity under a high external magnetic field is enhanced  for \nthe annealed sample , which indicates the presence of a Si-rich interfaci al layer. Figure 4(b)  \nshows  VEMF against  H-HFMR for the annealed sample , measured  at θH = 0° and 180°. The \nmicrowave excitation power was 200 mW. T he signal shape is seen to be significantly different \nto that for the as-grown sample shown in Fig. 1(c). The magnitude of VISHE was estimated to be \n12.6 V/mm , which is about one-fifth of that for the as -grown sample . To investigate the effect \nof an intentionally inserted additional FM layer near the interface with the substrate, as in Fig. \n3(c), two further samples were fabricated. In these samples, a 2 -nm-thick layer of either Fe 4Si or \nFe2Si was grown  on the substrate before growth of the single -Fe3Si layer. Despite the large \ncomposition change, no evidence was found that the presence of such an inte rfacial layer \naffected the epitaxial growth of the single -Fe3Si layer, and the interfaces remained atomically \nflat. Figure 4(c) shows VEMF against  H-HFMR for these two samples.  The magnitude of VISHE was \ndrastically reduced  to 13.2 V/mm for  the sample with  Fe4Si and 16.2 V/mm for the sample \nwith Fe 2Si, despite the  presence of  atomically flat interface s. The results  shown in Fig. 4 \nstrongly support the idea that to realize highly efficient spin injection using spin pumping \ntechniques, magnet ic propert ies of the FM layer near the interface should be carefully \nconsidered. These findings are likely to have a major impact in the field of spintronics. We \nbelieve that further enhancement of the spin injection efficiency can be realized by using \ncompletely uniform FM metals with  a much  small er  value , such as Co -based Heusler alloys.26  Methods  \nUndoped Si (111)  wafer s were  used as substrate s for growing  single -crystal Fe3Si, and Co6Fe4 \nlayers. After cleaning the substrate with an aqueous HF solution (HF:H 2O=1:40 ), a heat treatment \nwas carried out at 450 °C for 20 min in a molecular beam epitaxy ( MBE ) reaction chamber with a \nbase pressure of 2 10−9 Torr. Transmission electron microscopy observations revealed that interface s \nin the  Fe3Si and  Co6Fe4 samples were atomically flat . Polycrystalline Py and Fe 3Si layer s were \nform ed on thermally oxidized  Si(100) substrates (oxide thickness 500 nm ) using electron beam \nevaporation and pulse laser deposition, respectively . After the substrates  were cleaned  with acetone \nand isopropanol, a 25-nm-thick Py or Fe 3Si layer w as formed at room temperature. After d eposition \nof the ferromagnetic layer, a polycrystalli ne nonmagnetic layer such as Pd and Al  was form ed using \nelectron beam evaporation at room temperature. The dimension s of the samples  were 2  mm × 1 mm.  \nTwo lead wires for measuring the electromotive force were attached to the edge of the \nnonmagnetic layer using Ag paste. The sample was placed near the center of a TE102 microwave \ncavity  in an electron spin resonance (ESR) system (Bruker EMX10/12) , where the magnetic -field \ncomponent was a maximum and the electric -field component was a minimum. Microwave s with a \nfrequency  of 9.610.01 GHz, and  a static external magnetic field w ere applied to the samples. The \nelectromotive force was measured using a nanovoltmeter (KEITHLEY 2182A) and all \nmeasurements were performed at room temperature.  \n  Acknowledgements  \nThis research was supported  in part  by a Grant -in-Aid for Scientific Research from the MEXT, \nJapan , by STARC,  by the Adaptable & Seamless Technology Transfer Program through \nTarget -driven R&D  from JST , and by the Toray Science Foundation.  \n \nAdditional information  \nThe authors declare no competing financial interests.  \n \n  References  \n1. Wolf, S. A. et al.  Spintronics: A spin -based electronics vision for the future. SCIENCE  294, \n1488 -1495 (2001).  \n2. Parkin, S. S. P. et al.  Giant tunnelling magnetoresistance  at room temperature with MgO \n(100) tunnel barriers. Nature Mater . 3, 862 -867 (2004).  \n3. Yuasa, S., Nagahama, T., Fukushima, A., Suzuki, Y . & Ando, K. Giant room -temperature \nmagnetoresistance in single -crystal Fe/MgO/Fe magnetic tunnel junctions. Nature M ater. 3, \n868-871 (2004).  \n4. Johnson, M. & Silsbee, R. H. Interfacial charge -spin coupling - injection and detection of \nspin magnetization in metals. Phys Rev. Lett . 55, 1790 -1793 (1985).  \n5. Jedema, F. J., Filip, A. T. & van Wees, B. J. Electrical spin inje ction and accumulation at \nroom temperature in an all -metal mesoscopic spin valve. Nature  410, 345 -348 (2001).  \n6. Fukuma, Y . et al. Giant enhancement of spin accumulation and long -distance spin precession \nin metallic lateral spin valves. Nature Mater . 10, 527-531 (2011).  \n7. Kimura, T. et al . Room -temperature generation of giant pure spin currents using epitaxial \nCo2FeSi spin injectors. NPG ASIA MATERIALS  4, e9 (2012).   \n8. Mizukami, S., Ando, Y . &  Miyazaki, T. Effect of spin diffusion on Gilbert damping for a very \nthin permalloy layer in Cu/permalloy/Cu/Pt films. Phys. Rev. B  66, 104413 (2002).  \n9. Tserkovnyak, Y ., Brataas, A. & Bauer, G. E. W. Enhanced Gilbert Damping in Thin \nFerromagnetic Films. Phys Rev. Lett . 88, 117601 (2002).  \n10. Brataas, A., Tserkovnyak, Y ., Bauer, G. E. W. & Halperin, B. I. Spin battery operated by \nferromagnetic resonance. Phys Rev. B  66, 060404(R) (2002).  \n11. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin  current into charge \ncurrent at room temperature: Inverse spin -Hall effect. Appl. Phys. Lett.  88, 182509 (2006).  12. Ando, K. & Saitoh, E. Inverse spin -Hall effect in palladium at room temperature. J. \nAppl.Phys . 108, 113925 (2010).  \n13. Ando, K., et al . Electrically tunable spin injector free from the impedance mismatch \nproblem. Nature Mater . 10, 655 -659 (2011).  \n14. Shikoh, E., et al . Spin -pumping -induced spin transport in p -type Si at room temperature. \nPhys Rev. Lett . 110, 127201  (2013) . \n15. Koike, M., et al. Dynamical Spin Injection into p -type Germanium at Room Temperature. \nAppl. Phys. Express  6, 023001 (2013).  \n16. Tang, Z., et al. Dynamically generated pure spin current in single -layer graphene. Phys. Rev. \nB 87, 140401(R) (2013).  \n17. Uchi da, K., et al. Observation of the spin Seebeck effect. Nature  455, 778 -781(2008).  \n18. Uchida, K., et al. Long -range spin Seebeck effect and acoustic spin pumping . Nature Mater.  \n10, 737-741(2011 ). \n19. Hamaya, K., Ueda, K., Kishi, Y ., Ando, Y ., Sadoh, T. & Miyao, M. Epitaxial ferromagnetic \nFe3Si/Si(111) structures with high -quality heterointerfaces. Appl. Phys. Lett.  93, 132117 (2008).  \n20. S adoh , T. et al . Atomically controlled molecular beam epitaxy of ferromagnetic silicide \nFe3Si on Ge. Appl. Phys. Lett.  89, 182511 (2006).  \n21 Herfort , J. et al. Epitaxial growth of Fe 3Si/GaAs(001) hybrid structures. Appl. Phys. Lett.  83, \n3912 -3914 (2003).  \n22. Ando, Y ., et al . Magnetic properties of epitaxially grown Fe 3Si/Ge(111) layers with \natomically flat heterointerface s. J.Appl. Phys . 105, 07B102 (2009).  \n23. Egan, W. G. and Juretschke, H. J. DC Detection of Ferromagnetic Resonance in Thin Nickel \nFilms. J.Appl. Phys . 34, 1477 (1963).  \n24. Huang, S. Y ., et al . Intrinsic Spin -Dependent Thermal Transport. Phys Rev. Lett . 107, 216604 (2011).  \n25. Weiler, M., et al. Local Charge and Spin Currents in Magnetothermal Landscapes. Phys Rev. \nLett. 108, 106602 (2012).  \n26. Maeda, Y ., Hamaya, K., Yamada, S., Ando, Y ., Yamane, K. & Miyao, M., Appl. Phys. Lett.  \n97, 192501 (2010).  \n27. Kitamura, Y., et al.  Vertical Spin Transport in Al with Pd/Al/Ni 80Fe20 Trilayer Films at \nRoom Temperature by Spin Pumping. In press  (arXiv:1212.0283 ). \n28. Tsukahara, A., et al.  Observation of the intrinsic inverse spin Hall effect from ferromagnet. \n(arXiv:1301.3580 ). \n29. Ando Y ., et al.  Electric -field control of spin accumulation signals in silicon at room \ntemperature , Appl. Phys. Lett.  99, 132511  (2011). \n30. Ando Y . , et al. Temperature evolution of spin accumulation detected electrically in a \nnondegenerated silicon channel. Phys. Rev. B  85, 035320 (2012).  \n \n  Figure legends   \nFigure 1 | Electromotive force measurement s for  Pd/Fe 3Si/Si sample.  \na) Schematic illustration of the Pd/Fe 3Si/Si sample  structure . The lateral dimension s of the \nFe3Si layer were 2 mm ( w) × 1 mm and the thickness, d, was 25 nm. Two contact wires were \nattached to  the P d layer using Ag paste. The e lectrode separation, w, was 1.0 mm. The \nstatic external magnetic field, H, was applied at an angle of θH to the Fe3Si film plane. b) \nFMR spectra, d I(H)/dH, for the Fe 3Si sample  at θH = 0, 8 0, 110 and 180° as a function of \nH-HFMR, where I is the microwave absorption intensity in arbitrary un its. The microwave \npower was 200 mW. The FMR  field, HFMR, for θH=0 was estimated to be 92.9 mT.  c) \nDependence  of the electromotive force, VEMF, on H for θH=0, 80, 110 and 180. d) \nDependence of the electromotive force, VEMF, on H for θH=0. The open circles are \nexperimental data, and the green  solid line  is a fit obtained using  Eq. (1)  considering  the \ncontribution s from the ISHE and AHE . The  red and blue line s are fits for th e ISHE signal \nfrom the Pd layer  and the AHE signal  from the Fe 3Si layer , respectively. e) Dependence of  \nVISHE and VAHE on the magnetic field ang le, θH, where VISHE and VAHE are the electromotive \nforces due to the ISHE and the AHE, respectively. f) Dependence of V on H for different \nmicrowave powers at θH=0. The inset shows the microwave power dependence of VISHE \nand VAHE. \n \nFigure 2 | Electromotive force measurements for different  ferromagnetic samples.  \nThe Ni 80Fe20 (Py) and polycrystalline Fe 3Si layer s were deposited on thermally oxidized \nSi(100)  substrate s (oxide thickness 500 nm)  using electron beam evaporation and pulse \nlaser deposition, respectively, at room temperatu re. The single -crystal Co 6Fe4 layer  was \ngrown  by molecular beam epitaxy at room temperature . The microwave power was 200 mW. H dependence of the (top) FMR signal, d I(H)/dH, and the (bottom) electromotive force, VEMF, \nat θH=0 and 180 for a) Pd/Py/SiO 2/Si, b) Pd/poly -Fe3Si/SiO 2/Si, and c) Pd/Co 6Fe4/Si \nsamples . The microwave excitation power was 200 mW. The FMR and EMF  measurement \nprocedures  and the sample geometry were  the same as those for the Pd/ single -crystal \nFe3Si/Si sample  shown in Fig. 1 (a). HFMR was estimated to be  131.6, 89.9, and 51.2 mT for  \nthe Py, poly -Fe3Si, and Co6Fe4 sample, respectively.  \n \nTable 1 | Summary of physical parameters.  \nPhysical parameters for estimating 𝐽𝑠0 and 𝑔𝑟↑↓ for different  ferromagnetic samples . Ms and \n were obtained from HFMR and the linewidth of the FMR spectrum, respectively. The \nconductivity and spin diffusion length for the Pd layer are 4.08×106 -1m-1 and 9 nm, \nrespectiv ely, as reported in Ref . 28. \n \nFigure 3| Schematic illustration of spin -current  flow under FMR condition s for \nsamples with different interface  structures . \nDifferent interface  structures  between FM layer and substrate , a) an atomically flat interface , \nb) rough interface , and  c) an atomically flat interface with an interfacial  FM2 layer whose \nsaturation magnetization is different from that of the FM1 layer . The schematics show  the \nspin current  flow under FMR conditions for the FM1 layer. The upper figure represents ideal \nspin pumping  conditions .  \n \nFigure 4 | Electromotive force measurement s for Pd/Fe 3Si/Si sample s with different  \ninterfacial layer s.  \na) FMR spectra for as -grown  Pd/Fe 3Si/Si sample  and sample annealed at 350 C for 30 min in an Ar atmosphere. Slight intermixing between the Fe3Si layer and Si substrate is known to \noccur at around 300 C. After annealing, a 5-nm-thick Pd layer was deposited. In order to \nhighlight the differences in the FMR  field between these two samples, the microwave \nabsorption intensity, I, rather than d I/dH, is plotted as a function  of H-HFMR. HFMR for the \nas-grown and annealed sample s is estimated to be 88.0 and 87.7 mT, respec tively . b) DC \nEMF for the annealed sample at θH = 0 and 180. The microwave excitation power was 200 \nmW. The EMF  measurement  procedure and the  sample  geometry were  the same as those \nfor the Pd/single -Fe3Si/Si sample. The open circles are experimental data and the solid line \nis a fit obtained using Eq. (1) . VISHE and VAHE were estimated to be 12.6  and 10.4 V/mm, \nrespectively . c) DC EMF  for Pd/Fe 3Si(23 nm)/Fe 3-XSi1+X(2 nm)/Si samples at θH = 0. The \nmicrowave excitation power was 200 mW. The composition of the 2 -nm-thick interfac ial \nlayer is  either Fe 4Si (blue) or Fe 4Si (green). The interfaces were confirmed to be  atomically  \nflat. A representative VEMF-H curve for the Pd/Fe 3Si(25 nm)/Si sample is displayed  using red \ncircles . The open circles are experimental data and the solid line is a fit obtained using Eq. \n(1). \n \n   \n-400 0 400\nH-HFMR (Oe) VEMF / w (V/mm)H = 0o \nH = 80o \nH = 180o H = 110o \n-400 0 400Intensity (arb. unit)\nH-HFMR (Oe) H = 0o \nH = 80o \nH = 180o H = 110o a\nb c\nFig.1  Y . Ando50 \n-80-4004080\n0 60 120 180V / w (V/mm)\nH (degree) \n-200 0 200VEMF / w (V/mm)\nH-HFMR (Oe)20 \n0204060\n0 100 200V / w  (V/mm)\nPMW  (mW) \n-200 0 200-80-60-40-20020\nH-HFMR (Oe) VEMF / w (V/mm)H=0oAHE\nISHE\nISHE\n+AHEd\nef\nAHE\nISHEAHEISHE\n200 mW\n159 \n127 \n100 \n80\n50\n25\n10Fe3Si \nSi(111) HH\n V\nPd\n25 nm 5 nm Js\n Jc\nR.T. R.T.R.T.\nH=0o\nR.T.H=0o\nR.T.hrf+- \n \n-200 0 200Intensity (arb. unit)\nH-HFMR (Oe)0o \n180o Pd/Py Pd/Co6Fe4a c\n-200 0200Intensity (arb. unit)\nH-HFMR (Oe)0o \n180o \n-200 0200\nH-HFMR (Oe)VEMF / w (V/mm)0o \n180o \n-200 0 200\nH-HFMR (Oe)VEMF / w (V/mm) 0o \n180o \n-200 0 200Intensity (arb. unit)\nH-HFMR (Oe)0o \n180o b\n-200 0 200\nH-HFMR (Oe)VEMF / w (V/mm)0o \n180o \nPd/Poly -Fe3Si\n5 2R.T. R.T. R.T.\nR.T. R.T. R.T.\nFig.2  Y . Ando5   \nTable 1 Y . Ando\nMs\n(emu/cc) σF\n(-1m-1)Js0\n(J/m2)gr↑↓\n(m-2)\nFe3Si Single crystalline 828 0.0087 1.3×1062.75×10-86.2×1020\nFe3Si Polycrystalline 860 0.0050 1.3×1065.76×10-92.5×1019\nPy Polycrystalline 535 0.0149 2.5×1061.25×10-95.2×1019\nCo6Fe4 Single crystalline 1600 0.0227 5.0×1061.76×10-93.1×1020 \n  \nSubstrateFM 1NM \nSpin \ncurrentHFMRVISHE\nHFMRVISHESubstrateFM 1NM \nSpin \ncurrent HFMRVISHE\nSubstrateFM 2FM 1NM \nSpin \ncurrentHFMRVISHE\nHFMRVISHEa\nb\nc\nFig.3 Y. Ando   \n-80-60-40-20020\n-400 -200 0 200 400V EMF/ w (V / mm) \nH-HFMR (Oe)\n-100 0 100Annealed \n350 oCAs-grownI (arb. unit)\nH-HFMR (Oe)H = 0o a b\nc\nSi Sub.Fe4SiFe3SiPd\n2nm\nSi Sub.Fe2SiFe3SiPd\n2nm\nPd/Fe3Si/Si\nFig.4  Y . Ando\n-400 0 400V EMF/ w (V / mm) \nH-HFMR (Oe)H = 0o \nH = 180o 20 \nH=0o\nR.T.R.T.R.T.Supplementary Information  \n \nGiant enhancement of spin pumping efficiency from Fe 3Si ferromagnet   \n \nY. Ando1#, K. Ichiba1, S. Yamada2, E. Shikoh1, T. Shinjo1, K. Hamaya2, M. Shiraishi1 \n \n1 Graduate School of Engineering Science, Osaka University, Osaka, Japan    \n2Department of Electronics, Kyushu University, Fukuoka, Japan  \n \nA. Theoretical prediction of the microwave power dependence of VISHE \nAs expressed in Eqs. (1) –(3), VISHE has an h2 dependence, where h is the microwave magnetic \nfield. Since h has a linear relationship with \nPMW , VISHE is expected to increase linearly with PMW. \nTherefore , the existence of a linear relationship between the microwave power and VISHE is a reliable \nindicator of  successful spin pumping.  \n \nB. Contribution of spurious  effects  \nSince it is possible that extrinsic  EMF s unrelated to the inverse spin Hall effect  in the Pd layer \nmay also be detected using our experimental setup, these need to be taken into consideration . For \nthis reason,  control experiments  were carried out to investigate  the anisotropic magnetoresistance \n(AMR) effect,  the Seebeck effect, the spin Seebeck effect, and the anomalous Nernst -Ettingshausen \neffect.1-3 The AMR effect produces signals with Lorentzian line shape s in the VEMF-H curve when a \ncharge  current is generated in  the FM layer. The main origin of the charge current is induction due to \nthe microwave and external magnetic field s.4-6 In order to investigate the contribution of the AMR \neffect,  the EMF  was first measured  for a single -Fe3Si layer without a Pd layer , as shown in the inset of Fig. S1 (a). VEMF–H curves for θH = 0° and 180°  are shown in the main panel of Fig. S1 (a). The \nmeasurements were performed at room temperature  and t he microwave excitation power was 200 \nmW. As can be seen, t he signal shapes are quite different from those  for the Pd/single -Fe3Si/Si \nsample  shown in Fig. 1 (c). VISHE and the VISHE/VAHE ratio are estimated to be 2.7 V/mm  and 0.35 \nrespectively , indicating that  the intensity of signals with a Lorentzian shape  is drastically reduced \ndue to absence of the Pd layer. If VISHE for the Pd/Fe 3Si sample was mainly due to the AMR effect  in \nthe Fe 3Si layer , VISHE should remain the same or increase in the absence of a  Pd layer . It can \ntherefore be  conclude d that the contribution of an EMF due to the AMR effect is negligibly  small. \nThe clear difference in the θH dependence  of VISHE also supports this conclusion . \nThe contribution s of the spin Seebeck and anomalous Nernst -Ettingshausen effect s are next \nconsidered . These effects occur  when a vertical thermal gradient exists under the FMR  condition , as \nshown in Fig. S1 (d).1-3 In fact, the  θH dependence  of the EMF induced by the anomalous \nNernst -Ettingshausen effect  or the spin Seebeck is the same as that for the inverse spin Hall effect . \nHere,  the EMF is compared for the Pd/Fe 3Si/Si, Fe 3Si/Si and  Al/Fe 3Si/Si sample s. The thermal \nconductivities of Pd, Al , air and Si are reported to be 71.8 , 200, 0.026, and 149 Wm-1K-1,7-10 \nrespectively. Therefore, the  vertical thermal gradient in the Pb/Fe 3Si/Si sample is expected to be \nsmaller  than that in the Fe 3Si/Si sample and to be in the opposite direction to that in the Al/Fe3Si/Si \nsample. If the anomalous Nernst -Ettingshausen effect or the spin Seebeck effect had a dominant \ninfluence on  VISHE, VEMF for the Pd/Fe 3Si/Si sample should be smaller than that for the Fe 3Si/Si \nsample  and should have the opposite polarity to that for the Al/Fe 3Si/Si sample.  However, as shown \nin Fig. S1 (b), VISHE for the Al/Fe 3Si/Si sample was significantly smaller than that for the Pd/Fe 3Si/Si \nsample , and almost the same as that for the Fe 3Si/Si sample.  This cannot be explained if VISHE is \nmainly due to the anomalous Nernst -Ettingshausen effect or the spin Seebeck effect.  Only the \ninverse spin Hall effect  can produce such behavior  because spin -orbit interaction s are much weaker  in Al  than in Pd. \nFinally, the contribution of the Seebeck effect  is considered.  This plays an important role when a \nlateral thermal gradient exists , as shown in Fig. S1(e). A sample  was fabricated in which  a Pd layer \nwas deposited only in the contact area , as shown in the inset Fig. S1 (c). Since  the Pd layer between \nthe contact s is missing, no ISHE occurs  but any contribution from  the Seebeck effect would not be \naffected . As shown in Fig. S1 (c), VISHE is estimated to be 2.4 V/mm, which is small  enough  to \nconclude  that the Lorentz ian signal observed in the  Pd/Fe 3Si/Si sample is not  due to the Seebeck \neffect . \n \nC. Electromotive force  in the Al/ Fe3Si/Si sample  \nSince the conductivity of the Al layer is 9.42×106 m-1, which is more than twice as large as that \nof the Pd layer  (4.08×106 m-1 ), it is necessary  to take into account the change in  conductivity of \nthe entire  sample, 𝑑𝑁𝜎𝑁+𝑑𝐹𝜎𝐹, in Eq. (2) that occurs when the  NM layer  is changed . The \n𝑑𝑁𝜎𝑁+𝑑𝐹𝜎𝐹 values for the Al/single -Fe3Si/Si and Pd /single -Fe3Si/Si sample s are calculated to be \n0.125 and 0.060 respectively. On the other hand, the VISHE values are estimated to be 4.20 and \n67.1 V/mm , respectively. Such  a large discrepancy  cannot be explain ed only in terms of the high  \nconductivity of the Al layer. Therefore, it is clear that VISHE strongly depends on the strength of \nspin-orbit interaction s in the NM layer.  \n \nReferences  \n1. Uchida K., et al. Observation of the spin Seebeck effect . Nature  455, 778 -781 (2008).  \n2. Huang, S. Y ., et al. Intrinsic Spin -Dependent Thermal Transport. Phys Rev. Lett . 107, 216604 \n(2011).  \n3. Weiler, M., et al. Local Charge and Spin Currents in Magnetothermal Landscapes, Phys Rev. Lett. 108, 106602 (2012).  \n4. Juretschke, H. J. Electromagnetic Theory of dc Effects in Ferromagnetic Resonance.  J.Appl. \nPhys . 31, 1401 (1960).  \n5. Egan, W. G. and Juretschke , H. J. DC Detection of Ferromagnetic Resonance in Thin Nickel \nFilms. J.Appl. Phys . 34, 1477 (1963).  \n6. Azevedo, A. , et al. Spin pumping and anisotropic magnetoresistance voltages in magnetic \nbilayers: Theory and experiment. Phys. Rev. B  83, 144402 (2011).    \n7. Parker, W. J., et al.  Flash method of determining thermal diffusivity, heat capacity, and \nthermal conductivity. J. Appl. Phys . 32, 1679 (1961).  \n8. Parker, W. J., et al.  Flash method of determining thermal diffusivity, heat capacity, and \nthermal conductivity. J. Appl. Phys . 32, 1679 (1961).  \n9. Lemmon, E. W. & Jacobsen, R. T. Viscosity and Thermal Conductivity Equations for \nNitrogen, Oxygen, Argon, and Air. International J ournal of Thermophysics  25, 1(2004).   \n10. Slack, G. A. Thermal Conductivity of Pure and Impure Silicon, Silicon Carbide, and Diamond . J. \nAppl. Phys . 35, 3460  (1964).  \n11. Kimura, T. et al. Room -temperature generation of giant pure spin currents using epitaxial \nCo2FeSi spin injectors. NPG ASIA MATERIALS  4, e9 (2012).  \n12. Hamaya , K. et al.  Estimation of the spin polarization for Heusler -compound thin films by \nmeans of nonlocal spin -valve  measurements: Comparison of Co 2FeSi and Fe 3Si. Phys. Rev. B  85, \n100404 (2012).  \n13. Ando, K. & Saitoh, E. Inverse spin -Hall effect in palladium at room temperature. J. \nAppl.Phys . 108, 113925 (2010).  \n14. Kitamura, Y., et al.  Vertical Spin Transport in Al with Pd/Al/Ni 80Fe20 Trilayer Films at \nRoom Temperature by Spin Pump ing. In press (  arXiv:1212.0283 ).  Figure legends  \nFigure S1 | Control experiments for evaluat ing contribution  of spurious effects . \nH dependence of the electromotive force when θH=0 and 180 for a) single -Fe3Si/Si sample \nwith no Pd layer, b) 10-nm-thick Al/single -Fe3Si/Si sample, and c) single -Fe3Si/Si sample \nwith Pd contacts. The me asurements were performed at room temperature.  The microwave \nexcitation power was 200 mW. The FMR and EMF measurement procedures and the \nsample geometry were the same as those  for the Pd/Fe 3Si/Si sample. Schematic \nillustrations of  a FMR -induced  thermal gradient along  the d) vertical  direction  and e) lateral \ndirection . \n  \n   \n-400 -200 0 200 400VEMF / w (V/mm)\nH-HFMR (Oe) H = 0o \nH = 180o \n-400 -200 0 200 400VEMF / w (V/mm)\nH-HFMR (Oe) H = 0o \nH = 180o 10\n-400 -200 0 200 400VEMF / w (V/mm)\nH-HFMR (Oe) H = 0o \nH = 180o 10\nPd\nV\nV\nFig.S1 Y . Ando\nDV ≠ 0\n DV ≠ 0\nMd ea b c\nR.T. R.T.10\nV\nR.T."    },    {        "title": "2212.04697v1.Spin_momentum_properties_of_the_paraxial_optical_beams.pdf",        "content": "Spin\n/\nmomentum \nproperties\n \nof\n \nthe \nparaxial optical \nbeam\ns\n \nPeng Shi, Heng Li, Luping Du\n*\n, Xiaocong Yuan\n*\n \nInstitute of Micro/Nano Optoelectronics, \nShenzhen University, 518060, China\n \n*\nAuthors to whom correspondence should be addressed: \nlpdu@szu.edu.cn\n, \nand \nxcyuan@szu.edu.cn\n \n \nAbstract:\n \nSpin angular momentum, an elementary dynamical property of classical electromagnetic \nfields, plays an important role in spin\n–\norbit and light\n–\nmatter interactions, especially in near\n-\nfield \noptics. The research on optical spins has led to the discovery of phe\nnomena such as optical spin\n–\nmomentum locking and photonic topological quasiparticles, as well as applications in high\n-\nprecision detection and nanometrology. Here, we investigate spin\n–\nmomentum relations in paraxial \noptical systems and show that the optical \nspin angular momentum contains transverse and \nlongitudinal spin components simultaneously. The transverse spin originates from inhomogeneities \nof field and governed by the vorticity of the kinetic momentum density, whereas the longitudinal \nspin parallel to\n \nthe local canonical momentum is proportional to the polarization ellipticity of light. \nMoreover, the skyrmion\n-\nlike spin textures arise from the optical transverse spin can be observed in \nparaxial beams, and their topologies are maintained free from the in\nfluence of the Gouy phase during \npropagation. Interestingly, the optical singularities, including both phase and polarization \nsingularities, can also affect the spin\n–\nmomentum properties significantly. Our findings describe the \nintrinsic spin\n–\nmomentum prope\nrties in paraxial optical systems and apply in the analysis of the \nproperties\n \nof spin\n–\nmomentum in optical focusing, imaging, and scattering systems.\n \n \nK\neywords\n.\n \nSpin angular momentum of light\n, \nspin\n-\nmomentum locking,\n \noptical \nskyrmions, \nparaxial \nlight,\n \nstructured light\n \n \nI. Introduction\n \nSpin angular momentum (SAM) is a fundamental dynamical property of elementary particles and \nclassical wave fields\n1\n–\n14\n \nand plays a critical role in understanding spin\n–\norbit\n15\n–\n29\n \nand wave\n–\nmatter\n1,2\n,30\n–\n32\n \ninteractions and predicting the behavior of interacting systems. For a classical electromagnetic (EM) \nfield, the SAM associated with circular or elliptical polarization can be orientated in an arbitrary \ndirection\n15,16,33,34\n. From this general perspe\nctive, for plane\n-\nwave solutions of Maxwell’s equations, the \nSAM component oriented along the propagating direction (defined by the wavevector or canonical \nmomentum) is considered as the longitudinal spin\n33\n \n(L\n-\nspin), whereas the SAM component oriented \nperpendicular to the wavevector is the transverse spin\n34\n \n(T\n-\nspin)\n.\n \nTo date, T\n-\nspins have been intensively \ninvestigated in various EM systems, including confined optical systems such as evanescent waves\n35\n–\n40\n, \ngu\nided modes\n41\n–\n45\n, and free\n-\nspace optical systems such as the Gaussian focused fields\n46\n–\n50\n, interference \nfields\n51\n, nondiffracting fields\n52\n, and unpolarized fields\n53\n–\n56\n. Remarkably, in confined optical systems, the \nT\n-\nspins of evanescent and guided modes featu\nre a well\n-\nknown property referred to as spin\n–\nmomentum \nlocking\n36,41,42,57,58\n \nand their spin\n–\norbit couplings raise a large class of remarkable phenomena, such as \nunidirectional guided wave\n59\n–\n67\n \nand photonic topological quasiparticles\n68\n–\n78\n, and offer potentia\nl \napplications in angular\n-\nmomentum\n-\nbased optical manipulation\n37,49,50,79,80\n, imaging\n81\n–\n93\n, detection\n94\n–\n96\n, \nmetrology\n97,98\n, and on\n-\nchip quantum technologies\n99\n.\n Previously, in free space, only the spin properties of special optical beams were \ninvestigated\n46\n–\n52\n. \nHowever, for an arbitrary structured light which carries the inhomogeneities of intensity, phase, \npolarization and singularities\n100\n–\n105\n, a unified methodology is lacking in describing the dynamical \nevolving of momentum and angular moment\num, especially regarding paraxial focusing, imaging, and \nscattering systems. Moreover, although optical beams in the free space can form skyrmionic beams\n106\n–\n118\n, the Gouy phase governs the continuous evolution of polarization structures in the propagating \nbeam, \nwhereas SAM is a good candidate in describing the topological invariants of propagating optical beams. \nTherefore, understanding the spin\n–\nmomentum properties of paraxial optical fields is meaningful in \nproviding a guide for spin\n-\nstate manipulation.\n \nHe\nre, we investigate the spin\n–\nmomentum properties of paraxial optical systems and present a \nunified methodology to perform the decomposition of the optical spin into T\n-\nspin and L\n-\nspin for various \ntypes of paraxial optical beams. The results reveal that there\n \nare optical T\n-\nspins that govern optical spin\n–\nmomentum locking and L\n-\nspins determined by the polarization ellipticities (helicities) that are parallel \nto the canonical momentum in paraxial optical beams. Remarkably, from the spin\n–\nmomentum locking \nderived f\nrom optical T\n-\nspin, skyrmion\n-\nlike spin textures can form in paraxial optical beams and their \ntopologies are maintained during propagation, free from the influence of the Gouy phase. We furthermore \ninvestigate the influence of optical singularities on the s\npin\n–\nmomentum \nproperties\n \nin paraxial optical \nsystems and discover an extraordinary SAM component. The direction of its vector, which is \nperpendicular to that of canonical momentum, is not determined by the vorticity of the kinetic momentum \nbut by the polari\nzation topological charge of the vector vortex beam. This SAM component is \ncharacterized by ℤ\n4\n \ntopological invariants and should be considered as a L\n-\nspin. Our findings are general \nfor high\n-\norder structured light beams constructed through the linear superpositions of paraxial optical \nmodes and can help in the understanding of the dynamic\nal properti\nes\n \nof light and broadens the study of \ntopological quasiparticles in paraxial systems.\n \n \nII\n \nTheory\n \nG\neneral \ntheoretical results\n \nFor monochromatic, time\n-\nharmonic EM waves in the paraxial approximation with complex electric field \nstrength \nE\n \nand magnetic field strength \nH\n \nhaving angular frequency dependence \nω\n, \nwe demonstrate that \nthe kinetic momentum \n\u0000\n=\nRe\n{\n\u0000\n∗\n×\n\u0000\n}\n2\n\u0000\n\u0000\n⁄\n, with \nc\n \nthe velocit\ny of light in vacuo, and the SAM \n\u0000\n=\nIm\n{\n\u0000\n(\n\u0000\n∗\n×\n\u0000\n)\n+\n\u0000\n(\n\u0000\n∗\n×\n\u0000\n)\n}\n4\n\u0000\n⁄\n , with \nε\n \nand \nμ\n \nthe permittivity and permeability of the vacuum and \nsuperscript \n*\n \nsignifying complex conjugation\n119\n–\n123\n, have an inherent relationship given by\n \n2\n1\n2\nT\nk\n \nS p\n \nand\n \nL T\n \nS S S\n. \n  \n    \n \n \n               \n(1)\n \nHere, \nk\n \n= \nω\n/\nc\n \ndenotes the wave number, \nS\nT\n \nand \nS\nL\n \ndenote the optical T\n-\nspin and L\n-\nspin, respectively. \nThese equations show that, in the paraxial optical systems, the T\n-\nspin stems from the field \ninhomogeneities and given by the vorticity of the kinetic momentum density. The transversality of the \noptical T\n-\nspin is confirmed from the identity \n∇\n·(\n∇\n×\nA\n) = 0. However, it does not mean that the curl of the \nkinetic momentum is always perpendicular to the kinetic momentum. By decomposing the structured \nlight into a superposition of plane waves, the perpendicularity\n \nof the momentum vector and the optical \nT\n-\nspin vector is satisfied for each plane wave basis, as for example the evanescent plane wave\n35,36\n. \nRemarkably, the T\n-\nspin is capable of producing spin\n–\nmomentum locking\n36,41,42,57\n, i.e., if the kinetic \nmomentum is r\neversed, the T\n-\nspin is inverted correspondingly. In addition, the T\n-\nspin is a classical \nphysical quantity, and the kinetic momentum describing the group velocity of photons is considered as the current of light\n119,121\n. Thus, spin\n–\nmomentum locking from the \nT\n-\nspin, which may also be considered \nas \nspin\n–\ncurrent locking, originates from the intrinsic spin\n–\norbit property inherent in Maxwell’s equations \nand is different from the quantum spin Hall effect in condensed matter physics\n124\n.\n \nIn contrast, the L\n-\nspin is ex\ntracted by subtracting the T\n-\nspin from the total SAM. In the following, \nwe shall demonstrate that this difference, which is related to the three\n-\ndimensional (3D) polarization \nellipticities along the local wavevector\n15\n–\n17\n \nand the Berry curvature of paraxial optical fields\n15\n, is \ndefinitely the L\n-\nspin. Here, we primarily focus our attention on the spin\n–\nmomentum properties of the \nparaxial modes, the solutions of the Helmholtz equation, in both Cartesian and cylindrical coordi\nnates, \nand high\n-\norder structured light beams constructed from linear superpositions of these modes\n125\n–\n130\n.\n \n \nSpin\n-\nmomentum property of Hermite\n-\nGaussian beams\n \nWe first consider a Hermite\n-\nGaussian (HG) beam, the solution to the Helmholtz equation in the parax\nial \napproximation in Cartesian coordinates (\nx\n, \ny\n, \nz\n) with unit vector (\n\u0000\n\u0000\n, \n\u0000\n\u0000\n, \n\u0000\n\u0000\n), propagating along the \nz\n \naxis. \nThe electric and magnetic field components can be expressed as\n \ni\nHG HG HG HG\n1\nˆ\nˆ\nˆ\n, ,\ni\nkz\nx y x y\nu u u e\nk x y\n   \n\n\n \n \n \n    \n \n \n \n \n \nE x y z\n  \n  \n           \n(2)\n \nand\n \ni\nHG HG HG HG\n1\nˆ\nˆ\nˆ\n, ,\ni\nkz\ny x y x\nk\nu u u e\nk x y\n   \n\n\n\n \n \n \n    \n \n \n \n \n \nH x y z\n,  \n  \n         \n(3)\n \nwhere \nη\nx\n \nand \nη\ny\n \nare arbitrary complex constants describing the relative strength, \nIm\n{\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n}\n \nspecifies the \npolarization ellipticity (helicity) of the paraxial HG beam, and the superscript T indicates the transpose \nof the matrix. The electric field \nE\nHG\n \nand magnetic field \nH\nHG\n \nalso satisfy Gauss’s law (\n∇\n·\nE\nHG\n \n= 0 and \n∇\n·\nH\nHG\n \n= 0) in the paraxial approx\nimation (\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n\u0000\u0000\n\u0000\n≪\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n\u0000\u0000\n≪\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n). The complex amplitude \nu\nHG\n \nis \ngiven by\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n2 2\n2 2\n1\n0\nHG,\n2\n2 2\nexp i exp i 1 tan\n2\nmn m n\nR\nk x y\nw\nx y x y z\nu H H m n\nw z w z w z R z z\nw z\n\n \n\n     \n \n\n \n     \n \n   \n \n \n \n   \n \n     \n \n. (4)\n \nHere, \nH\nm\n(\nx\n) is the Hermite polynomial with non\n-\nnegative integer index \nm\n, \n\u0000\n\u0000\n=\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n⁄\n \nthe \nRayleigh \nrange, \n\u0000\n(\n\u0000\n)\n=\n\u0000\n\u0000\n\u0000\n1\n−\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n⁄\n \nthe beam width of the propagating wave, \nw\n0\n \nthe beam radius at the beam \nwaist, \nR\n(\nz\n) the radius of curvature of the wavefronts, \nλ\n \nthe wavelength, and the last factor \nexp\n\u0000\n−\ni\n(\n1\n+\n\u0000\n+\n\u0000\n)\ntan\n\u0000\n\u0000\n(\n\u0000\n\u0000\n\u0000\n⁄\n)\n\u0000\n \nis the Gouy phase\n107\n. From Eq\nuation \n2, 3, and 4, one finds that \nthe \nparaxial HG beam displays both intensity and phase inhomogeneities\n, whereas the polarization is \nhomogeneous in the transverse propagating plane (\nxy\n-\nplane)\n.\n \nEmploying Eq\nuation\n \n2 and 3, the kinetic momentum of \nparaxial HG beam\n \nis\n \n\n\n\n\n\n\nHG HG HG\nHG\nHG HG HG\nHG\nHG HG\n1\nˆ\ni\n1\nˆ\nRe\n2 i\n1\nˆ\ni 0\ni\nx x y y x y\nx x y y x y\nx x y y\nu u u\nu\nk x y\nu u u\nk\nu\nk y x\nk u u\nk\n     \n\n     \n\n   \n\n   \n\n   \n  \n \n \n \n  \n \n \n \n \n \n \n \n \n \n   \n \n \n \n \n \n \n \n  \n \n \n \nx\np y\nz\n. \n     \n \n \n     \n(5)\n From the spin\n–\norbit decomposition theory of the kinetic momentum for classical EM fields\n41,42,119,121\n, the \nkinetic momentum of an optical field can be decomposed into canonical and spin momentum \ncomponents \n(\np\n \n= \np\no\n \n+ \np\ns\n). One recognizes the first term in Eq\nuation \n5 as the canonical momentum (\n\u0000\n\u0000\n∝\n⟨\n\u0000\n|\ni\n∇\n|\n\u0000\n⟩\n \nwith \nψ\n \nthe 6\n-\nvector photonic wave function\n131,132\n), and the \nz\n-\ncomponent of the canonical momentum as \nbeing proportional to the energy \ndensity (\n\u0000\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n+\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n∗\n\u0000\n\u0000\u0000\n ). In contrast, the second term in \nEq\nuation \n5 is the Belinfante spin momentum \np\ns\n \n= \n∇\n×\nS\n/\n2\n119,121\n, in which the SAM is proportional to the \npolarization ellipticity \nIm\n\u0000\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n∗\n\u0000\n\u0000\u0000\n33\n. Generally, for structured light beams, the Belinfante spin \nmomentum is nonzero, and thus the kinetic momentum is not parallel to the canonical momentum, which \ndeterm\nines the local wavevector \nk\n \nthrough relation \n\u0000\n\u0000\n=\nℏ\n\u0000\n.\n \n \nFigure 1.\n \nMomentum properties of paraxial HG\n12\n \nbeams in the pre\n-\nfocal, focal and post\n-\nfocal planes: (a) \nx\n-\ncomponent \np\nx\n, (b) \ny\n-\ncomponent \np\ny\n, and (c) \nz\n-\ncomponent \np\nz\n \nof the kinetic momentum at plane \nz\n \n= −100\nλ. \nBy analyzing their vector structure, the horizontal kinetic momentum components contain multiple \nvortex structures. The \nz\n-\ncomponent kinetic momentum is always nonzero, hence the photons of this HG \nbeam precess d\nuring propagation. (d\n–\nf) as for (a\n–\nc), but at plane \nz\n \n= 0.001\nλ; (\ng\n–\ni) same as (a\n–\nc), but at \nplane \nz\n \n= +100\nλ\n. \nHere, \nη\nx\n \n= 1, \nη\ny\n \n= 2i, \nw\n0\n \n= 8 µm, \nλ = 0.6328 µ\nm.\n \nTo understand the momentum properties of paraxial HG beams, we plot the three components of \nthe ki\nnetic momentum densities for the HG\n12\n \nmode in the pre\n-\nfocal plane, focal plane, and post\n-\nfocal \nplane in Figure 1. From Figure 1a\n-\nb, d\n-\ne, and g\n-\nh, the horizontal kinetic momenta differ slightly through \nconvergence in the pre\n-\nfocal plane and divergence in th\ne post\n-\nfocal plane compared with those in the \nfocal plane. By analyzing the vector properties of these horizontal kinetic momentum densities, one can \nfind that they contain multiple vortex structures \n(\nFigure 3a\n)\n. Moreover, from Figure 1c, f and i, one find\ns \nthat the \nz\n-\ncomponent kinetic momenta in these planes are parallel. Thus, the photons of this paraxial HG \nmode precess around the \nz\n-\naxis and these spiral trajectories of the photons give rise to the geometric \nphase\n133\n-\n135\n. Note that the Gouy\n \nphase in Eq\nuation \n4 does not affect the properties of the kinetic momenta \nwhen the beam passes through the focal plane, and thus the vortex structures of the kinetic momenta \nremain unchanged during propagation.\n \n \n \n \nFigure 2.\n \nSpin properties of paraxial HG\n12\n \nbeams in the focal plane. The total spin is three dimensional \nwith: (a) \nx\n-\ncomponent \nS\nx\n, (b) \ny\n-\ncomponent \nS\ny\n, and (c), \nz\n-\ncomponent \nS\nz\n. The total spin decomposes into a \nT\n-\nspin that contributes to spin\n–\nmomentum locking and a L\n-\nspin determined by the polarization \nellipticities. The (d) \nx\n-\n \nand (e) \ny\n-\ncomponent T\n-\nspins originate from the inhomogeneity of the kinetic \nmomentum density and the (\nf) \nz\n-\ncomponent T\n-\nspin relates to the Berry curvature. For the L\n-\nspin, the \nx\n-\n \nand \ny\n-\ncomponents, (g) and (h) respectively, are considered as projections of the \nz\n-\ncomponent (i) onto the \nhorizontal axes. In the plane \nz\n \n= 0.001\nλ, \nboth components are too small a\nnd can be ignored. The beam \nparameter settings are same as given in Figure 1.\n \nTo exhibit the spin property of the paraxial HG beam, the total SAM and the decomposed T\n-\nspin \nand L\n-\nspin given in Eq\nuation \n1 were calculated,\n \n\n\n\n\n\n\n\n\nHG HG HG HG\nHG HG\nHG HG HG HG\nHG HG\nHG HG\n1\nˆ\ni\n1\nˆ\nIm\n4 i\nˆ\n2\nx x y y x y x y\nx x y y x y x y\nx y y x\nu u u u\nu u\nk y x x\nu u u u\nu u\nk x y y\nu u\n       \n\n       \n\n   \n \n    \n \n    \n  \n \n \n \n  \n    \n \n \n \n  \n \n \n \n \n \n \n \n  \n \n     \n \n \n \n  \n \n \n \n \n \n \n\n\n \n\n \nx\nS y\nz\n\n\n,\n   \n  \n(6)\n \n\n\n\n\n\n\n\n\n\n\n\n\nHG HG\nHG HG\n2\n2\nHG HG HG HG\n2\n1\nˆ\ni\n1 1\nˆ\nIm\n4 i\n2\n1 1\nˆ\n2\nx x y y\nT x x y y\nx x y y x y y x\nz\nu u\nk y\nu u\nk x\nk\nu u u u\nk\n   \n\n   \n\n       \n\n \n\n \n     \n\n \n \n\n \n \n \n\n \n \n \n \n\n \n     \n \n \n\n \n \n \n \n \n     \n \n \n \n \nx\nS p y\nz\n, \n \n(7)\n \nand\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\nHG HG\nHG HG\nHG HG\nHG HG\nHG HG\n2\nHG HG HG HG\n2 2\n1\nˆ\ni\n1\nˆ\nIm\n4 i\n2\nˆ\n1 1\n2\nx y x y\nL x y x y\nx y y x\nx x y y x y y x\nz\nu u\nu u\nk x x\nu u\nu u\nk y y\nu u\nu u u u\nk k\n   \n\n   \n\n   \n       \n\n  \n\n  \n  \n     \n\n\n\n \n \n \n\n  \n \n \n\n \n \n \n \n \n \n \n   \n \n\n \n \n \n \n \n \n\n \n \n      \n \n \nx\nS y\nz\n\n\n\n\n \n \n \n\n \n \n \n \n \n \n \n.\n   \n \n(\n8\n)\n \nThe total SAM given in Eq\nuation \n6 are plotted in Figure 2a\n-\nc. The first terms of the horizontal SAMs are \nT\n-\nspins \n(\nEq\nuation \n7), which originate from the \ninhomogeneities of the EM fields and are proportional to \nthe transverse gradients of the kinetic momenta in the paraxial approximation\n17,41,42\n; see Figure 2d\n-\nf. The \nother terms comprise the L\n-\nspin, which is determined by the polarization ellipticity of the\n \nEM field; see \nFigure 2g\n–\ni. The horizontal components of L\n-\nspin in Eq\nuation \n8 can be understood further by considering \nthe complex amplitude \nu\nHG\n \nas an approximate superposition of plane waves (exp(i(\nk\nx\nx\n+\nk\ny\ny\n)))\n125\n, and \nhence satisfy \n\u0000\n\u0000\n\u0000\u0000\n\u0000\u0000\n⁄\n=\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n, \n\u0000\n\u0000\n\u0000\u0000\n\u0000\u0000\n⁄\n=\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n \nwith \nk\nx\n/\nk\n \nand \nk\ny\n/\nk\n \nrepresenting the horizontal \ndirectional vector components. Thus, these horizontal SAM components can be regarded as projections \nof the L\n-\nspin onto the horizontal axes when the EM wave is either converging or diver\nging. In this way, \nthe horizontal SAM components are nearly zero at the focal plane; see Figure 2g\n–\nh.\n \nMoreover, from the analysis above, the photons of paraxial HG modes possess spiral trajectories, \nleading to strong spin\n–\norbit coupling, as evident in the \nz\n-\ncomponent T\n-\nspin\n \n(\nEq\nuation \n9\n)\n. Therein, the \nfirst term (\nIm\n{\n∇\n\u0000\n\u0000\u0000\n∗\n×\n∇\n\u0000\n\u0000\u0000\n}\n∝\n∇\n×\n\u0000\n\u0000\n ) has a similar form to the Berry curvature of the optical \npotential\n73,136\n, which is related to the evolution of the geometric phase in a paraxial optical system. The \nlast \nterm in Eq\nuation \n7 originates from the vorticity of spin momentum \np\ns\n, i.e., \n∇\n×\n\u0000\n\u0000\n=\n−\n∇\n\u0000\n\u0000\n2\n⁄\n≈\n−\n\u0000\n\u0000\nIm\n\u0000\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n−\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n\u0000\n∇\n\u0000\n\u0000\n(\n\u0000\n\u0000\u0000\n∗\n\u0000\n\u0000\u0000\n)\n \nand \n\u0000\n≈\n\u0000\n\u0000\n2Im\n\u0000\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n−\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n∗\n\u0000\n\u0000\u0000\n. Here, the transverse Laplace \noperator is \n∇\n\u0000\n\u0000\n=\n\u0000\n\u0000\n\u0000\u0000\n\u0000\n⁄\n+\n\u0000\n\u0000\n\u0000\u0000\n\u0000\n⁄\n. The physical \nmeaning of this quantity can be further understood in \na more general context. Employing the spin\n–\norbit decomposition to calculate the vorticity of the kinetic \nmomentum directly, one obtains\n41\n \n\n\n\n\n\n\n\n\n\n\n2\n2 2 2\n2\n1 1 1 1 1\n2\n2 2 2\n1\nRe\n2\n8\nt o s\ni\nk k k\n \n\n\n\n \n\n \n \n             \n \n \n \n       \n \n \n \n       \n \n \nS p p p S\nE H E H\nS\nH E H E\n,\n \n \n \n      \n(9)\n \nwhere\n 1 2 1 2 1 2\n1 2 1 2 1 2 1 2\n1 2 1 2 1 2\nx x y x z x\nx y y y z y\nx z y z z z\n \n \n \n \n \n \nr r\n.\n  \n \n \n  \n \n                  \n(10)\n \nThe T\n-\nspin given by the vorticity of kinetic momentum has a similar structure to the quantum 2\n-\nform\n137\n \nthat generates the Berry phase associated with a circuit. Thus, the \nz\n-\ncomponent of T\n-\nspin \n(\nEq\nuation \n7\n)\n \ncan be regarded as the Berry curvature of the optical potential that originates from the spiral trajectories \nof photons. If polarization ellipticity is \nabsent (\nη\nx\n \n= 0 or \nη\ny\n \n= 0 or the \nη\nx\n \nand \nη\ny\n \nare in\n-\nphase), the photons \ndo not undergo spiral trajectories and thus the \nz\n-\ncomponent T\n-\nspin is nearly zero, i.e., no Berry phase is \ngenerated in this instance. These Berry curvature\n-\nrelated terms are also include\nd in the expression for L\n-\nspin \n(\nEq\nuation \n8\n)\n \nbecause they describe the evolution of the polarization ellipticities during propagation. \nNote that for paraxial optical systems the conditions imposed are \n\u0000\n\u0000\n\u0000\u0000\n\u0000\n⁄\n≪\n\u0000\n\u0000\n\u0000\u0000\n⁄\n≪\n\u0000\n\u0000\n  \nand \n\u0000\n\u0000\n\u0000\u0000\n\u0000\n⁄\n≪\n\u0000\n\u0000\n\u0000\u0000\n⁄\n≪\n\u0000\n\u0000\n \n(as considered in Figure 2) and the integrals of the horizontal component of \nthe L\n-\nspin and the Berry curvature\n-\nrelated term on the transverse plane are zero; one then concludes that \nthe SAMs are conserved in the paraxial approximation, in agreement with \nr\nef\nerence\ns\n138\n-\n141\n. In summary, \nEq\nuation \n8, which represents the projection of the polarization ellipticity onto the 3D axes and includes \nthe Berry curvature, is definitely the L\n-\nspin component.\n \n \nFigure 3.\n \nMomentum and spin textures of the paraxial HG\n12\n \nbeam: (a) The vector diagram of kinetic \nmomentum at the focal plane. The horizontal components contain multiple vortex structures, and thus \nthe photons precess around the \nz\n-\naxis. (b) The normalized spin vector of the total spin. For each texture, \nthe spin\n \nvector varies from the center “up” state to the boundary “horizontal” state, and thus the skyrmion \nnumber of each spin texture is −1/2 (half\n-\nskyrmion). (c) The normalized spin vector of the T\n-\nspin. The \nT\n-\nspin vectors vary from the center “up” state to the\n \nboundary “down” state and rotate in helical spirals, \nwhich is a manifestation of a Bloch\n-\ntype skyrmion. The skyrmion number of each skyrmion\n-\nlike spin \ntexture is equal to −1. Skyrmion\n-\nlike spin textures are formed in each transverse plane through spin\n–\nmom\nentum locking of the T\n-\nspin and their topologies are maintained during propagation, free from the \ninfluence of the Gouy phase. Here, the topological number (skyrmion number) is calculated from \n\n\n4\nN x y\n\n\n     \n\nm m m\n68\n, where \nm\n \n= \nS\n/|\nS\n| is the normalized\n \nspin vector.\n \nIn the confined optical field, a vortex structure of the kinetic momentum may result in a skyrmion\n-\nlike spin texture\n41,42,68\n. Here, if there is a nonvanishing polarization ellipticity (\nIm\n\u0000\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n−\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n\u0000\n≠\n0\n), a \nsimilar vortex structure for the horizontal kinetic momentum is observed \n(\nFigure 3a\n)\n. The vortex structure \nof the horizontal kinetic momentum causes the formation a skyrmion\n-\nlike spin texture\n \nwith T\n-\nspin \nthrough spin\n–\nmomentum locking of the T\n-\nspin\n \n(Fi\ngure 3\nb\n, \nc\n)\n. In contrast to the confined field case in \nwhich the spin vector of spin texture rotates in cycloidal spirals (Néel\n-\ntype skyrmion), the spin texture \nof the paraxial HG beam possesses a Bloch\n-\ntype configuration\n \n(Figure 3c)\n, for which the spin ve\nctor \nrotates in helical spirals. Moreover, the skyrmion\n-\nlike spin texture manifest spin\n–\nmomentum locking and \nthe kinetic momentum is maintained during propagation. Thus, the topology of the skyrmion\n-\nlike spin \ntexture is also unchanged during propagation an\nd is unaffected by the evolution of the Gouy phase, in \ncontrast to that of a paraxial skyrmionic beam\n107\n. Note that if there is a local disturbance (such as a \nnanoparticle) causing a kinetic momentum deflection, the topology of the skyrmion\n-\nlike spin textu\nre \nbreaks down. The skyrmion number of each skyrmion\n-\nlike spin texture is −1 when the polarization \nellipticity \nIm\n\u0000\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n−\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n\u0000\n>\n0\n.\n \n \nSpin\n-\nmomentum property of Laguerre\n-\nGaussian beams carrying phase singularit\nies\n.\n \nNext, we consider a paraxial Laguerre\n-\nGaussian (LG) beam\n142\n \npropagating along the \nz\n \naxis in the \ncylindrical coordinates (\nρ\n, \nφ\n, \nz\n) with unit vector (\n\u0000\n\u0000\n, \n\u0000\n\u0000\n, \n\u0000\n\u0000\n). The electric and magnetic field components \nbecome\n \n\n\n\n\ni\nLG LG\nLG LG LG\ni\n1\nˆ\nˆ\nˆ\n, , cos sin sin cos\nkz\nx y x y x y\nu lu\nu u e\nik\n         \n \n\n \n\n \n     \n \n \n\n \n \nE x y z\n,\n \n \n \n(11)\n \nand\n \n\n\n\n\ni\nLG LG\nLG LG LG\ni\n1\nˆ\nˆ\nˆ\n, , sin cos cos sin\nkz\ny x x y x y\nu lu\nk\nu u e\nik\n         \n  \n\n \n\n \n     \n \n \n\n \n \nH x y z\n.\n \n(12)\n \nHere, \nη\nx\n \nand \nη\ny\n \nare arbitrary complex constants describing the relative strength and polarization ellipticity, \nrespectively. We use these expressions to ensure the polarization state is homogeneous in the transverse \nplane. The electric field \nE\nLG\n \nand magnetic field \nH\nLG\n \nsa\ntisfy Gauss’s law (\n∇\n·\nE\nLG\n \n= 0 and \n∇\n·\nH\nLG\n \n= 0) in \nthe paraxial approximation. The complex amplitude \nu\nLG\n \nis given by\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n2 2 2\n1\n0\nLG,\n2 2\n2 2\nexp exp exp 1 2 tan\n2\nl\nl\npl p\nR\nw\nk z\nu L i il i p l\nw z w z R z z\nw z w z\n   \n\n\n   \n   \n \n     \n \n \n \n \n \n \n \n \n \n \n   \n   \n.\n \n \n(13)\n \nHere, \nL\nl\np\n(\nx\n) denotes the generalized Laguerre polynomial, \nl\n \nthe topological charge of vortex phase, and\n \nexp\n\u0000\n−\ni\n(\n1\n+\n2\n\u0000\n+\n|\n\u0000\n|\n)\ntan\n\u0000\n\u0000\n(\n\u0000\n\u0000\n\u0000\n⁄\n)\n\u0000\n  \nis the Gouy phase. From Eq\nuation \n11 and 12, one obtains the \nkinetic momentum of the paraxial LG beam,\n \n\n\n\n\n\n\nLG LG LG\nLG\nLG LG LG\nLG\nLG LG\n1 1\nˆ\ni\n1 1 1\nˆ\nRe\n2 i\n1\nˆ\ni 0\ni\nu u u\nu\nk\nu u u\nk\nu\nk\nk u u\nk\n     \n     \n   \n     \n  \n     \n   \n   \n\n   \n\n   \n  \n \n \n \n  \n \n \n \n \n \n \n \n \n \n   \n \n \n \n \n \n \n \n  \n \n \n \nρ\nP\nφ\nz\n,\n   \n \n \n \n \n   \n  \n(14)\n \nwhich has a similar form to that of Eq\nuation 5\n. Here, we use substitutions \n\u0000\n\u0000\n=\n\u0000\n\u0000\ncos\n\u0000\n+\n\u0000\n\u0000\nsin\n\u0000\n \nand \n\u0000\n\u0000\n=\n−\n\u0000\n\u0000\nsin\n\u0000\n+\n\u0000\n\u0000\ncos\n\u0000\n. Thus, through energy conservation, the polarization ellipticities are \n\u0000\n\u0000\n∗\n\u0000\n\u0000\n−\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n=\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n−\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n \nand \n\u0000\n\u0000\n∗\n\u0000\n\u0000\n+\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n=\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n+\n\u0000\n\u0000\n∗\n\u0000\n\u0000\n. In this \ninstance, \nη\nρ\n \nand \nη\nφ\n \nare \nφ\n-\ndependent \nwith \n\u0000\n\u0000\n\u0000\n\u0000\u0000\n⁄\n=\n\u0000\n\u0000\n \nand \n\u0000\n\u0000\n\u0000\n\u0000\u0000\n⁄\n=\n−\n\u0000\n\u0000\n. The kinetic momentum of the paraxial LG\n21\n \nbeam is found \nfrom Figure 4a, b. The first and second terms of Eq\nuation \n14 are the canonical momentum and the \nBelinfante spin momentum,\n \nrespectively, and \n\u0000\n\u0000\n\u0000\u0000\n∗\n\u0000\n\u0000\u0000\n\u0000\u0000\n⁄\n=\n0\n. The paraxial LG beam carries phase \nsingularities, which are associated with the orbital angular momenta (OAMs) of light\n142\n. In general, the photons of paraxial LG beam possess spiral trajectories, which derive from \nboth the SAM and OAM of \nlight.\n \n \nFigure 4. \nSpin\n–\nmomentum properties of paraxial LG\n21\n \nbeams in the focal plane z = 0.001\nλ: (\na) azimuthal \np\nφ\n \nand (b) z\n-\ncomponent \np\nz\n \nkinetic momenta result in spiral trajectories of the photons (the radial SAM \nvanishes in the focal plane); (c) azimuthal \nS\nφ\n \nand (d) \nz\n-\ncomponent \nS\nz\n \nspin angular momenta decompose \ninto T\n-\nspin and L\n-\nspin; (e) azimuthal T\n-\nspin stems from the radial gradient o\nf \nz\n-\ncomponent kinetic \nmomentum density whereas (f) \nz\n-\ncomponent T\n-\nspin is related to the Berry curvature; (g) azimuthal L\n-\nspins are considered projections of (h), the \nz\n-\ncomponent L\n-\nspin onto the azimuthal direction. Here, \nη\nx\n \n= \n1, \nη\ny\n \n= 2i, \nw\n0\n \n= 8 µm, \nλ = 0.6\n328 µ\nm.\n \nTo exhibit the spin \nproperty\n \nof the paraxial LG beam, the total SAM and the decomposed T\n-\nspin \na\nnd L\n-\nspin are expressed in the form\n \n\n\n\n\n\n\nLG LG\nLG LG\nLG LG LG LG\nLG LG\nLG LG\n1\nˆ\n0\ni\n1 1 1\nˆ\nIm\n4 i\nˆ\n2\nu u\nu u\nk\nu u u u\nu u\nk\nu u\n   \n       \n   \n   \n \n\n       \n     \n   \n\n   \n \n    \n  \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n  \n \n     \n \n \n \n  \n \n \n \n \n \n \n\n \n \n \n \nρ\nS\nφ\nz\n,\n \n \n \n \n(1\n5\n)\n \n\n\n\n\n\n\n\n\n\n\nLG LG\n2\n2\nLG LG LG LG\n2\nˆ\n0\n1 1\nˆ\nIm\n4 i\n2\n1 1\nˆ\n2\nT\nz\nu u\nk\nk\nu u u u\nk\n   \n       \n\n   \n \n       \n\n \n     \n\n \n \n \n \n \n\n \n     \n \n\n \n \n \n \n \n     \n \n \n \n \nρ\nS p\nφ\nz\n, \n \n(1\n6\n)\n \nand\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLG LG\nLG LG\nLG LG\nLG LG\nLG LG\n2\nLG LG LG LG\n2\n1\nˆ\ni\n1 1 1\nˆ\nIm\n4 i\n2\n1 1\n2\nL\nu u\nu u\nk\nu u\nu u\nk\nu u\nu u u u\nk\n   \n   \n   \n       \n   \n \n\n   \n    \n   \n       \n\n  \n\n  \n  \n     \n\n \n \n \n  \n \n \n \n \n \n \n \n \n \n   \n \n \n \n \n \n \n \n \n \n \n \n      \n\n \n \n \nρ\nS\nφ\nˆ\n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \nz\n.\n \n  \n(\n1\n7\n)\n \nThe \ntotal spin components, Eq\nuation \n15, are shown in Figure 4c, d. The first terms of the horizontal \nSAMs, which are proportional to the transverse gradients of the kinetic momenta, are T\n-\nspins, Eq\nuation \n16, which are plotted in Figure 4e, f, whereas the \nother terms comprise the L\n-\nspin, Eq\nuation \n17, which \nare plotted in Figure 4g, h. The horizontal L\n-\nspin components are regarded as projections of the total L\n-\nspin onto the horizontal axis as evident from Eq\nuation \n8.\n \nAs mentioned above, for the paraxial LG m\nodes, if the total angular momentum is nonzero, the \nphotons exhibit spiral trajectories\n142\n. Interestingly, either the SAM or the OAM associated spiral \ntrajectories generates a Berry phase. Thus, the \nz\n-\ncomponent of the T\n-\nspin in Eq\nuation \n16 and the last two\n \nterms in Eq\nuation \n17 are related to the Berry curvature of the optical potential associated with the spiral \ntrajectories of photons. If polarization ellipticity is absent (\nη\nx\n \n= 0 or \nη\ny\n \n= 0 or \nη\nx\n \nand \nη\ny\n \nare in\n-\nphase), the \nphotons of the paraxial LG modes a\nlso possess spiral trajectories and a nonvanishing Berry curvature \n(\nIm\n{\n∇\n\u0000\n\u0000\u0000\n∗\n×\n∇\n\u0000\n\u0000\u0000\n}\n≠\n0\n ) also exists. This is the primary distinction between the paraxial HG beam \nwithout optical singularities and the paraxial LG beam carrying phase \nsingularities.\n \n \nFigure 5. \nMomentum and spin vectors of the paraxial LG\n21\n \nbeam: (a) The vector diagram of kinetic \nmomentum at the focal plane. The horizontal components contain multiple vortex structures (along the \nradial direction), and thus the photons p\nrecess around the \nz\n-\naxis. (b) The normalized spin vector of the \ntotal spin. The spin vector varies from the center “up” state to boundary “horizontal” state in the radial \ndirection, and thus the skyrmion\n \nnumber of each spin texture is −1/2 (half\n-\nskyrmion). (c) The normalized \nspin vector of the T\n-\nspin. For the center spin texture, the T\n-\nspin vectors vary from the center “up” state \nto boundary “down” state and rotate in helical spirals. The skyrmion number \nof this skyrmion\n-\nlike spin \ntexture is equal to −1. As the radius increases, the \nz\n-\ncomponent T\n-\nspin is too small and the spin texture \nis unidentifiable.\n \nIn addition, for the paraxial system, the conditions \n\u0000\n\u0000\n\u0000\u0000\n\u0000\n⁄\n≪\n\u0000\n\u0000\n\u0000\u0000\n⁄\n≪\n\u0000\n\u0000\n  \nand \n1\n\u0000\u0000\n⁄\n≪\n1\n \napply, the integr\nals of the horizontal component of L\n-\nspin and the Berry curvature\n-\nrelated term on the \ntransverse plane are zero, and the SAMs are conserved in the paraxial approximation. Overall, Eq\nuation \n17, which describes the 3D polarization ellipticity and Berry curva\nture, definitively defines the L\n-\nspin. \n \nFrom Eq\nuation \n14\n-\n17 and the above analysis, one finds that the introduction of phase singularities \nproduces an additional contribution to spin\n–\nmomentum locking, specifically, an OAM\n-\nassociated Berry \ncurvature in the \nT\n-\nspin. This OAM\n-\nassociated Berry curvature was also reported in \nr\nef\nerence\ns\n126,127\n.\n \nIn \naddition, the \nkinetic\n \nmomentum of paraxial LG mode contains the vortex structure (Figure 5a), and hence \nthe skyrmionlike spin texture constructed by T\n-\nspin appears at th\ne center (Figure 5b, c).\n \n \nSpin\n-\nmomentum property of Bessel\n-\nGaussian beams carrying \npolarization\n \nsingularit\nies\n.\n \nFinally, we consider a paraxial Bessel\n-\nGaussian (BG) mode\n143\n \ncarrying polarization singularities \npropagating along the \nz\n \naxis in cylindrical coordinates (\nρ\n, \nφ\n, \nz\n) with unit vectors (\n\u0000\n\u0000\n, \n\u0000\n\u0000\n, \n\u0000\n\u0000\n). The electric \nand magnetic field components become\n \n\n\nBG\ni\nBG\nBG BG BG\n1\n1 1\nˆ\nˆ\nˆ\ncos ,sin , cos\ni\nkz\nn u\nu\nu u e\nk\n\n  \n\n\n \n\n \n\n    \n \n \n\n \n \n \nE\nρ\nφ\nz\n,\n \n \n \n \n \n   \n(18)\n \nand\n \n\n\nBG\ni\nBG\nBG BG BG\n1\n1 1\nˆ\nˆ\nˆ\nsin , cos , sin\ni\nkz\nn u\nu\nk\nu u e\nk\n\n   \n\n \n\n \n\n     \n \n \n\n \n \n \nH\nρ\nφ\nz\n.\n \n \n \n \n \n \n \n(19)\n \nHere, we use the substitution \nΦ = [(\nn\n−\n1)\nφ\n \n+ \nφ\n0\n] \nwith \nφ\n0\n \nthe initial angle of the polarization state. When \nn\n \n= 0, the electric/magnetic field distributions prescribe a paraxial Gaussian beam carrying a \nhomogeneous polarization. For \nn\n \n≠ 0, the polarization state is inhomogeneous in the transverse \npropagating plane and depends on the azimuthal coordinate \nφ\n. \nThere \nis a polarization singularity at the \ncenter of the BG beam. The electric field \nE\nBG\n \nand magnetic field \nH\nBG\n \nalso satisfy Gauss’ law (\n∇\n·\nE\nBG\n \n= \n0 and \n∇\n·\nH\nBG\n \n= 0) in the paraxial approximation. The complex amplitude \nu\nBG\n \nis given by\n143\n \n\n\n2\n2 2\n0\nBG\n2\n1\nexp exp\n1 1 1 1\nn\nR R R R\ni z k\nw\nu J\ni z z i z z i z z i z z\n\n\n\n \n \n \n  \n \n \n \n   \n \n \n \n \n,\n \n  \n \n  \n \n \n    \n(20)\n \nwhere \nβ\n \ndenotes a constant that determines the beam profile (\nβ\n \nis positive for this study), and \nJ\nn\n \nthe \nn\n-\norder Bessel function of the first kind. \nFrom the expressions for the electric and magnetic fields of these \nBG modes, one \nobtains a kinetic momentum of\n \nBG\nBG BG BG\n1 1 1\nˆ\nˆ\nˆ\nRe ,0 , i\n2 i i\nu\nk\nu ku u\nk k\n \n \n \n\n \n \n  \n \n \n \n\n \n \np\nρ\nφ\nz\n.\n   \n \n  \n         \n \n(21)\n \nIn the case, there is only canonical momentum, and Belinfante spin momentum is zero approximately \n(\nFigure 6a, e\n)\n. Moreover, the photons do not possess spiral \ntrajectories (\np\nφ\n \n= 0) \nand no Berry phase is \ngenerated. The total SAM and its T\n-\nspin a\nnd L\n-\nspin components are\n \nBG BG BG BG\n1\nˆ\nˆ\nˆ\nIm 0 , 2 ,0\n4 i\nu u u u\nn\nk\n\n  \n\n \n \n \n\n  \n \n \n\n \n \n \nS\nρ\nφ\nz\n,\n   \n  \n  \n \n          \n \n(22)\n \nBG BG\n2\n1 1\nˆ\nˆ\nˆ\nIm 0 , ,0\n4 i\n2\nT\nu u\nk\nk\n\n \n\n\n \n \n\n    \n \n \n\n \n \n \nS p\nρ\nφ\nz\n,\n   \n  \n            \n \n(23)\n \nand\n \nBG BG\n1\nˆ\nˆ\nˆ\nIm 0 , 2 ,0\n4 i\nL\nnu u\nk\n\n \n\n\n \n \n \n \n \n \n \n \nS\nρ\nφ\nz\n.\n  \n \n \n \n                  \n(24)\n In Eq\nuation \n22, one finds that the total SAM (Figure 6b, f) contains two terms: the azimuthal SAMs, \nwhich are proportional to the transverse gradients of kinetic momenta, are T\n-\nspins \n(\nFigure 6c, g\n)\n \nas given \nin Eq\nuation \n23, whereas th\ne other terms are determined by the polarization topological charge. When the \npolarization topological charge \nn\n \n= 0, the polarization state is homogeneous; the result is consistent with \nthat of Eq\nuation \n8, i.e., no L\n-\nspin exists. However, if \nn\n \n≠ 0, the dir\nectional vector of this SAM component \nis determined by the inherent property of the mode, namely, the polarization topological charge \n(\nFigure \n6d, h). Thus, this SAM component in Eq\nuation \n24 can be regarded as an L\n-\nspin, although it is localized \nin the tran\nsverse plane of the kinetic momentum. This extraordinary L\n-\nspin, which does not generate \nspin\n–\nmomentum locking but is a ℤ\n4\n \ntopological invariant, is interesting and was discovered quite \nrecently\n44,57\n. Obviously, no skyrmion\n-\nlike spin textures arise because the \nz\n-\ncomponent SAM vanishes. In \nsummary, the introduction of polarization singularities in the paraxial beam creates an extraordinary L\n-\nspin but no skyrmion\n-\nlike spin textures.\n \n \nFigure 6. \nSpin\n-\nmomentum properties of para\nxial BG beams in the focal plane z = 0: (a) \nz\n-\ncomponent \nkinetic momentum \np\nz\n, (b) azimuthal optical spin \nS\nφ\n, (\nc) optical T\n-\nspin, (d) optical L\n-\nspin for the +3\n-\norder \nparaxial BG beams, (e) z\n-\ncomponent kinetic momentum \np\nz\n, (f) azimuthal optical spin \nS\nφ\n, (\ng) o\nptical T\n-\nspin, and (h) optical L\n-\nspin for the −3\n-\norder paraxial BG beam. The other components are zero. From (a) \nand (e), the kinetic momenta are the same, and thus the T\n-\nspin are also the same; see (c) and (g). However, \nthe L\n-\nspin is inverted when the pol\narization topological charge switches between +3 and −3. This spin, \nwhich is independent of the vorticity of kinetic momentum but depends on the polarization singularity, \nis the L\n-\nspin (and is a ℤ\n4\n \ntopological invariant). Here, \nw\n0\n \n= 8 µm, \nλ = 0.6328 µ\nm.\n \n \nIII \nConclusion\n \nTo summarize, we presented a unified methodology to describe the spin\n–\nmomentum properties in the \nparaxial optical systems. The theory uncovers the underlying physical difference between T\n-\nspin and L\n-\nspin. Moreover, t\nhe decomposition of T\n-\nspin is consistent with the Helmholtz decomposition theory\n144\n \nand can be generalized to other classical wave fields\n44,145\n. We investigated the influence of optical \nsingularities on the spin\n–\nmomentum properties of paraxial optical beam\ns. For the HG beam without \noptical singularities, skyrmion\n-\nlike spin textures may arise from the T\n-\nspins when the beam elliptically \npolarized. Then, for a LG beam with a phase singularity, the OAM property of the phase singularity \nresults in a Berry curvat\nure term in the T\n-\nspin, and thus a skyrmion\n-\nlike spin texture arises from the T\n-\nspins when the total angular momentum of the beam is nonzero. Moreover, for the BG mode with a \npolarization singularity, no skyrmion\n-\nlike spin texture is formed. Nevertheless, \nwe discovered a SAM \ncomponent, the direction of which is not determined by the kinetic momentum but rather the polarization \ntopological charge of vector vortex beam. This SAM component is a ℤ\n4\n \ntopological invariant and should \nbe considered as an extraordin\nary L\n-\nspin, although its vector is perpendicular to the \ncanonical \nmomentum/wavevector. 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Photon.\n \n2022\n, \n16\n, 261\n-\n262\n.\n \nDOI:\n \n10.1038/s41566\n-\n022\n-\n00982\n-\n4\n "    },    {        "title": "1709.04253v1.Ultrafast_spin_lattice_relaxation_in_ferromagnets_including_effective_spin_orbit_fields.pdf",        "content": "Ultrafast Spin-Lattice Relaxation in Ferromagnets Including\nE\u000bective Spin-Orbit Fields\nKai Leckron, Svenja Vollmar,\u0003and Hans Christian Schneidery\nPhysics Department and Research Center OPTIMAS,\nKaiserslautern University, P. O. Box 3049, 67663 Kaiserslautern, Germany\n(Dated: October 25, 2019)\nAbstract\nWe investigate ultrafast demagnetization due to electron-phonon interaction in a model band-\nferromagnet. We show that the microscopic mechanism behind the spin dynamics due to electron-\nphonon interaction is the interplay of scattering and the precession around momentum-dependent\ne\u000bective internal spin-orbit magnetic \felds. The resulting magnetization dynamics can only be\nmimicked by spin-\rip transitions ifthe spin precession around the internal \felds is su\u000eciently fast\n(compared to the scattering time) so that it averages out the transverse spin components.\nPACS numbers: XXXXX\n1arXiv:1709.04253v1  [cond-mat.mtrl-sci]  13 Sep 2017INTRODUCTION\nIn 3d-ferromagnets, excitation by an ultrashort linearly polarized pulse can reduce the\nmagnetization, as observed by the magneto-optical Kerr e\u000bect [1, 2] or X-ray magnetic\ncircular dichroism [3], by 50% and more, even reaching a complete \\quenching\" of the\nmagnetization for high \ruence. The spin angular momentum, as determined experimentally,\nis thus dramatically reduced. Experimental evidence [4, 5] points mainly to the importance\nof electronic scattering and transport for this e\u000bect.\nAs the transport contribution to magnetization can be suppressed, there must be an ad-\nditional microscopic mechanism that contributes to the observed magnetization dynamics,\nwhich is still under debate [6{9]. Here we focus on the mechanism that has long been re-\ngarded as the most probable explanation of demagnetization dynamics: electronic spin-\rip\nscattering with phonons, which is often called the Elliott-Yafet demagnetization mecha-\nnism [10, 11] after a spin relaxation mechanism for electrons in semiconductors [12{15].\nThe original Elliott-Yafet mechanism was developed for a pair of degenerate bands , whose\nnon-pure spin states are of the general form jk;\u0006i=a(\u0006)\nkj\"i+b(\u0006)\nkj#idue to spin-orbit\ncoupling. The labels \\+\" and \\ \u0000\" indicate whether a state is predominantly spin-up or spin\ndown, depending on which coe\u000ecient jaj2orjbj2is larger. In ferromagnets, the majority and\nminority states are of the same general form, but due to the spin splitting the labels \\+\" and\n\\\u0000\" now also refer to the spin eigenvalues with respect to a quantization axis. Due to spin-\norbit coupling this quantization axis is kdependent and the spin structure belonging to the\n+/\u0000states is essentially noncollinear. The conventional Elliott-Yafet mechanism determines\nthe spin dynamics due to electron-phonon interactions from transition rates between + and\n\u0000states and is therefore incapable of describing deviations from the spin quantization axis,\ni.e., spin coherences. However, these coherences are always present and may be expected to\nbe particularly important if the noncollinearity is prounounced, for instance, at spin-orbit\nhybridization points in band ferromagnets [16].\nIn the present paper, we include consistently the in\ruence of spin-orbit coupling and\nexchange splitting on electron-phonon scattering dynamics and numerically study the case\nof a model ferromagnet. We obtain the ensemble magnetization dynamics from the micro-\nscopic spin-density matrix, which allows us to include spin coherences in noncollinear (i.e.,\nnon-trivially k-dependent) spin structures. While we do not present a complete theory of\n2demagnetization dynamics, we demonstrate how spin-\rip transitions, as they are assumed\nin the conventional Elliott-Yafet mechanism and believed to play an important role in the\ndemagnetization process in ferromagnets, can result from the interplay of precessional spin\ndynamics and spin-independent electron-phonon interaction. Further, we uncover a demag-\nnetization regime for which the conventional Elliott-Yafet mechanism fails.\nBefore we discuss our approach in detail we draw attention to di\u000berences from other\nElliott-Yafet-like treatments. As we neglect the small explicitly spin-dependent electronic\ninteraction with phonons and as the longitudinal acoustic phonons, which are most important\nfor electron-phonon scattering, do not carry angular momentum, the phonons do not take\naway spin in a scattering transition as envisaged in Ref. 17. Thus, it is the spin-orbit coupling\nin the equilibrium electronic and ionic con\fguration (i.e., the lattice), which acts both as a\nspin sink and a spin source. Our approach is also fundamentally di\u000berent from a recent study\nemploying spin coherences for demagnetization dynamics [18], which inconsistently combines\npure spin states with explicitly spin-dependent electron-phonon interaction matrix elements.\nMODEL\nWe employ a ferromagnetic Rashba model because it leads to simple analytical expression\nfor the electronic single-particle states including both spin-orbit coupling and a Stoner mean-\n\feld splitting. Compared to an ab-initio approach , our model is much simpler and works, for\nnumerical simplicity, with a two-dimensional kspace, but, in principle, is not restricted to\nthis particular model. We use the following 2 \u00022 e\u000bective, i.e., k-dependent, hamiltonian [19]\n^He(k) =~2k2\n2me\u000b1+\u000b(kx^\u001by\u0000ky^\u001bx)\u0000\u0001^\u001bz; (1)\nwhich determines the Bloch u-functions at \fnite kin a two-band model. Here, the \frst term\nis a spin-diagonal kinetic contribution with e\u000bective mass me\u000b, the second is a Bychkov-\nRashba spin-orbit term with Rashba parameter \u000band the last term is the mean-\feld ex-\nchange splitting. The Pauli matrices are denoted by ^ \u001b\u000b, with\u000b=x,y,z. The single-\nparticle states of this model at each kpoint are 2-spinors, which we denote by jk;\u0006iwith\ntwo-dimensional k= (kx;ky). These states are used to de\fne the reduced density matrix\nby\u001a\u0017\u00170\nk=hcy\nk\u0017ck\u00170i, wherecy\nk\u0017andck\u0017, respectively, create and annihilate an electron in the\nsingle-particle state jk;\u0017i. The ensemble average is denoted by h\u0001\u0001\u0001i . The density matrix\n3contains the distribution functions nk\u0017=\u001a\u0017\u0017\nkand the coherence \u001a+\u0000\nk. As thejk;\u0006istates\nare non-pure spin states with a kdependent spin mixing, one can compute the kdependent\nexpectation values\nh^\u001b\u000bik=X\n\u0017\u00170hk;\u0017j^\u001b\u000bjk;\u00170i\u001a\u0017\u00170\nk (2)\nThe ensemble spin expectation value is determined by Sz= (~=2)P\nkh^\u001bzik\u0011(~=2)h^\u001bzi.\nThe ferromagnetic character of the model comes from a Stoner model, for which we\nassume an e\u000bective Coulomb energy Ue\u000b. It determines the mean \feld contribution in (1)\nvia \u0001 = (2 =3)Ue\u000bh^\u001bzi. A larger Ue\u000bleads to more robust ferromagnetism with a larger\nexchange splitting. The eigenenergies \u000f\u0006\nk=~2\n2me\u000bk2\u0006p\n\u00012+\u000b2k2are isotropic and the\neigenstates take the form\njk;\u0006i=1\nN\u00060\n@\u0000i\u000bke\u0000i'k\n\u0001\u0006p\n\u00012+\u000b2k21\nA (3)\nwithk=jkj, polar angle 'k, and normalization factor N\u0006=\u0002\n\u000b2k2+\u0000\n\u0001\u0006p\n\u00012+\u000b2k2\u00012\u00031=2.\nThe e\u000bective hamiltonian is formally identical to that of a spin in an external magnetic\n\feld. The Rashba contribution points in the direction ( \u0000kykx0)Tin thex-yplane and\nthe Stoner mean \feld in the zdirection. The sum of the two contributions gives the k\ndependent e\u000bective magnetic \feld. Our choice of parameters is such that the e\u000bective \feld\nis dominated by the mean-\feld exchange and the Rashba contribution only adds a small\ndeviation from the zdirection. The resulting band and spin structure is shown in Fig. 1\nfor parameters Ue\u000b= 720 meV and \u000b= 30 meV nm\u00001, which will be used in the numerical\ncalculations below.\nIn equilibrium, we compute the single-particle states/energies together with the equilib-\nrium density matrix self consistently for a given temperature Teqand density ne. To this\nend, we assume that the equilibrium density matrix is diagonal with n(eq)\nk\u0006=f(\u000f\u0006\nk\u0000\u0016), and\nwe restrict the self-consistent calculation to the z-direction as preferred spin orientation,\nand thus obtain the equilibrium chemical potential \u0016eq. With our choice of Ue\u000bthejk;+i\nstates are mainly spin-up and the jk;\u0000iare mainly spin-down, as shown in Fig. 1. The\nsame parameters are used for all calculations in the paper. Because of the parabolic band\nstructure, the mean-\feld equilibrium realizes a weak ferromagnet.\nThe time-development of the electronic spin-density matrix due to scattering with\nphonons is described in Markov approximation by the dynamical equation\n4\u0000i~@\n@t\u001a\u0017\u00170\nk= (\u000f\u0017\nk\u0000\u000f\u00170\nk)\u001a\u0017\u00170\nk\n+X\nk1\u00171n\ngk1\u00171;k\u0017hX\n\u00172\u00173gk\u00172;k1\u00173\u0010(1 +Nk1\u0000k)(\u000e\u00172\u00170\u0000\u001a\u00172\u00170\nk)\u001a\u00171\u00173\nk1\u0000Nk1\u0000k\u0000\n\u000e\u00171\u00173\u0000\u001a\u00171\u00173\nk1\u0001\n\u001a\u00172\u00170\nk\n\u000f\u00171\nk1\u0000\u000f\u00170\nk\u0000~!k1\u0000k+i~\u0000\n\u0000(1 +Nk\u0000k1)(\u000e\u00171\u00173\u0000\u001a\u00171\u00173\nk1)\u001a\u00172\u00170\nk\u0000Nk\u0000k1(\u000e\u00172\u00170\u0000\u001a\u00172\u00170\nk)\u001a\u00171\u00173\nk1\n\u000f\u00171\nk1\u0000\u000f\u00170\nk+~!k\u0000k1+i~\u0000\u0011i\n\u0000\u0000\n\u0017$\u00170\u0001\u0003o\n(4)\nfor the reduced density matrix. This is a standard expression that is derived, e.g., in\nRef. 15, under the assumption that the phonons are in equilibrium and are described by\na Bose-Einstein distribution Nqfor a given phonon wave vector q. An accurate calculation\nof the electron-phonon matrix element gk\u0017;k0\u00170for a ferromagnetic metal can be done ab\ninitio [20, 21], but for the simple model considered in this paper we make the simplifying as-\nsumption [15] that the matrix element can be related to a deformation potential constant D\naccording to gk\u0017;k0\u00170=p\n~=2MioncDp\njk0\u0000kjhk;\u0017jk0;\u00170i. This is for the interaction with\nacoustic phonons with a linear dispersion ~!q=~cq, wherecis the sound velocity. We\nchoose the candMionvalues for iron. We assume D= 3:2 eV at \frst and study its in\ru-\nence on the dynamics below. The electrostatic electron phonon interaction is obviously spin\nindependent , but there are electron-phonon \\spin-\rip\" matrix elements gk+;k0\u0000because the\nspin-orbit coupling gives rise to nonvanishing overlaps hk;+jk0;\u0000i6= 0.\nThe e\u000bective hamiltonian (1) does not commute with the spin operator ^ \u001bzbecause of the\nspin-orbit coupling, and enters Eq. (4) directly via the spin splitting between the electronic\nenergies\u000f+\nk\u0000\u000f\u0000\nk. This spin splitting leads to a contribution to the equation of motion (4)\nfor the coherence \u001a+\u0000that describes the precession of the spin expectation value (2) around\nthek-dependent e\u000bective internal \feld. Even though the precessional contribution is always\npresent, it only leads to an oscillatory dynamics if it is not counteracted by the scattering\nterm in (4) .\nRESULTS\nFor the calculation of the spin and charge dynamics we do not attempt to model the\ndetails of ultrashort-optical-pulse excitation here, but choose the following simple initial\nconditions. We take the self-consistently determined equilibrium states/energies and change\n5FIG. 1. Band structure of the ferromagnetic Rashba model for an equilibrium temperature Teq=\n70 K, Stoner parameter Ue\u000b= 720 meV, and Rashba parameter \u000b= 30 meV nm\u00001. The expectation\nvalueshk\u0017j~^\u001bjk\u0017iat some kpoints are indicated by arrows and the energy by a color code: low\n(dark blue) to high (light yellow).\nthe electronic distributions in the equilibrium density matrix instantaneously to an elevated\ntemperature Te> T eqwhile keeping the electron density \fxed, which results in a small\nchange of the ensemble spin. In the following we always assume Teq= 70 K.\nFigure 2 shows the ensemble spin and charge dynamics resulting from Eq. (4) for an\nexcited electronic kinetic energy (temperature) of Te= 1000 K. The numerical evaluation\nof (4) assumes an in\fnitesimal broadening ~\u0000!0. We characterize the charge dynamics\nby \ftting the non-equilibrium distributions nk;\u0006with Fermi-Dirac functions where the \ft\nparameters are a common temperature and di\u000berent chemical potentials for the two bands.\nFig. 2(a) shows a demagnetization in about 100 fs, followed by a slower remagnetization. The\nmicroscopic charge scattering dynamics is re\rected in the change of the e\u000bective temperature\n(determined for all electrons in the system), shown in Fig. 2(b).\nAn important result of our calculation is that, microscopically, demagnetization dynam-\nics due to electron-phonon interaction in ferromagnets occurs via the interplay between\n60.430.440.45Sz/¯h\niBK2 UTbVTUEV\n00.40.81.21.620300600FIG. 2. Dynamics of the zcomponent of the ensemble spin Sz, and the electronic temperature\nTof the full EOM (4) for parameters as in Fig. 1 and an excitation temperature Tex= 1000 K.\nEquilibrium values are indicated by dotted horizontal lines.\nscattering and k-dependent internal magnetic \felds. This can be demonstrated as follows:\nAnalytically one can show that the scattering contribution in (4) alone does not lead to\nany spin/magnetization dynamics, i.e., for the ensemble spin Szwe \fnddSz=dt= 0 [22], if\nwe leave out the coherent precession [23]. We have checked this numerically by switching\no\u000b the precessional term, i.e., the \frst term on the RHS of (4), which results in a similar\ncharge scattering dynamics after excitation, but no magnetization dynamics at all. Fur-\nther, the precessional term alone, can, in principle, change the magnetization by dephasing\nas it describes precession around the e\u000bective magnetic \felds, but results only in a very\nsmall magnetization change that is four orders of magnitude smaller than what is shown in\nFig. 2(a). It therefore has to be the combination of both contributions. In semiconductors,\na similar spin relaxation mechanism was \frst identi\fed by Wu and coworkers [23, 24].\nWe elucidate the interplay of scattering and precession in Fig. 3 by investigating the (spin)\nexpectation value (2) for a kspace cell at a particular point ~k, which lies just above the\nFermi wave vector of the \\ \u0000\" band in positive kxdirection. In equilibrium, the expectation\nvalue (2) points in the direction of the local e\u000bective \feld, which is also the direction indicated\nby the arrows in Fig. 1 at each kpoint. At ~k, in particular, this direction lies in y-z\nplane. We plot the change \u0001 h^\u001b\u000bi~k(t) =h^\u001b\u000bi~k(t)\u0000h^\u001b\u000bi(eq)\n~kwith respect to the equilibrium\nexpectation value. In addition, we plot the contribution to \u0001 h^\u001b\u000bi~k(t) arising from the\n7spin coherences, i.e., the \u00176=\u00170terms in (2), only, and we discuss these \frst. For our\nexcitation conditions, the coherences are initially zero and the scattering contribution to\nEq. (4) is needed to get the coherences started. However, their dynamics is also in\ruenced\nby the precessional contribution in (4) due to the e\u000bective magnetic \feld. Note that the\nprecessional contribution is always there (with a period of about 7 fs mainly due to the Stoner\ncontribution to the spin splitting), but it results in oscillatory motion of the coherences\nwith that period only during the \frst 50 fs. The \frst 50 fs, during which also the main\ndemagnetization happens, is the time scale on which the system reaches a quasi-equilibrium\nin the sense that electrons in both bands can be described by the same e\u000bective temperature\n(but the respective densities are di\u000berent from their equilibrium values). After that, the\nscattering suppresses the oscillatory motion of the coherences, but the spin dynamics stay\nnoncollinear, as can be seen from the xcomponent at ~k. It is nonzero not only on the time\nscale of Fig. 3, but also during the remagnetization, and its existence demonstrates that the\nspin expectation value does not point into the direction of the e\u000bective magnetic \feld at ~k.\nNote that, while the spin precession contribution from the coherences is important for the\nspin dynamics, it is essentially invisible on the scale of \u0001 h^\u001byi~k(t) and \u0001h^\u001bzi~k(t), which are\ndominated by the distributions n\u0006k.\nFinally, we would like to explore the connection between spin precession and spin-\rip scat-\ntering (Elliott-Yafet mechanism) in Fig. 4 for di\u000berent electron-phonon coupling strengths.\nTo do this we compare the computed dynamics using the full Eq. (4) for the reduced density\nmatrix\u001ato that neglecting the coherences \u001a+\u0000\nk(and therefore also the precession) in Eq. 4.\nIn the latter case, we keep only the diagonal parts of the density matrix, i.e., the distri-\nbution functions nk\u0006whose dynamics are governed by conventional Boltzmann scattering\nintegrals [2, 20, 25]. Note that without the coherences, only components of the spin expecta-\ntion value in the direction of the local e\u000bective magnetic \feld are included and the ensemble\nspin is given by Sz= (~=2)P\nk\u0017hk;\u0017j^\u001bzjk;\u0017ink\u0017.\nFig. 4 shows that, for the set of parameters used so far, the calculations with and without\ncoherences lead to almost identical spin dynamics. The reason for this similarity lies in\nthe precessional contribution to the equation of motion (4) for the coherences discussed in\nconnection with Fig. 3, even though the precessional dynamics is only visible for short times\ndue to the in\ruence of the scattering. For the electron-phonon coupling strength used so\nfar, the precession period period of 7 fs is short compared to the energy relaxation time\n8250◊ 250◊\n250◊25◊\n1000 ◊10◊\u0000101\n\u000010\u0000Èˆ‡xÍ˜k\u0000Èˆ‡yÍ˜k\ntime (ps)\u0000Èˆ‡zÍ˜k\n0 0.05 0.1 0.15 0.2\u000010FIG. 3. Computed spin change \u0001 h^\u001b\u000bi~kat~k(solid line). At this point in k-space the e\u000bective\ninternal magnetic \feld points in the y-zdirection. Dotted red line: contribution to (2) only from\nthe coherences \u001a+\u0000\nk. Note that in the lower two graphs the di\u000berent curves are multiplied by\ndi\u000berent factors as indicated, whereas both curves are identical in the top graph. The parameters\nare the same as in Fig. 2.\ntime [ps]∆S[%]\n0 0.02 0.04 0.06 0.08 0.1−6−4−20\nFIG. 4. Computed normalized spin dynamics \u0001 S= (Sz\u0000S(eq)\nz)=S(eq)\nzcalculated from Eq. (4) (solid\nlines) and without coherences, i.e., including only electronic distributions (dashed lines). The thin\nblue curves are the same as in Fig. 2, for the thick red curves we enlarged the deformation potential\nD2!10D2. All other parameters as in Fig. 2.\nof 120 fs, so that, at each kpoint, the part of the spin expectation value deviating from\nthe (local) magnetic \feld direction averages out. It therefore looks like as if the spin had\ninstantaneously changed (i.e., \\\ripped\") into its projection on the direction of the e\u000bective\nmagnetic \feld, as assumed in the conventional Elliott-Yafet mechanism. Even though the\n9spin dynamics are almost identical, the underlying microscopic dynamics are still di\u000berent.\nIn particular, the distribution functions n\u0006\nkin both calculation di\u000ber by a small amount.\nWe next turn to the spin dynamics when the deformation potential is increased D2!\n10D2. In this case, calculation with coherences leads to a reduced demagnetization with a\nsmaller minimum at slightly longer times, as compared to the calculation neglecting coher-\nences. As the energy relaxation time is shortened by an order of magnitude to 12 fs, the\nprecessional contribution to (4) no longer just leads to an e\u000bective projection of hsikon the\ndirection of the e\u000bective \feld. For this case, the conventional Elliott-Yafet-type picture with\nspin \rips fails, and cannot be replaced by a simpler description. Only if the electron-phonon\ncoupling strength were increased more so that the scattering becomes faster than the pre-\ncession, one enters a collision dominated regime that could be described by the classical\nDyakonov-Perel type picture with motional narrowing.\nOur results show that calculations of the spin/magnetization dynamics using the conven-\ntional Elliott-Yafet mechanism [20, 21] can only overestimate the demagnetization due to\nelectron-phonon scattering. Thus the conclusion of these earlier papers that this process is\nnot e\u000ecient enough to explain the experimentally observed ultrafast magnetization dynam-\nics remain unchanged [26]. However, even if the calculations with and without coherences\nyield similar spin dynamics, the presence of spin coherences gives a slightly di\u000berent picture\nof how the lattice acts a spin sink and spin source via the precession around internal ef-\nfective magnetic \felds. This understanding is in line with the variety of spin-orbit induced\ntransport e\u000bects [27].\nCONCLUSION\nWe investigated ultrafast magnetization dynamics in a ferromagnetic model system in-\ncluding spin-orbit coupling and electron-phonon scattering. By computing the reduced\nspin density matrix for itinerant electrons we showed how the magnetization change oc-\ncurs due to the interplay of spin precession around internal e\u000bective magnetic \felds and\nspin-independent scattering. If the precession period around the exchange \feld is short\ncompared to typical scattering time, the precessional contributions are e\u000bectively averaged\nout and one obtains good agreement with the magnetization computed using spin-\rip tran-\nsition rates, as assumed in the conventional Elliott-Yafet mechanism. 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Lett. 107, 207201 (2011).\n[22] A numerical solution of (4) for that case also shows that the temperature does not completely\nreturn to its equilibrium value at later times.\n[23] M. W. Wu and C. Z. Ning, Eur. Phys. J. B 18, 373 (2000).\n[24] J. H. Jiang, M. W. Wu, and M. Q. Weng, Physics Reports 493, 61 (2010).\n[25] B. Y. M uller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschlimann, H. C. Schneider, and\nB. Rethfeld, Phys. Rev. B 111, 167204 (2013).\n[26] If one includes a time-dependent (mean-\feld) exchange splitting [25], the e\u000bect of spin-\rip\nscattering of electrons with phonons and other electrons can be ampli\fed to reach magnitudes\nobserved in experiments.\n[27] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys.\nRev. Lett. 92, 126603 (2004).\n12"    },    {        "title": "0910.0951v2.Diffusive_and_precessional_spin_dynamics_in_a_two_dimensional_electron_gas_with_disorder__a_gauge_theory_view.pdf",        "content": "arXiv:0910.0951v2  [cond-mat.other]  18 May 2010Diﬀusive and precessional spin dynamics in a two-dimension al\nelectron gas with disorder: a gauge theory view\nI.V. Tokatly1,2,4and E. Ya. Sherman3,4\n1European Theoretical Spectroscopy Facility (ETSF),\nDepartamento de Fisica de Materiales,\nUniversidad del Pa´ ıs Vasco UPV/EHU and Centro\nMixto CSIC-UPV/EHU, San Sebastian, Spain\n2Moscow Institute of Electronic Technology, Zelenograd, 12 4498 Russia\n3Department of Physical Chemistry,\nUniversidad del Pa´ ıs Vasco UPV/EHU, 48080 Bilbao, Spain\n4IKERBASQUE Basque Foundation for Science,\nAlameda Urquijo 36-5, 48011, Bilbao, Bizkaia, Spain\nAbstract\nWe develop a gauge theory for diﬀusive and precessional spin d ynamics in two-dimensional\nelectron gas with disorder. Our approach reveals a direct co nnections between the absence of the\nequilibrium spin current and strong anisotropy in the spin r elaxation: both eﬀects arise if the spin-\norbit coupling is reduced to a puregauge SU(2) ﬁeld. In this case, by a gauge transformation in the\nform of a local SU(2) rotation in the spin subspace the spin-orbit coupling ca n be removed. The\nresulting spin dynamics is exactly described in terms of two kinetic coeﬃcients: the spin diﬀusion\nand electron mobility. After the inverse transformation, f ull diﬀusive and precessional spin density\ndynamics, including the anisotropic spin relaxation, form ation of stable spin structures, and spin\nprecession induced by a macroscopic current, is restored. E xplicit solutions of the spin evolution\nequations are found for the initially uniform spin density a nd for stable nonuniform structures.\nOur analysis demonstrates a universal relation between the spin relaxation rate and spin diﬀusion\ncoeﬃcient.\nPACS numbers: 72.25.-b\n1I. INTRODUCTION.\nDescription of the spin dynamics of a two-dimensional electron gas is one of the most\nimportant problems for fundamental and applied modern spintronic s.[1–3] Two mutually\nrelated problems in this ﬁeld attract a great deal of attention: spin current and spin relax-\nation. Spin currents describe how the spin density pattern change s with time mainly due\nto the spin transfer between diﬀerent parts of the electron gas. Since the spin dynamics\nof interest occurs usually in systems out of equilibrium, spin relaxatio n becomes important\nand contributes strongly into the evolution of spin density pattern .\nThe key for understanding these properties is the spin-orbit coup ling, making the orbital\nand spin degrees of freedom mutually dependent. Spin-orbit couplin g has many crucial\ninﬂuences on the properties of the systems where it occurs: the t ypical examples are nuclei,\nelementary particles, atoms, and electrons in solids. Spin-orbit cou pling, in general, makes\nthe spin a non-conserved quantity, thus leading to a spin relaxation . It causes mutually\ndependent spin, charge, and mass ﬂows in solids and quantum liquids.[4 –6] In addition,\nspin-orbit coupling leads to a spin response to an external electric ﬁ eld, providing an ability\nof spin manipulation by the electric ﬁeld driving the dynamics in the orbit al degrees of\nfreedom.[7]\nThe general techniques for calculation of spin relaxation and spin cu rrent out of equilib-\nrium arethe classical or quantum Boltzmann-like equation for the sp in-density matrix [8–12]\nand nonequilibrium Green functions.[13, 14] In this approach, the de scription of the electron\ndynamics takes into account possible relaxation processes due to t he presence of spin-orbit\ncoupling, disorder, phonons, and electron-electron collisions. The experimentally observable\nspin dynamics is due to the spin-orbit (spin-momentum) coupling. At t he equilibrium, the\nexpectation value of the spin current can be calculated directly. Su rprisingly, such a direct\ncalculation demonstrated that the spin current can exist even in th e equilibrium state of a\ntwo-dimensional electron gas with spin-orbit coupling.[15] This obser vation brought a puzzle\nfor the understanding of the basic phenomena in spin transport sin ce the equilibrium spin\ncurrent is not related to any spatial spin accumulation that can be s een experimentally.\nOn the other hand, the spin dynamics due to the spin-orbit coupling c an be understood\nin terms of a theory where the coupling is treated as a non-Abelian ga uge ﬁeld, and the\ncorresponding formalism can be applied [16, 17]. On a single-electron scale, for example, for\n2electrons in quantum dots, the gauge transformation of the spin- orbit ﬁeld was employed in\nRefs.[18,19]andusedfortheanalysisofexperimental resultsons pinmanipulationbyelectric\nﬁeld in quantum dots in Ref.[20]. Other interesting theoretical examp les of applications of\nthe non-Abelian gauge ﬁeld approach for single-electron spin trans port and electrons in\nquantum dots, were found and studied (for example, Refs.[21]-[26]) .\nWhen applied to the two-dimensional electron gas, the approach ba sed ona formal SU(2)\ngauge invariance of the spin-orbit Hamiltonian (i. e. the symmetry wit h respect to local\nrotations in the spin subspace) proved that the equilibrium spin curr ent is the diamagnetic\nresponse to the eﬀective non-Abelian spin-orbit magnetic ﬁeld.[27] I f the spin-orbit ﬁeld is\na pure gauge and, thus, can be removed by a gauge transformatio n, the eﬀective SU(2)\nmagnetic ﬁeld is zero, and the equilibrium spin current vanishes.\nHere we present a theory based on the gauge transformation, fo r spin dynamics in a two-\ndimensional electron gas in the case when the spin-orbit ﬁeld can be c ompletely removed by\nsuch a gauge transformation. We show that the absence of the eq uilibrium spin current is\ndirectly related to the giant anisotropy in the spin relaxation rate, w hen the relaxation does\nnot occur for certain spin directions.[28, 29] After gauging away th e spin-orbit coupling,\nthe entire nonequilibrium dynamics of a transformed spin becomes almost trivial and can\nbe described phenomenologically exactly by only two transport coeﬃ cients which can be\ndetermined experimentally, or calculated theretically to any desired level of accuracy. The\nﬁrst is the spin diﬀusion coeﬃcient and the second is the electron mob ility required only\nwhen a constant electric ﬁeld is applied. With the inverse transforma tion to the initial\ndynamical variables, we recover the full nontrivial dynamics of the physicalspin, including\nthe absence of the spin relaxation for certain spin directions, that is a strong anisotropy\nin the spin relaxation, stable spin conﬁgurations forming persistent spin helices, and spin\nprecession due to a charge current in a constant external electr ic ﬁeld. In addition, this\napproach allows making predictions for more general cases of spin- orbit coupling, including\nnonuniform spin-orbit ﬁelds.\n3II. SPIN CURRENT AND SPIN RELAXATION: THE CONVENTIONAL AP-\nPROACH.\nWe begin with the conventional Hamiltonian of spin-orbit coupling in two -dimensional\nelectron gas:\nHso=1\n2/summationdisplay\nj(αja(ρ)kj+kjαja(ρ))σa, (1)\nwhereαjais the coordinate-dependent spin-orbit coupling ﬁeld, kj=−i∂/∂xjis the mo-\nmentum operator, Cartesian subscript indices j=x,ycorrespond to the electron coordinate\nρ= (x,y),andσaare the Pauli matrices with the upper Cartesian indices correspond ing\nto three directions x,y,zin the spin subspace. We use the system of units with /planckover2pi1≡1 and\nsum up over repeating indices. Interaction Hsoarises in a two-dimensional electron gas from\nvarioussources. Two originsareconsidered asthemost important . The ﬁrst one, arising due\nto the inversion asymmetry of the crystal unit cell, is described by t he Dresselhaus model.\nThe other one is the Rashba ﬁeld,[30] where the coupling originates fr om the macroscopic\nasymmetry of the structure hosting the two-dimensional electro n gas.[31] Due to various\nphysical origins, including material, structure, doping, and possible mechanical strain, nu-\nmerical values of parameters αjavary strongly from system to system ranging from 10−12\neV·cm for Si- to 10−9eV·cm for GaAs-based structures and will not be discussed here.\nFor the coordinate-independent spin-orbit ﬁeld kjαja(ρ) = 0, and the Hamiltonian (1)\ncan be presented as:\nHso=/summationdisplay\njαj/parenleftbig\nh[j]·σ/parenrightbig\nkj, (2)\nwhereh[j]is a unit-length vector, and αjis the corresponding spin-orbit coupling constant\nforgiven component ofmomentum; its contribution tothe Hamiltonia nis, therefore, propor-\ntionaltothespinprojectionontothe h[j]axis. Thecouplingleadstoamomentum-dependent\nspin splitting of the electron states. As a results, the Fermi line of t he electron gas becomes\nspin-dependent and two Fermi lines in the electron gas appear. The HsoHamiltonian makes\nthe velocity spin-dependent:\nvj=kj\nm+i[Hso,ρj] =kj\nm+αj/parenleftbig\nh[j]·σ/parenrightbig\n, (3)\nwithmbeing the electron eﬀective mass.\n4With the spin-dependent velocity in Eq.(3) we deﬁne the operator of the spin current in\nthe form:\nJa\nj=1\n2/summationdisplay\nkC†\nk(vjτa+τavj)Ck, (4)\nwhere the SU(2) group generators τa=σa/2,andC†\nk,Ckare the corresponding spinors.\nFor example, let |Φ/an}b∇acket∇i}htbe the ground state wave function. The resulting expectation valu e of\nthe total spin current, summed up over all electrons in the gas is:\n/angbracketleftbig\nJa\nj/angbracketrightbig\n=/an}b∇acketle{tΦ|Ja\nj|Φ/an}b∇acket∇i}ht. (5)\nIn the absence of special symmetry relations between the compon ents of the Hamiltonian\nHso,the expectation values of spin current/angbracketleftbig\nJa\nj/angbracketrightbig\nare not zero, leading to the puzzling equilib-\nrium spin current without any measurable spin transport. Therefo re, spin current can be a\ncharacteristic of the equilibrium states of two-dimensional electro n gas. In the conventional\ncalculation of/angbracketleftbig\nJa\nj/angbracketrightbig\ndue to the spin-doubling of the Fermi line, contributions to/angbracketleftbig\nJa\nj/angbracketrightbig\ncome\nfrom two subsystems: single- and double occupied states at a given electron momentum.\nThese two contributions have opposite signs and almost compensat e each other, yielding\nthe results in the third order of the coupling constants α3\nj. This third-power dependence\nis expected from perturbation theory:/angbracketleftbig\nJa\nj/angbracketrightbig\nshould be an odd function of the spin-orbit\ncoupling and vanish in the ﬁrst order since in the ground state withou t spin-orbit coupling\nthe Fermi-line is not spin-split and all states with given kare doubly occupied.\nAnother important feature of the electron gas with spin-orbit cou pling is the spin relax-\nation. Assume that one has initially produced a nonequilibrium state Φ Sof a uniform spin\ndensity with the components:\nSa=/an}b∇acketle{tΦS|/summationdisplay\nkC†\nkτaCk|ΦS/an}b∇acket∇i}ht. (6)\nThen, the state |ΦS/an}b∇acket∇i}htwill relax to the equilibrium through all possible interactions and spin-\norbit coupling. The ﬁrst stage of the process, the momentum relax ation, is fast. If the\nspin-orbit coupling is weak compared to the random interactions cau sing the momentum\nrelaxation, asitisassumedfortherestofthispaper, thefollowings pinrelaxationisrelatively\nslow. As a result, at the second stage the spin components decrea se with relaxation rates\ndescribed by a symmetric tensor Γab:\ndSa\ndt=−ΓabSb. (7)\n5The components of Γabdepend on the spin-orbit coupling and all possible interactions of\nelectrons with disorder, phonons, and other electrons in the syst em.[32, 33] If the spin-orbit\ncoupling vanishes, Γab= 0.\nIII. SPIN-ORBIT COUPLING AS A GAUGE FIELD: PURE GAUGE.\nNow we write the Hamiltonian of two-dimensional electron gas in the pr esence of spin-\norbit coupling as:\nH=1\n2m/integraldisplay\ndxdyΨ+(i∂i+Ai)2Ψ+W/parenleftbig\nΨ+,Ψ/parenrightbig\n(8)\nwhereW(Ψ+,Ψ) contains all explicitly spin-independent terms, including electron- electron\ninteractions and possibly, the eﬀect of the external potential. Th e general non-Abelian\ntwo-component potential is given by 2 ×2 matrices:\nAj≡Aa\njτa= 2mαj(ρ)ha\n[j](ρ)τa. (9)\nThe expression (9) is valid for any arbitrary nonuniform spin-orbit c oupling. Let us now\nperform at a given ρ−point a local SU(2)−rotation [27] in the spin subspace by\nU= exp[iθa(ρ)τa], (10)\nwith the transformation of the ﬁeld operators:\n/tildewideΨ+U−1= Ψ+,/tildewideΨ =UΨ. (11)\nThis transformation renders the spin-independent quantities suc h as the charge density and\nthe charge current density, invariant. In contrast, the spin den sity operators,\nS=Saτa, (12)\ntransforming as\nS=U/tildewideSU−1, (13)\nexemplify covariant observable quantities. This diﬀerence between the physical quantities\nwhich transform invariantly and covariantly under a local SU(2) rotation is crucial for the\nunderstanding of the spin dynamics.\n6For the matrix U= exp[iθ(h·τ)],wherehis a unit length vector, the τb−matrices,\ntransformed according to Eq.(13), acquire the form:\n/tildewideτb=hb(h·τ)+cosθ[τb−hb(h·τ)]+sinθεabchaτc, (14)\nwhereεabcis the Levi-Civita tensor. This equation shows that the product h·τis unaﬀected\nby the transformation (10). Therefore, if we present Sas the sum of longitudinal and\ntransverse components S=S/bardbl+S⊥withS/bardbl=h(S ·h),the longitudinal component (spin\nprojection onto the h−axis) remains constant, while the S⊥does not; it rotates by the angle\nθaround the h−axis). This simple observation will be important for the further analy sis in\nthis paper.\nThe Hamiltonian preserves its form under a local SU(2) rotation of the fermionic ﬁelds\nif the vector-potential is transformed as follows\n/tildewideAi=U−1(i∂iU)+U−1AiU. (15)\nIndeed, after the transformation the Hamiltonian acquires the fo rm:\nH=1\n2m/integraldisplay\ndxdy/tildewideΨ+/parenleftBig\ni∂i+/tildewideAi/parenrightBig2/tildewideΨ+W/parenleftBig\n/tildewideΨ+,/tildewideΨ/parenrightBig\n, (16)\nwhich is identical to that of Eq. (8), but with Ψ and Aibeing replaced by the transformed\nquantities, /tildewideΨ and/tildewideAi, respectively.\nAssume now that Aiin the original Hamiltonian (8) corresponds to a pure gauge vector-\npotential, that is both AxandAycan be removed by the above transformation such that\n/tildewideAx=/tildewideAy= 0. In this case there exists a local rotation determined by three c oordinate-\ndependent functions θa\nA(x,y):\nUA= exp[iθa\nA(ρ)τa], (17)\nsuch that the initial components Aican be presented in the form\nAi=UA/parenleftbig\ni∂iU−1\nA/parenrightbig\n. (18)\nA vector-potential of this form is gauged away by the transforma tion (10) with U=UA:\n/tildewideAi=U−1\nA/parenleftbig\ni∂i+UA/parenleftbig\ni∂iU−1\nA/parenrightbig/parenrightbig\nUA= 0. (19)\nIf the spin-orbit ﬁeld can be removed by a gauge transformation, t he subsequent spin dy-\nnamics is simpliﬁed considerably and in certain regimes, like the drift-diﬀ usion processes\n7considered below, the problem becomes elementary. The inverse SU(2)−rotation trans-\nforms the spin components to the actual values and we recover th e full dynamics of the\nphysical spin. We will follow this procedure in the present paper.\nWe mention a textbook example of a similar approach. When the motion of a relativistic\nelectron in static perpendicular electric ﬁeld Eand magnetic ﬁeld His considered, there\nexists a reference frame, where, after the Lorentz transform ation, the smaller of these ﬁelds\nvanishes. In this frame the equations of the electron motion are ve ry simple, and in the case\nH < E,they are essentially, one-dimensional. The inverse Lorentz transf ormation provides\nthe full description of the electron motion in the presence of both ﬁ elds.[34]\nIn the pure gauge ﬁeld after the local SU(2) transformation UA= exp[iθa\nA(ρ)τa] the\nHamiltonian takes the form:\nH=−1\n2m/integraldisplay\ndxdy/tildewideΨ+∆/tildewideΨ+W/parenleftBig\n/tildewideΨ+,/tildewideΨ/parenrightBig\n, (20)\nwith no spin-orbit coupling present. As mentioned above, the spin dy namics with this\nHamiltonian can be formulated in general terms phenomenologically an d then by inverse\ntransformation, returned to the formwhere the coupling and full spin dynamics arerestored.\nVector-potential is a pure gauge, allowing removal six terms in Ax,Ay,with the trans-\nformation UAbased on the three functions θa\nA(ρ) given certain relations between the Ax\nandAycomponents. The corresponding conditions are naturally formulat ed in terms of a\nnon-Abelian ﬁeld strength tensor Fij: the vector potential is locally a pure gauge if the\nﬁeld strength is zero,\nFij=∂iAj−∂jAi−i[Ai,Aj] = 0. (21)\nFor the spatially uniform case, using Eq.(9) this condition is reduced t o [Ai,Aj] = 0,that\nis:\n(i) either αiαj= 0,or\n(ii)/bracketleftbig\nh[i]·τ,h[j]·τ/bracketrightbig\n= 0 ifαiαj/ne}ationslash= 0.\nThe commutation relation\n/bracketleftbig\nh[i]τ,h[j]τ/bracketrightbig\n=iτ·/parenleftbig\nh[i]×h[j]/parenrightbig\n, (22)\ndemonstrates that the spin projections commute only for the sam e axis, that is h[i]=±h[j].\nTherefore, the solution to Eq. (22) has the form (we assume below h[i]=h[j]in the case (ii)\n8FIG. 1: (Color online) Illustration of two cases of the pure g auge spin-orbit ﬁeld. (a) one of the\ncoupling constants αjis zero, case (i) (b) both coupling constants are not zero, th e directions\nof corresponding magnetic ﬁelds coincide, case (ii). The di rection of the spin-orbit ﬁeld remains\nconstant for any electron momentum k= (kx,ky).\nfor deﬁniteness):\nAj= 2mανj(h·τ) (23)\nwhereh=h[i]=h[j]ifαiαj/ne}ationslash= 0 orh=h[f]for nonzero αf, wheref=xorf=y, as\nillustrated in Fig.(1). Here α=/parenleftbig\nα2\nx+α2\ny/parenrightbig1/2,andνis a unit vector. The corresponding\ngauge transformation is:\nUA= exp[2imαρjνj(h·τ)]exp[2imαρiνi(h·τ)] = exp[2 imα(h·τ)(ρ·ν)].(24)\nFrom this condition we immediately conclude that the projection of th e total spin at the\nh[i]=h[j]axis commutes with Hso, and, therefore, remains constant with time for arbitrary\ndynamics. Experimentally, this fact corresponds to the vanishing r elaxation for this spin\ndirection; this conclusion crucial for the design and application of sp in-based devices. If\nαiαj= 0,the problem immediately becomes one-dimensional, trivial from the dia magnetic\nresponse interpretation of the equilibrium spin current, [27] since o ne-dimensional systems\ndo not demonstrate this kind of response. The same situation occu rs in quantum wires,\nwhere the motion of electrons is strictly one-dimensional, no equilibriu m spin current exists,\nand the spin projection along the h[f]axis is conserved. In the Appendix, for illustration,\nwe perform a conventional calculation of the equilibrium spin current in a two-dimensional\nelectrongaswiththepuregaugespin-orbitcouplingandinaquantum wire, anddemonstrate\nthat the spin current vanishes in both systems.\nThere are two widely studied realizations of the above discussed pur e gauge ﬁeld. The\nαiαj= 0 case is realized for the Dresselhaus model for the electron gas c onﬁned in the\nquantum wells of GaAs grown along the [110] crystal axis. The couplin g constant αin this\n9FIG. 2: (Color online) Illustration of coordinate-depende nt mutual orientation of /tildewideSandSvectors\ncorresponding to Eq.(26) for a structure grown along the [11 0] crystal axis. Vectors handνare\nshown in the Figure. The angle between /tildewideSandSis determined solely by the x-component of\nρ−vector.\nsystem [35] is approximately inversely proportional to the square o f the quantum well width\nw. In this case the vector-potential and the corresponding trans formations are:\n(Ax,Ay) = (2mατz,0),UA= exp[2imαxτz], (25)\nwhere the z−axis is oriented along the growth direction and the x-axis is that of the unit\ncell. Here we use transformation (24) with h=(0,0,1),ν= (1,0),andθ(x,y) = 2mαxto\nobtain:\n/tildewideτz=τz,/tildewideτx= cosθτx+sinθτy,/tildewideτy= cosθτy−sinθτx. (26)\nWe illustrate the resulting relations between Sand/tildewideSfor this simple situation in Fig.(2):\nwhen/tildewideSremains constant in space, Sturns by the angle θ(x,y) around the z−axis.\nTheαiαj/ne}ationslash= 0case is realized in thecompensated Dresselhaus-Rashba model f or the GaAs\nstructure grown along the [001] crystal axis. Here\n(Ax,Ay) = (2mα(τx−τy),2mα(τx−τy)), (27)\nUA= exp[2imα(x+y)(τx−τy)]. (28)\nHere we obtain with h=(1,−1,0)/√\n2,ν= (1,1)/√\n2,andθ= 2√\n2mα(x+y):\n/tildewideτz= cosθτz−1√\n2sinθ(τx+τy), (29)\n/tildewideτx= cos2θ\n2τx−sin2θ\n2τy+1√\n2sinθτz,/tildewideτy= cos2θ\n2τy−sin2θ\n2τx+1√\n2sinθτz.\n10Equations (26), (29) illustrate a general feature of the relations between the original S=\n(Sx,Sy,Sz) and gauge-transformed /tildewideSspin densities and vice versa . For the spin-orbit ﬁeld\ncharacterized by the direction h,for a uniform coordinate-independent S,the components\nS/bardbland/tildewideS/bardblcoincide. The /tildewideS⊥-component forms a periodic structure on the spatial scale of the\norder ofLso= 1/mα, or/planckover2pi12/mαwhentheunits arerestored. The meanvalue/angbracketleftBig\n/tildewideS⊥(x,y)/angbracketrightBig\n= 0\nfortheinﬁnitelylargesystems considered here, where thebounda ryconditionsdonotchange\nthespindynamics. Themeaningofthelength Lsocanbeunderstoodasfollows. Hamiltonian\n(2) shows that the spin-orbit coupling Hsocauses for an electron moving with the velocity v,\nspin precession around hwith the rate of the order of αmv.The corresponding precession\nangle is of the order of αmL,whereL=vtis the electron displacement. Thefore, Lsocan\nbe viewed as the travel distance at which the electron spin can unde rgo a full rotation.\nAnother circumstance is, however, more important: the spin rota tion angle depends only on\ntheelectrondisplacement andnotonthedetailsofitsmotionbetwee n initialandﬁnalpoints,\nleading to the appearance of stable spin structures, discussed be low. Here a numerical value\nof typical Lsocan be of interest. For GaAs with m= 0.067m0,wherem0is the free electron\nmass, and αof the order of 10−7meV·cm,Lsois of the order of several microns.\nIn both these systems, the observed spin relaxation rate is stron gly anisotropic with\none spin component having lifetime orders of magnitude longer than t he others. The weak\nrelaxation rate for these components is determined by the mechan isms diﬀerent from the\nhomogeneous spin-orbit coupling, most probably, related to the dis order in the spin-orbit\ncoupling. [36–38]\nIV. SPIN DYNAMICS: DIFFUSION, PRECESSIONAL BEHAVIOR, AND DRIFT\nCONTRIBUTIONS.\nAfter the gauge transformation, the spin-orbit interactions is sw itched oﬀ. Therefore, on\na time scale much longer than the momentum relaxation time, the spin d ynamics becomes\ncombination of pure spin diﬀusion and spin drift:\n∂t/tildewideS=D∆/tildewideS+µEj∂j/tildewideS, (30)\nwhereDis the spin-diﬀusion coeﬃcient, µis the electron mobility, and Eis the two-\ndimensional applied electric ﬁeld [39] as illustrated in Fig.(3). In Eq.(30) we have taken\n11FIG. 3: (Color online) Nonuniform spin density evolving by d iﬀusion and drift dynamics. Small\nsquare with an arrow illustrates the eﬀect of the external ele ctric ﬁeld on the nonuniform spin\ndynamics.\ninto account that the uniform velocity of electrons is −µE. These two parameters fully\ndescribe the drift-diﬀusive spin dynamics in the absence of spin-orb it coupling. Macroscopic\nmotion of electrons (electric currents) can drag nonuniform spin d ensity between diﬀerent\nparts of the electron gas. This eﬀect leads to the µEj∂j/tildewideSterm in∂t/tildewideS. The initial spin\ndensity eventually vanishes due to diﬀusion, however, the total int egrated spin polarization\nwill remain constant. The diﬀusive evolution of the transformed spin densityD∆/tildewideSoccurs\nif the electron free path of the order of ℓ=vτpis much less than the spatial scale of the\ninhomogeneity: ℓ≪Lso.This condition can be formulated as Ω soτp≪1,meaning that the\nspin-orbit coupling is relatively weak. The spatial inhomogeneity of th e order of LsoandD\nof the order of v2τpset the time scale of the diﬀusion smearing of the /tildewideSas/tildewidetD∼L2\nso/Don\nthe order of Ω−2\nsoτ−1\np,and, therefore, the same spin relaxation time for real spin S.\nThe evolution of the physical measurable spin density:\nS=UA/tildewideSU−1\nA, (31)\nis due to the diﬀusion and spin precession since the transition of elect ron from point ρ1to\npointρ2is accompanied by the rotation of its spin, dependent only on the disp lacement\nρ2−ρ1. Irregular motion in the diﬀusion process is seen in the spin relaxation , and regular\ndrift causes spin precession, with these two processes being mutu ally related.\nMotion of Sis described, therefore, by the following equations for the time evo lutions\n12of the spin density, which are obtained by applying the inverse trans formation (31) to the\ndrift-diﬀusion equation (30),\n∂tS=DUA/bracketleftbig\n∆/parenleftbig\nU−1\nASUA/parenrightbig/bracketrightbig\nU−1\nA+µEjUA/bracketleftbig\n∂j/parenleftbig\nU−1\nASUA/parenrightbig/bracketrightbig\nU−1\nA. (32)\nThe resulting most general equation of motion valid for any pure gau ge spin-orbit ﬁeld takes\nthe form\n∂tS −D∆S −µEj∂jS\n=D/braceleftbig\n2/bracketleftbig\nUA∇U−1\nA,∇S/bracketrightbig\n−2/parenleftbig\nUA∇U−1\nA/parenrightbig\nS/parenleftbig\nUA∇U−1\nA/parenrightbig\n+/parenleftbig\nUA∆U−1\nA/parenrightbig\nS+S(∆UA)U−1\nA/bracerightbig\n+µEj/bracketleftbig/parenleftbig\nUA∂jU−1\nA/parenrightbig\n,S/bracketrightbig\n. (33)\nThe totalexpression forlocal evolution ofspindensity component s canbeobtainedfromthis\nequation by multiplying both sides by τaand taking the trace using the identity: tr/parenleftbig\nτaτb/parenrightbig\n=\nδab/2.The result can be presented as the sum:\n∂tSa=D∆Sa+µEj∂jSa+Bab\nj∂jSb−HabSb−ΓabSb. (34)\nThe general expressions for non-diagonal tensors of kinetic coe ﬃcients entering this equation\nare:\nBab\nj=−Bba\nj= 4Dtr/braceleftbig\nτa/bracketleftbig\nUA∂jU−1\nA,τb/bracketrightbig/bracerightbig\n, (35)\nHab=−Hba= 2µEjtr/braceleftbig\nτa/bracketleftbig\nUA∂jU−1\nA,τb/bracketrightbig/bracerightbig\n, (36)\nΓab= 4D/parenleftbigg\ntr/braceleftbig\nτa/parenleftbig\nUA∂jU−1\nA/parenrightbig\nτb/parenleftbig\nUA∂jU−1\nA/parenrightbig/bracerightbig\n−1\n2tr/braceleftbig\nτa/parenleftbig\nUA∆U−1\nA/parenrightbig\nτb+τa(∆UA)U−1\nAτb/bracerightbig/parenrightbigg\n(37)\nNow we can study the physical meaning of the obtained non-diagona l tensors and simplify\nthe expressions for the time derivatives for uniform spin-orbit cou pling with:\nUA= exp[2imα(h·τ)(ν·ρ)],U−1\nA=U+\nA= exp[−2imα(hτ)(ν·ρ)],(38)\nWith the given form in Eq.(38) of UAandU−1\nAwe obtain:\nUA∇U−1\nA=−2imα(h·τ)ν,∆UA=−4m2α2UA,∆U−1=−4m2α2U−1\nA.(39)\nWith formulas (38),(39) we obtain for the diﬀusion-related coeﬃcien ts:\nBab\nj=−2mαjDεabchc, (40)\nΓab= 4m2α2D/parenleftbig\nδab−hahb/parenrightbig\n. (41)\n13The corresponding drift-dependent contribution:\nHab=−mαµ(ν·E)εabchc, (42)\ndescribes the spin precession.\nThe answer for the spin density Swith components ( Sx,Sy,Sz) has the form of three\nterms of diﬀerent order in α:\n∂tS=∂tS|0+∂tS|1+∂tS|2. (43)\nThese terms have diﬀerent meaning and can be expressed as:\n∂tS|0=D∆S+µEj∂jS, (44)\n∂tS|1= 4mDα(ν·∇)(h×S)+2mαµ(ν·E)(h×S), (45)\n∂tS|2=−4m2α2D(S−h(h·S)). (46)\nThe∂tS|0term describes the standard drift-diﬀusion spin dynamics for zero spin-orbit cou-\npling.\nThe∂tS|1term corresponds to the spin precession due to the spin-orbit cou pling. The\nmobility-determined contribution in ∂tS|1is the precession in the macroscopic spin-orbit\nﬁeld arising due to the uniform velocity of electrons. When the electr ic current is induced,\nthemomentum distribution functionisshifted such that themoment um hasanonzero value.\nAs a result, the Hamiltonian Hsoforms a macroscopic spin-orbit Zeeman ﬁeld [40] and, as\na result, a regular spin precession ∂tS=2mαµ(ν·E)(h×S).If (ν·E) = 0, contributions\nof the momentum changes along the xandy-axes in the macroscopic spin-orbit “magnetic”\nﬁeld compensate each other, and no regular precession occurs. T hus, Eq.(45) reproduces\nthe diﬀusive and non-diﬀusive spin precession.\nThe∂tS|2term is the Dyakonov-Perel’ mechanism of spin relaxation,[41] which c an be\nseen from the fact that Dis determined by /an}b∇acketle{tv2/an}b∇acket∇i}htτp, wherevis the electron velocity (see, also\nin Ref. [13]). Taking into account that electron momentum is mv,one can see that ∂tS|2\ncorrespondstotheDyakonov-Perel relaxationwiththerelaxatio nrateontheorderof α2k2τp.\nThe obtained relation between the spin relaxation rate and diﬀusion c oeﬃcient is universal.\nFortwo diﬀerent systems withthesamesample-dependent mαparameter, theratioofΓab/D\nremains constant. Since the parameters ΓabandDcan be measured independently, this\nuniversality can be veriﬁed experimentally. For example, in the measu rements performed at\n14the same sample at diﬀerent temperatures, the ratio Γab/Dis expected to remain constant\nfor degenerated and non-degenerated electron gas.\nEqs.(45) and (46) show that S/bardbl=h·Sdoes not change with time, as expected, and the\nentire dynamics is solely due to the S⊥−component. As a simple illustration we consider\nthe evolution of an initially homogeneous spin density. By solving equat ions (43)-(46) with\nthe initial condition S(ρ,t= 0) =S0we ﬁnd the spin dynamics\nS(t) =h(h·S0)+{cos(ΩEt)[S0−h(h·S0)]+sin(Ω Et)(h×S0)}e−Γt,(47)\nwhere Ω E= 2αmµ(ν·E) is the precession frequency in a drift-induced spin-orbit Zeeman\nﬁeld, and Γ = 4 α2m2Dis the diﬀusion related relaxation rate. From Eq. (47) we see that th e\nspin precesses with the frequency Ω Eabout the h-axis and its transverse component decays\nat the rate Γ in such a way that the projection of the spin at hremains stationary. By\ncomparing the characteristic time scales of the drift-induced prec ession and the diﬀusion-\ninduced relaxation, we can estimate the external electric ﬁeld at wh ich the role of the\ndrift-dependent terms becomes important; in particular, the pre cession becomes visible at\nthe scale of the relaxation time. From the condition of still visible prec ession Ω E∼Γ we\nﬁnd the corresponding electric ﬁeld E∼αmD/µ. In this ﬁeld, the precession rate Ω Eis of\nthe order of Ω2\nsoτp, making the contributions of regular and random motion in the prece ssion\nangle of the same order. If the spin diﬀusion is dominated by the impur ity scattering,\nthenDandµare proportional to the momentum relaxation time τp, and this electric ﬁeld\nis disorder-independent. However, it can change with the tempera ture since in the non-\ndegenerated gas Dapproaches the electron diﬀusion coeﬃcient [39] and, therefore , by the\nEinstein relation D=µT.\nAnother interesting eﬀect of a spin-orbit coupling, which follows str aightforwardly from\nour formulation – the existence of stable spatially inhomogeneous sp in conﬁguration. It is\neasy to verify that a general stationary ( ∂tS=0) solution to the equations (43)-(46) is of\nthe form\nS(ρ) =h(h·S0)+cos(2 mα(ν·ρ))[S0−h(h·S0)]−sin(2mα(ν·ρ))(h×S0),(48)\nwhereS0isanarbitraryconstantvector. Thisspatiallyinhomogeneousstat ionarysolutionto\nthedrift-diﬀusionequationarisesduetothesymmetryofthesyst em. Asitwasdemonstrated\nfor the particular case of the model with balanced Rashba and Dres selhaus couplings, the\n15symmetry can protect electron spins from relaxation [29] and allows for the persistent spin\nhelix structures [42, 43] of the form of Eq.(48). The fact that the shape of this conﬁguration\ndoes not depend on the diﬀusion coeﬃcient Dshows that the persistent spin structure is\ninsensitive to the spin-independent disorder, in agreement with Ref .[42] The analysis of the\nspinhelixstabilityinthepresenceofdisorderedspin-orbitcouplingca nbefoundinRef.[44]It\nis interesting to note that the helix structure is also insensitive to th e presence of the electric\nﬁeldand, therefore, to themobilityandpresence ofatransportc harge current, atleast inthe\nlinear Ohm’s law regime. This seemingly counterintuitive result follows fr om the fact that\nthe drift of the helix governed by the second term in (44) is exactly c ompensated by the spin\nprecession in the current-induced eﬀective Zeeman ﬁeld, the seco nd term in (45). A similar\ncancellation occurs in the diﬀusion channel. A diﬀusive spreadout of t he helix, the ﬁrst\nterm in (44), and the relaxation of the transverse component of t he spin, (46), are balanced\nby the “gradient-precession“ contribution, the ﬁrst term in (45) . The persistent spin helix\nconﬁguration (48) has an extremely simple interpretation in terms o f the transformed spin\ndensity/tildewideS. The general stationary solution of the standard drift-diﬀusion e quation (30) is\nsimply a constant /tildewideS=/tildewideS0. The relation between the physical and transformed spin density\ncomponents yield the conservation /tildewideS0·h=S·h. The perpendicular /tildewideS0,⊥is transformed\naccording to Eq. (31) as S=UA/tildewideSU−1\nAwithUA= exp[2imα(h·τ)(ρ·ν)] from Eq. (24)\naccording to Eq. (14). The sum of the transformed terms is precis ely the persistent spin\nhelix of Eq. (48).\nIt is also instructive to look at the precession and relaxation of a spa tially homogeneous\nspinS(t), Eq. (47), from the point of view of the dynamics of the transfor med spin density\n/tildewideS. The initial condition S(ρ,t= 0) =S0for the physical spin is mapped to the initial\nconﬁguration for /tildewideSin a form of a spin helix that is similar to Eq. (48). The subsequent\nevolution of /tildewideSis governed by the standard drift-diﬀusion equation (30). Theref ore the\ndynamical behavior is obvious -the initial helix for the transformed s pin moves with the\ndrift velocity vdrift=mαµ(ν·E), and washes out diﬀusively. When transformed to the\nphysical spin, the drift of the helix translates to the precession, w hile its diﬀusive decay is\nmapped to the relaxation of the physical spin. This interpretation c learly explains why the\nrelaxation time of the transverse components of the spin is univers ally determined by the\ndiﬀusion coeﬃcient. The relaxation of the physical spin components is gauge-equivalent to\na purely diﬀusive process of washing out the initial helix conﬁguration of the transformed\n16spin.\nV. CONCLUSIONS.\nWe developed a gauge theory of macroscopic spin dynamics in a two-d imensional electron\ngas when the spin-orbit coupling can be described as a pure gauge, a nd, therefore, removed\nby a local SU(2) rotation in the spin subspace. We have shown that for a pure sp in gauge,\nequilibrium spin current vanishes and a selected axis of conserved sp in projection appears\nsimultaneously, demonstrating gauge-related symmetry relation o f these eﬀects. After re-\nmoving the spin-orbit coupling, we considered macroscopic phenome nological equations of\nspin dynamics, including spin diﬀusion and spin drift in an external elect ric ﬁeld. By the\ninverseSU(2) rotation we obtained the full system of partial diﬀerential equ ations for the\ntime- and spatial measured spin density evolution. This system repr oduces the physics of\nspin precession, stable spin conﬁgurations such as the persistent spin helix, and the result-\ning strongly anisotropic spin relaxation. Since we described the syst em without spin-orbit\ncoupling phenomenologically, our approach is valid at any temperatur e and electron concen-\ntration. It predicts that the ratio of the spin relaxation rate to th e spin diﬀusion coeﬃcient\nremains temperature- and electron concentration-independent if the coupling constants do\nnot depend on these two system parameters.\nWe presented explicit equations for the spatially uniform spin-orbit c oupling and their\nsolutions describing stable nonuniform structures, the precessio n and the relaxation of uni-\nform spin polarization. These equations can be explicitly generalized f or nonuniform two-\ndimensional electron gas in macroscopic systems. We mention two of them. The ﬁrst one\nis the GaAs quantum well grown along the [110] direction with a modulat ed width w(x),\nwhere the spin-orbit α(x) originated from the Dresselhaus coupling, varies as 1 /w2(x). The\ncorresponding spin-orbit ﬁeld Ax= 2mα(x)τz,Ay= 0, with ∂Ax/∂y= 0 remains a pure\ngauge. The other example is the balanced Rashba-Dresselhaus mod el with the coordinate-\ndependent Rashba and Dresselhaus parameters remaining exactly equal or exactly opposite\neverywhere. As in the [110] structure, variation in the Dresselhau s term is due to the con-\ntrolled variation in the structure width, while the control of the Ras hba coupling is achieved\nby a coordinate-dependent bias across the well. A diﬀerent kind of in homogeneity occurs\nin mesoscopic systems where the eﬀect of the boundary conditions for the coupled spin-\n17charge dynamics becomes important.[45–47] Generalization of the g auge theory approach\nfor the dynamics at the sample boundaries can be an interesting ext ension of our analysis\nfor the inﬁnite systems. Spin dynamics in these systems is of interes t for the fundamental\nunderstanding of spin transport and for applications in spintronics devices.\nVI. ACKNOWLEDGMENT.\nIVT acknowledges funding by the Spanish MEC (FIS2007-65702-C0 2-01), ”Grupos Con-\nsolidados UPV/EHU del Gobierno Vasco” (IT-319-07), and the Eur opean Community\nthrough e-I3 ETSF project (Grant Agreement: 211956). EYS is g rateful to the Univer-\nsity of Basque Country UPV/EHU for support by the Grant GIU07/ 40.\nVII. APPENDIX.\nHere we show by a conventional calculation of the spin current that it vanishes at the\nequilibrium in the considered above pure gauge spin-orbit coupling in a t wo-dimensional\nelectron gas and, similarly, in one-dimensional quantum wires. We beg in with the pure-\ngauge two-dimensional Hamiltonian,\nH=k2\nx\n2m+αx(haσa)kx+k2\ny\n2m+αy(haσa)ky, (49)\nwherehis unit length vector and αx,αyare the corresponding spin-orbit coupling constants.\nThe spectrum of electrons described by Eq.(49) is the sum of kxandky-dependent terms for\nthe two spin branches ”+“ and ”-“:\nε±(kx,ky) =k2\nx+k2\ny\n2m±(αxkx+αyky). (50)\nEq. (49) demonstrates that the system with pure gauge spin-orb it coupling remains in a\ncertain sense, one dimensional: spin-orbit coupling does not couple d iﬀerent components\nof momentum in the spectrum. For illustration we consider only the x−component of\nmomentum and velocity:\n∂ε±(k)\n∂kx=kx\nm±αx, v x=i[H,x] =kx\nm+αx(haσa), (51)\n18FIG. 4: (Color online) Scheme of the spin-orbit split states in a quantum wire. EFis the Fermi\nenergy.\nyielding the spin current component:\nJb\nx=1\n4/summationdisplay\nkC†\nk/braceleftbig\nvx,σb/bracerightbig\nCk. (52)\nTaking into account that/braceleftbig\nσa,σb/bracerightbig\n= 2δab, we obtain the anticommutator:\n1\n2/braceleftbig\nvx,σb/bracerightbig\n=1\n2/braceleftbiggkx\nm+αx(haσa),σb/bracerightbigg\n=kx\nmσb+αxhb. (53)\nThe totalspin current isthe sumof contributions oftwo subsyste ms/angbracketleftbig\nJb\nx/angbracketrightbig\n=/angbracketleftbig\nJb\nx/angbracketrightbig\n++/angbracketleftbig\nJb\nx/angbracketrightbig\n−=\n2/angbracketleftbig\nJb\nx/angbracketrightbig\n+.Taking into account that for given branch/angbracketleftbig\nσb/angbracketrightbig\n±=±hb/h,the/angbracketleftbig\nJb\nx/angbracketrightbig\n+spin current\ncomponent becomes:\n/angbracketleftbig\nJb\nx/angbracketrightbig\n+=1\n2hb/integraldisplay\ndky/integraldisplay∂ε(k)\n∂kxdkx, (54)\nwhere the integration is perfromed over the area in momentum spac e occupied by electrons\nfrom the branch. The value of this integral is zero since this area is r estricted by the line of\nthe constant Fermi energy EF.\nFor one-dimensional case the situation is the same. We take the Ham iltonian:\nH=k2\n2m+α(haσa)k. (55)\nThe eigenstates of this Hamiltonian form two branches:\nε±=k2\n2m±αk, (56)\ncorresponding totwo parabolaswiththeminima at −k0andk0=αm,respectively, asshown\nin Fig.(4). To calculate the spin current directly, we perform integra tion over momenta and\nsummation over spin branches. The ground state expectation valu e is:\n/angbracketleftbig\nJb/angbracketrightbig\n=1\n2/bracketleftbigg/integraldisplay−k0+kF\n−k0−kF/parenleftbiggk\nm/angbracketleftbig\nσb/angbracketrightbig\n++hb/parenrightbigg\ndk+/integraldisplayk0+kF\nk0−kF/parenleftbiggk\nm/angbracketleftbig\nσb/angbracketrightbig\n−+hb/parenrightbigg\ndk/bracketrightbigg\n,(57)\n19wherekFis the Fermi momentum determined by the total concentration of e lectronsnas\nkF=πn/2. We obtain\n/angbracketleftbig\nJb/angbracketrightbig\n= 2kFhb+1\n2m/bracketleftbigg/integraldisplayk0−kF\n−k0−kFkhb\nhdk−/integraldisplayk0+kF\n−k0+kFkhb\nhdk/bracketrightbigg\n= 2kFhb−2\nmhb\nhkFk0. 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Sharma\nMax Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany\u0003\n(Dated: November 10, 2021)\nA recently proposed exchange-correlation functional with in density functional theory, which\nensures that the exchange-correlation magnetic \feld is source-free, is shown to give non-zero internal\nspin-torque. This spin-torque is identically zero for all conventional local and semi-local functionals.\nExtension of this source-free functional to the time domain is used to study the e\u000bect of the internal\nspin-torque on the laser induced spin-dynamics in bulk Co, Ni and interfaces of these metals with\nPt. It is shown that the internal spin-torque contribute signi\fcantly to spin-dynamics only when the\nmagneto crystalline anisotropy energy is small, as in the case of cubic bulk materials. For surfaces or\ninterfaces, where the anisotropy energy is large, these torques are too small to cause any signi\fcant\nprecession of spins in early times ( <100fs). Further more it is shown that the spin-dynamics caused\nby the internal spin-torque is slow compared to the inter-site spin transfer and spin-orbit mediated\nspin-\rips.\nI. INTRODUCTION\nThe possibility of controlling electronic spins by light\no\u000bers a future of highly e\u000ecient devices and fast (sub\nfemtosecond) memory storage. In light of this a large\namount of research is being devoted to the study of\nlaser induced dynamics of spins{ spin-injection[1{5], spin\ntransfer torque[6{9] across tailored interfaces, all-optical\nswitching[10{12], ultra-fast demagnetization[13{26] to\nname but a few examples.\nTheoretically, ab-initio methods for treating this laser-\ninduced spin-dynamics is the non-collinear spin-polarized\nextension time-dependent density functional theory (TD-\nDFT). The requirement of non-collinearity stems from\nthe fact that, to leading order, light couples to spins\nvia the spin-orbit (SO) coupling term, which requires\nthat the Kohn-Sham wave-functions be two component\nspinors. In principle, TD-DFT is an exact method, but in\npractice the quality of results depends upon the approx-\nimation used for exchange-correlation (xc) energy func-\ntional.\nUsually, in time-dependent case, the adiabatic exten-\nsion of ground state xc functionals is used for time-\npropagating the Kohn-Sham system. Most of the xc func-\ntionals like the local spin density approximation (LSDA)\nor generalized gradient approximation(GGA) are de-\nsigned for collinear systems and a non-collinear exten-\nsion of these functionals is performed using the Kubler-\nSandratskii method[27, 28]{ at each point in space and\ntime the densities (charge density \u001a, magnetization den-\nsitym), which are 2\u00022 complex matrices in spin-space,\nare \frst diagonalized and then the corresponding xc po-\ntentials (vxc,Bxc) are calculated via functional deriva-\ntives of energy wrt these diagonal densities. This im-\nmediately implies that by construction mandBxcare\nparallel at each point in space and time and torque\nm(r;t)\u0002Bxc(r;t) = 0, which is equivalent to saying that\n\u0003sharma@mpi-halle.mpg.dethe internal torque felt by the spins is identically 0. This\nis a serious limitation as once the external perturbation\n(like magnetic \feld or laser) have been switched o\u000b, it\nis these torques which contribute to the dynamics of the\nspins. This raises an interesting question: if one were to\ndesign a truly non-collinear functional, like the optimized\ne\u000bective potential[29], which gives a non-zero torque on\nthe spins, would the laser induced spin-dynamics be fun-\ndamentally di\u000berent from one observed for conventional\nfunctionals like adiabatic LSDA[30]?\nIn order to answer this question, in the present work we\nemploy time-dependent extension of our recently devel-\noped source-free functional[31] to study the laser-induced\nspin-dynamics. The source-free functional is a truly non-\ncollinear functional and, as we demonstrate in the present\nwork, it leads to a non-zero torque on the spins. We\n\fnd that for bulk systems (Ni and Co), where SO in-\nduced anisotropy is very small i.e. magneto crystalline\nanisotropy (MCA) energy is only 2 \u0016eV/atom, internal\ntorques on spins lead to precession of spins about the\neasy axis an e\u000bect which cannot be described by conven-\ntional functionals like ALSDA. For surfaces and inter-\nfaces, where the MCA is \u00181meV/atom, internal torques\ndo not cause much precession of spins and results for\nALSDA and source-free functional are almost the same.\nII. METHODOLOGY\nA. Time-dependent density functional theory\nThe Runge-Gross theorem [32] establishes that the\ntime-dependent external potential is a unique functional\nof the time dependent density, given the initial state.\nBased on this theorem, a system of non-interacting par-\nticles can be chosen such that the density of this non-\ninteracting system is equal to that of the interacting\nsystem for all times[33{35]. The wave function of this\nnon-interacting system is represented as a Slater deter-\nminant of single-particle orbitals. In what follows a fully\nnon-collinear spin-dependent version of these theoremsarXiv:1802.10382v1  [cond-mat.mtrl-sci]  28 Feb 20182\nis employed[36]. Then the time-dependent Kohn-Sham\n(KS) orbitals are 2-component Pauli spinors,  , deter-\nmined by the equations:\ni@ j(r;t)\n@t=\"\n1\n2\u0012\n\u0000ir+1\ncAext(t)\u00132\n+vs(r;t) (1)\n+1\n2c\u001b\u0001Bs(r;t) +1\n4c2\u001b\u0001(rvs(r;t)\u0002\u0000ir)\u0015\n j(r;t)\nwhere Aext(t) is a vector potential representing the ap-\nplied laser \feld, and \u001bare the Pauli matrices. The KS\ne\u000bective potential vs(r;t) =vext(r;t)+vH(r;t)+vxc(r;t)\nis decomposed into the external potential vext, the clas-\nsical electrostatic Hartree potential vHand the xc poten-\ntialvxc. Similarly the KS magnetic \feld is written as\nBs(r;t) =Bext(t) +Bxc(r;t) where Bext(t) is the mag-\nnetic \feld of the applied laser pulse plus possibly an ad-\nditional magnetic \feld and Bxc(r;t) is the xc magnetic\n\feld. The \fnal term of Eq. (1) is the spin-orbit cou-\npling term. It is assumed that the wavelength of the\napplied laser is much greater than the size of a unit cell\nand the dipole approximation can be used i.e. the spatial\ndependence of the vector potential is disregarded. The\n2-component Pauli spinors,  , are then used to construct\nthe magnetization density as:\nm(r;t) =NX\nj=1 y\nj(r;t)\u001b j(r;t); (2)\nmaking ma 2\u00022 matrix in the spin-space.\nB. Functional\nIn order to propagate Eq. (1) in time one needs to ap-\nproximatevxcandBxc. Usually adiabatic extensions of\nground-state functionals like LSDA are used in such time-\npropagation scheme[36{40]. LSDA is a collinear func-\ntional by design and requires only \u001a\"and\u001a#as input.\nThus onlyvxcandBz\nxcare obtained from the functional\nderivative. A non-collinear extension can be constructed\nby \frst diagonalizing the 2 \u00022 magnetization density in\nEq. (2) at each point in space and then calculating Bxc\nby taking functional derivative of energy with respect to\nthis diagonal magnetization density and reversing the di-\nagonalization . Such a Bxcis not curl of a vector \feld\nand contains unphysical magnetic monopoles (i.e. source\nterms). More importantly, by construction such a func-\ntional gives m(r;t)\u0002Bxc(r;t) = 0 i.e. the internal torque\non spins is zero at all times and in all space.\nRecently, it was shown that when these unphysical\nsource-terms are removed from ALSDA[31] the result-\ning functional is a truly non-collinear functional. This\nfunctional was able to successfully describe the correct\nground-state moment of pnictides, a problem which has\nbeen intractable for well over a decade. Interestingly,being a truly non-collinear functional the source-free\nALSDA gives torque m(r;t)\u0002Bxc(r;t)6= 0 and hence\nwould contribute to the dynamics of the spins. The con-\nstruction of this source-free functional requires following\n3 steps:\n\u000fThe LDA energy functional is modi\fed as\nExc[\u001a;m]!Exc[\u001a;sm] and a scaling of the re-\nsultant xc \feld is performed as BLDA\nxc!sBLDA\nxc\nin order to keep the functional variational with re-\nspect to m. The value of sis chosen empirically to\nbe 1.12.\n\u000fFollowing Poisons equation is then solved to calcu-\nlate\u001e\nr2\u001e(r;t) =\u00004\u0019r\u0001BLDA\nxc(r;t): (3)\n\u000fFinally the source term is removed from BLDA\nxcus-\ning:\nBSF\nxc(r;t)\u0011BLDA\nxc(r;t) +1\n4\u0019r\u001e(r;t): (4)\nIt is easy to show that r\u0001BSF\nxc= 0.\nIII. COMPUTATIONAL DETAILS\nAll the calculations in the present work are done using\nthe state-of-the art full potential linearized augmented\nplane wave (LAPW) method as implemented within the\nElk code[41] code. Within this method the core elec-\ntrons (with eigenvalues 95 eV below Fermi energy) are\ntreated fully relativistically by solving the radial Dirac\nequation while higher lying electrons are treated using\nthe scalar relativistic Hamiltonian in the presence of the\nspin-orbit coupling. To obtain the 2-component Pauli\nspinor states, the Hamiltonian containing only the scalar\npotential is diagonalized in the LAPW basis: this is the\n\frst variational step. The scalar states thus obtained\nare then used as a basis to set up a second-variational\nHamiltonian with spinor degrees of freedom [42]. This is\nmore e\u000ecient than simply using spinor LAPW functions,\nhowever care must be taken to ensure that a su\u000ecient\nnumber of \frst-variational eigenstates for convergence of\nthe second-variational problem are used.\nWe solve Eq. (1) for the electronic system alone. Cou-\npling of the electronic system to the nuclear degrees of\nfreedom is not included in the present work. Radiative\ne\u000bects, which can be included by simultaneously time-\npropagating Maxwell's equations, are also not included\nin the present work. At longer times scales these e\u000bects\nare expected to contribute signi\fcantly.\nA regular mesh in k-space of 8\u00028\u00021 for multi-\nlayers and 8\u00028\u00028 for bulk was used and a time step\nof \u0001t= 4:13fs was employed for the time-propagation\nalgorithm[43]. A smearing width of 0.027 eV was used.3\nLaser pulses used in the present work are linearly polar-\nized with a frequency of 1.55 eV (red). For all ground-\nstate calculations a full structural optimization was per-\nformed. For the case of Co/Pt(001) and Ni/Pt(001), the\nPt substrate was simulated by using 4 to 8 Pt mono lay-\ners (ML). We found that for Pt layer thickness greater\nthan 4 ML the results do not change signi\fcantly and\nhence all results presented here are for 3ML of Co or Ni\non 5ML of Pt(001).\nIV. RESULTS\nA. Ground-state and internal spin-torque\nThe \frst step is to determine the ground-state of the\nmaterial by using the LSDA and source-free functional.\nIn order to ensure an unbiased magnetic ground-state a\nfully unconstrained minimization was performed without\nimposing any magnetic symmetries. The moments thus\nobtained are shown in Table I. Then the laser pulse is\napplied to the material and the evolution of spins are\nstudied as a function of time using TD-DFT. It is clear\nfrom these results that the ground-state obtained using\nthe source-free functional is slightly more non-collinear\nthan the LSDA.\nTABLE I: Ground-state moments (in \u0016B) for all the\nmaterials studied in the present work. The results are\ncalculated using LSDA and source-free functionals.\nMaterial LSDA Source-free\nMxMyMzMx My Mz\nBulk Ni 0 0 0.67 0.02 0.37 0.56\nBulk Co 0 0 1.69 0 0.11 1.69\nNi@Ni/Pt 0 0.78 0 0.04 0.76 0\n0 0.71 0 0.04 0.67 0\n0.01 0.57 0 0.03 0.52 0\nCo@Co/Pt 0 1.89 0 0 1.87 0\n0 1.72 0 0 1.67 0\n0.11 1.66 0 0.1 1.62 0\nUnlike LSDA, source-free functional does not require\nBxcto be parallel to m. This results in the internal\nspin-torque, m(r)\u0002Bxc(r), being non-zero. These are\nplotted these spin-torques are plotted in Fig. 1 for bulk\nNi and a 3Ni/5Pt interface. In the ground-state, m(r)\u0002\nBxc(r) exactly cancels the divergence of spin-current[29].\nHowever, this term can contribute to the spin-dynamics\naway from the equilibrium. In the next section we look at\nthe e\u000bect of this internal spin-torque on the laser-induced\ndynamics of spins.\nB. Laser induced spin-dynamics\nWe now focus our attention on laser induced spin-\ndynamics. The results for bulk Ni and Co are shown\nFIG. 1: Top panel shows the m(r;t= 0)\u0002Bxc(r;t= 0)\nfor bulk Ni in (111) plane. Bottom panel shows the\nsame for 3Ni/5Pt in the (110) plane. The arows\nindicate the direction and colors the magnitude.\nin Fig. 2. The dynamics of the total moment,\nMtotal =q\nM2x+M2y+M2z(shown in red), obtained\nusing ALSDA and source-free functional are almost the\nsame. A closer look however reveals that the x,yand\nzprojected moments are strikingly di\u000berent for the two\nfunctionals. In the case of ALSDA there is no change in\nMxandMyas a function of time. On the other hand,\ninternal torques in the source-free functional cause the\nspins to rotate around the z-axis as a function of time{ as\nMx(t) increases, My(t) decreases. Mz(t) shows demagne-\ntization for both the functionals the reason for which, we\n\fnd to be, spin-orbit induce spin-\rips. It is also impor-\ntant to note that during the \frst \u001830fs (when the laser\npulse creates a non-equilibrium charge distribution) the\ne\u000bect of the internal spin-torque is small and all the spin-\ndynamics is caused by the SO induced spin-\rips. This\nindicates that the contribution of the torque term to the\nspin-dynamics is slower than the SO term.\nSince this internal spin-torque increases the non-4\nA-field FWHM=12.5fs, E=30mJ/cm2Ni bulk (a)\n-0.15-0.1-0.050\n∆ΜTotal\n∆Μx\n∆Μy∆Μz\n0 10 20 30 40 50 60 70 80 90 100\nTime(fs)-0.15-0.1-0.0500.05M(t)-M(t=0) [µΒ]\nA-field FWHM=12.5fs, E=30mJ/cm2Co bulk (b)\n-0.2-0.10\n∆ΜTotal\n∆Μx\n∆Μy∆Μz\n0 10 20 30 40 50 60 70 80 90 100\nTime(fs)-0.3-0.1500.15M(t)-M(t=0) [µΒ]\nFIG. 2: (a) Top panel shows the vector potential of the\napplied linearly polarized (along z-axis) laser pulse with\nfrequency=1.5eV, FWHM=12.5fs and \ruence =\n30mJ/cm2. Middle panel shows the total moment (red)\nand the bottom panel x(green),y(brown) and z(blue)\nprojected moments for bulk Ni as a function of time (in\nfs). Dashed line are the results obtained using ALSDA\nand full lines the results obtained using source-free\nfunctional. (b) The same as (a) but for bulk Co.\ncollinearity in the system, it will be interesting to know\nhow they e\u000bect the spin-dynamics on surfaces and inter-\nfaces, where lower symmetry has the e\u000bect of frustrat-\ning and thus enhancing the non-collinear nature of spins.\nSuch results are shown in Fig. 3 for 3ML of Co on 5ML\nof Pt(001) and 3ML Ni on 5ML of Pt(001). In both\ncases the ground-state moment points in-plane (see Ta-\nble I). The laser induced spin-dynamics from ALSDA as\nwell as source-free functional show that the internal spin-\ntorque does not signi\fcantly contribute to spin-dynamics\nin early times ( <100fs). In the case of interfaces the\ndemagnetization of Ni layers is caused by two distinct\nprocesses{ (i) spin injection in the Pt layers: optically\nexcited electrons make a inter-site spin transfer. This\ne\u000bect is called OISTR[12] and is caused due to optical\n(a)\nA-fieldFWHM=12.5fs, E=30mJ/cm23Ni/5Pt(001)\n0 10 20 30 40 50 60\nTime(fs)-2-1.5-1-0.50M(t)-M(t=0) [µΒ]\n∆Μy∆Μz∆Μx\n(b)\nA-fieldFWHM=12.5fs, E=30mJ/cm23Co/5Pt(001)\n0 10 20 30 40 50 60\nTime(fs)-4-3-2-10M(t)-M(t=0) [µΒ]\n∆Μy∆Μz∆ΜxFIG. 3: (a) Top panel shows the vector potential of the\napplied linearly polarized (along y-axis) laser pulse with\nfrequency=1.5eV, FWHM=12.5fs and \ruence =\n30mJ/cm2. Lower panel shows x(green),y(brown) and\nz(blue) projected moments for 3ML of Ni on 3ML of\nPt(001) as a function of time (in fs). Dashed line are\nthe results obtained using ALSDA and full lines the\nresults obtained using source-free ALSDA. (b) The\nsame as (a) but for 3ML of Co on 5ML of Pt(001).\ncharge excitations. These excitations lead to majority\nspin electrons being injected into the Pt layers. OISTR\ndominates the physics of demagnetization for the \frst\n25fs. (ii) Spin-orbit induced spin-\rips[36, 39, 40]: this\nprocess dominates after the \frst 25fs.\nAt the \frst sight these results look surprising { the\ninternal torque-induced spin-dynamics in bulk is much\nlarger than for surfaces or interfaces. However, this can\nbe explained based on ground-state energetics { in the\ncase of bulk Ni and Co the MCA (\u0001 E=Ez\u0000Exy,\nwhereEiis the total energy with spins pointing in the\ni-direction) is or the order of .. \u0016eV, making it easy for a\nsmall spin-torque (in Fig. 1) to rotate the electronic spin\nabout the easy axis. In the case of interfaces the MCA is5\nmuch larger,\u00181meV/atom, and small torque terms are\nnot su\u000ecient to overcome the energy barrier to rotate the\nspins signi\fcantly.\nV. SUMMARY\nIn the present work we explore the e\u000bect of using\nrecently derived source-free exchange-correlation func-\ntional on the ground-state and laser induced spin-\ndynamics of bulk Ni, Co and interfaces of these metals\nwith Pt. We compare the results obtained using source-\nfree functional to those obtained using unmodi\fed LSDA.\nOur key \fndings are: (a) the source-free functional in-\ntroduces an extra non-collinearity in the ground-state\ncompared to the conventional LSDA functional, (b) the\nspin-torque term, which is identically 0 for LSDA, is non-\nzero for source-free functional, (c) the e\u000bect of this inter-\nnal spin-torque on laser induced spin-dynamics is mostprominent for systems where the magneto crystalline\naninsotropy is small like bulk cubic materials. For such\nmaterials the small torque generated by the source-free\nfunctional are su\u000ecient to cause spin precession about\nthe easy axis and (d) the contribution of this spin rota-\ntion due to internal spin-torque is a slow process com-\npared to optical inter-site spin transfer and spin-orbit\ninduced spin-\rips.\nIn future it would be interesting to explore proper-\nties which are profoundly a\u000bected by spin-torque such\nas magnon response, spin-transfer-torque and spin-orbit\ntorque.\nA. Acknowledgments\nSharma would like to acknowledge SPP-QUTIF and\nSFB 762 for funding.\n[1] N. Laman, M. Bieler, and H. van Driel, J. Appl. Phys.\n98, 103507 (2005).\n[2] M. Cinchetti, K. Heimer, J.-P. Wuestenberg, O. An-\ndreyev, M. Bauer, S. Lach, C. Ziegler, Y. Gao, and\nM. Aeschlimann, Nat. Mater. 8, 115 (2009).\n[3] D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam,\nJ. M. Shaw, E. Turgut, P. Maldonado, S. Mathias,\nP. Grychtol, H. T. Nembach, T. J. Silva, M. Aeschli-\nmann, H. C. 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Commun. 209, 92 (2016)."    },    {        "title": "1904.01331v1.Electronic_spin_spin_decoherence_contribution_in_molecular_qubits_by_quantum_unitary_spin_dynamics.pdf",        "content": "Electronic spin-spin decoherence contribution in molecular qubits by quantum unitary\nspin dynamics\n1Alessandro Lunghi\u0003and1Stefano Sanvito\n1School of Physics, AMBER and CRANN, Trinity College, Dublin 2, Ireland\nThe realisation of quantum computers based on molecular electronic spins requires\nthe design of qubits with very long coherence times, T2. Dephasing can proceed over\nseveral di\u000berent microscopic pathways, active at the same time and in di\u000berent regimes.\nThis makes the rationalisation of the dephasing process not straightforward. Here we\npresent a computational methodology able to address spin decoherence processes for\na general ensemble of spins. The method consists in the propagation of the unitary\nquantum spin dynamics on a reduced Hilbert space. Then we study the dependence\nof spin dephasing over the magnetic dilution for a crystal of Vanadyl-based molecular\nqubits. Our results show the importance of long-range electronic spin-spin interactions\nand their e\u000bect on the shape of the spin-echo signal.\nI. INTRODUCTION\nIn recent years, the possibility of using quantum prop-\nerties of materials as active elements of new technolo-\ngies has emerged, setting the foundation for a quantum\nrevolution . Quantum computing, even though theorised\nseveral years ago, has only recently been implemented in\nseveral physical systems, ranging from Cooper's pairs in\nsuperconductors [1] to electronic spins [2]. In a nut-shell,\nthe vast dimension of a quantum system's Hilbert space\nand the quantum mechanical way to operate over it, al-\nlows one to perform operations at a speed much faster\nthan in classical computers. The basic working principle\nof this technology requires the ability to initialise, ma-\nnipulate and detect a many-body quantum state in its\nHilbert space [3]. The coherence over time of the infor-\nmation \\stored\" in the quantum Hilbert space, gener-\nally referred as the characteristic time, T2, is an essential\nproperty of a given system and depends on both intrin-\nsic [4] and extrinsic factors [5]. Quantum coherence has\nto be preserved long enough for the computation to be\ncarried out and it represents the basic requirement for\nany class of potential qubits.\nAmong the most promising paradigms to build\nquantum computers, transition-metal-based magnetic\nmolecules have been brought forward as potential qubits,\nthanks to their long spin-coherence times and easy\ntunability[6{13]. Indeed, molecular compounds represent\na rather rich materials platform, where several synthetic\nstrategies can be exploited to selectively tune speci\fc\ninter- or intra-molecular interactions, with the goal of\noptimising the compound and extracting the basic phys-\nical picture of the dephasing. In recent years, an ex-\ntensive amount of work has been performed in order to\nextend both the spin-lattice and spin-coherence times.\nSeveral guidelines to build more robust molecular qubits\nhad been proposed, highlighting the importance of both\n\u0003lunghia@tcd.iethe nature of the \frst coordination shell [4] and of sur-\nrounding nuclear spins density [5].\nFrom a rational design point of view, the optimisation\nprocess has to be led by a detailed knowledge of those mi-\ncroscopic mechanisms leading to spin dephasing and this\nrepresents the focus of this work. Restricting our study\nto the sole spin-spin interactions, i.e. assuming the tem-\nperature to be low enough to exclude spin-lattice relax-\nation processes, several sources of spin dephasing might\nbe operative. In a very general fashion, every spin in the\nsystem, other than the one addressed, cause a dynamical\n\ructuation of the spin local magnetic \feld, generating de-\ncoherence. In particular, three di\u000berent types of spin-spin\ninteractions are generally present in a crystal of molec-\nular magnets: electronic-electronic dipolar interactions,\nnuclei hyper\fne interaction and electronic-nuclear dipo-\nlar interactions. From a computational point of view,\nseveral approaches has been developed to study the cen-\ntral spin problem [14{17], where a single electronic spin\nloses its correlation due to the interaction with a bath of\nnuclear spins, \fnding application in both solid-state de-\nfect quantum bits [18] and molecular qubits [19]. In or-\nder to complement the existing picture of spin dephasing,\nwe present here a computational strategy to address the\nrole of electronic spin-spin interactions over the dephas-\ning time for an ensemble of generic spins. This approach,\nable to address up to hundreds of spins at the same time,\nis used to simulate the spin-echo decay experiment for a\ncrystal of Vanadyl qubits, revealing a linear dependence\nofT2with respect to the magnetic dilution. Moreover,\ninsights regarding the shape of the spin-echo envelop are\nreported, showing a transition from a Gaussian pro\fle,\nfor a perfectly ordered crystal, to a stretched exponential,\nfor the random-distributed magnetically diluted phase.\nII. UNITARY QUANTUM DYNAMICS AND\nSPIN ECHO\nA spin initialised in a speci\fc state is always subject\nto dephasing, unless it represents a perfect close sys-arXiv:1904.01331v1  [cond-mat.mtrl-sci]  2 Apr 20192\ntem, namely it is isolated from the rest of the universe.\nSuch a strict condition is never met in practice, so that\na quantum system is never fully isolated and thus is sub-\nject to some non-trivial dynamics. The study of this\nprocess, in principles, requires the knowledge of all the\nspin-\\external world\" interactions and the solution of the\nSchr odinger equation for the coupled system.\nIn order to formalise the problem, let us introduce the\ntotal spin Hamiltonian for a system of NScoupled spin\n1/2,S(i),\n^HS=NSX\ni\u0016B~B\u0001g(i)\u0001~S(i)+1\n2NSX\nij~S(i)\u0001Ddip(ij)\u0001~S(j);(1)\nwhere the interaction of the spins with an external \feld,\n~B, is gauged by the Bohr magneton, \u0016B, and theg-tensor,\ng(i), while the spin-spin interaction, Ddip, is assumed to\nbe only dipolar in nature. The state of the spin system\ncan be described in term of the spin density matrix, \u001a,\nwhose dynamics is regulated by the Liouville equation\nd^\u001a\ndt=\u0000i\n~[^Hs;^\u001a]: (2)\nThe matrix elements of ^HSare usually computed on a\nbasis set corresponding to the tensor product of all single\nspins ^Sz(i) operators' eigenkets\nfjSz(1);Sz(2);:::;Sz(N)ig=NO\ni=1fjSz(i)ig:(3)\nBy diagonalising HSone can solve numerically Eq. (2)\nto obtain an expression for the time evolution of \u001aabin\nterms of the eigenvalues of HS,Ea,\n\u001aab(t) =Uab;ab(t\u0000t0)\u001aab(t0) =e\u0000i!ab(t\u0000t0)\u001aab(t0);(4)\nwhere!ab= (Ea\u0000Eb)=~andUab;ab(t\u0000t0) is the unitary\nevolution operator.\nEq. 4 describes the exact dynamics of the NSinteract-\ning spins, but its use is limited by the size of the matrix\nthat one can actually diagonalise. The complexity of the\nproblem arises mainly from the exponential dependence\nof the size of the spin Hilbert space on NS. This limits\nthe solution to\u001816 spins on common parallel machines,\na general problem in quantum many-body physics. Nev-\nertheless, several approaches to overcome such limitation\nexist. The speci\fc problem concerning the correlation\nloss of a central spin embedded in a spins bath has been\nsuccessfully tackled with a range of methods falling un-\nder the name of cluster approximations [14{17], where\nthe total contribution to dephasing is decomposed into\nthe contribution of small spin clusters. Another possible\napproach, somehow based on a similar physical princi-\nple, consists in working on a truncated Liouvillian space,\nwhere the states considered are chosen among the ones\ndeemed relevant. The basic idea behind these methods\nconsists in assuming that only a small part of the totalLiouville space will be actually explored during the time\nevolution. Moreover, the states explored will be those\nclose enough to the states at time t= 0. Coupled clus-\nters [20], density-matrix renormalisation group [21] and\nadaptive state-space restriction [22] approaches all fall\ninto this category.\nIn this paper we will take advantage of a non-\nperturbative Hilbert space size reduction technique, be-\nlonging to the last class of methods and inspired by\nKuprov's works [22]. In the presence of an external mag-\nnetic \feld pointing along the zdirection and in the limit\nBz!1 , the ground state will tend to a state with all\nthe spins aligned towards the \feld. In this situation, the\nground state can be represented with only one vector in\ntheSzbasis set and all the excited states are represented\nby single, double, ..., N-th fold spin \rips. In the presence\nof spin-spin interactions a mixing between these eigen-\nstates is introduced, generating a non-trivial dynamics,\nleading to decoherence. However, as long as the Zeeman\nterm remains the leading interaction, the nature of the\neigenstates will be only slightly a\u000bected. Therefore, by\nassuming that the starting con\fguration can be gener-\nated close to the ground state of the system, we can re-\nstrict the size of the Hilbert space to those states within\na few spin \rips from the initial state. Such a restriction\nmakes the total number of spin states needed to inves-\ntigate the dynamics to scale polynomially with NSand\nthe problem expressed by Eq. (2) and Eq. (4) becomes\neasily tractable even for 102spins.\nThe method just outlined is here employed to study\nthe e\u000bect of electronic spin-spin interactions in the de-\ncoherence processes through the simulation of the spin\nEcho experiment. The spin Echo experiment is often use\nas a mesure of T 2, as it averages out inhomogeneous ef-\nfects, and it consists in a two pulses sequence. The \frst\npulse brings the magnetization along the x-axis, then, af-\nter a free evolution time \u001c, a\u0019pulse is applied. After the\nsecond pulse, another interval \u001cof free evolution time is\nwaited before reading out the x component of the residual\nmagnetization. This can be described by the path,\n\u001a(0)R(\u0019\n2)\u0000\u0000\u0000!\u001a\u0019\n2(0)U(\u001c)\u0000\u0000\u0000!\u001a\u0019\n2(\u001c)R(\u0019)\u0000\u0000\u0000!\u001a3\u0019\n2(\u001c)U(\u001c)\u0000\u0000\u0000!\u001a3\u0019\n2(2\u001c);\n(5)\nwhere ^\u001a(0) is the starting density matrix, U(t) is the time\npropagator operator as expressed in Eq. (4) and R(\u0012) is\na rotation operator that simulates the e\u000bect of a micro-\nwave electron paramagnetic resonance (EPR) pulse.\nIII. COMPUTATIONAL DETAILS\nThe implementation of the method just outlined fol-\nlows three steps: 1) the construction of the total Hamil-\ntonian matrix, 2) the diagonalization and 3) the propaga-\ntion of an initial density matrix according to the scheme\nintroduced in Eq. (5). The truncated basis set is gener-\nated by \ripping a given number of spins with respect to\na state where they are all aligned along the \feld direc-3\ntion. When possible, the spin Hamiltonian is rotated in\na block-diagonal form by forming a translation-invariant\nbasis set. The spin Hamiltonian diagonalisation is per-\nformed with scalapack libraries [23]. The initial spin-\ndensity matrix is generated as a pure state where all the\nspins are aligned along the external magnetic \feld. Tests\nhave also been performed with an initial spin-density ma-\ntrix equal to the canonical ensemble distribution, but\nno di\u000berences in the extracted T 2values have been ob-\nserved. The rotation operator is constructed by converg-\ning a Taylor expansion of R(\u0012) = exp[i(~S\u0001~n)\u0012=2\u0019], where\n~Sis the spin-vector operator of the spin to be rotated, ~n\nis the normal rotation axis and \u0012is the angle of rotation.\nThe time evolution of the expectation value for a general\noperator ^AishA(t)i= Trf^A^\u001a(t)g.\nWe have also tested an alternative to the diagonalisa-\ntion of the spin Hamiltonian matrix that involves the use\nof the original Szbasis set. In this basis set almost all the\noperators have a very sparse matrix representation that\ncan be conveniently exploited for storing purposes. How-\never, the time propagator ^U(t\u0000t0), being an exponential\noperator, has a dense matrix representation and must be\nmultiplied to the density matrix twice at each time-step.\nMoreover, the generation of the exponential operator in\nthis basis set requires the use of a very short time step\n\u0001t=t\u0000t0, usually requiring the calculation of more\nthan 106steps to reach the desired microsecond time-\nscale. Consequently, here we employ the direct diagonal-\nisation approach that makes it possible to set the desired\ntime step in the dynamics propagation. The gopera-\ntors, when not available from experiments, are computed\nby CASSCF calculations with the software Orca [24].\nAn active space including the delectrons and \fve 3 dor-\nbitals is used. The spin-spin distance used to compute\nthe dipole-dipole interaction is implemented using peri-\nodic boundary conditions in order to correctly reproduce\nthe solid state environment.\nIV. RESULTS\nIn this work we study two molecular systems belonging\nto the Vanadyl class of molecular quibits: VO(acac) 2[25],\nbeing acac=acetylacetonate, and VO(dmit) 2[4], with\ndmit = 1,3-dithiole-2-thione-4,5-dithiolate. These mag-\nnetic molecules have recently become subjects of intense\nresearch as they naturally show rather long coherence\ntimes, up to several \u0016seven at room temperature [26],\nsuggesting the possibility of further performance increase\nif speci\fcally tailored.\nThe \frst system investigated is the VO(acac) 2crystal.\nThe phase-coherence properties of this compound have\nnot been investigated experimentally, and here we use it\nto show the consistency of our computational method.\nThegtensor for this specie is calculated as explained in\nthe \\Computational Details\" section and it reproduces\nthe typical values for this class of molecules: gxx=\n1:9430,gxy= 0:0179,gxz=\u00000:0170,gyy= 1:9749,gyz= 0:0084 andgzz= 1:9735. The orientation of the\nmolecule in space is as found in the crystal's unit-cell\naligned with the crystallographic ~avector along xand\nthe~bvector lying in the xyplane. The lattice parameters\ncan be found in ref. [25]. Finally the external \feld was\nchosen aligned along the zdirection. Fig. 1 reports the\ntime evolution of the expectation value of the transverse\nmagnetisation, Mx, when the multiplicity of the spin ex-\ncitations included in the basis set of Eq. (3) is changed.\nThe study is performed on a 2 \u00022\u00022 super-cell of the\nVO(acac) 2crystal, containing a total of 16 independent\nand interacting spins. From the \fgure it is possible to\nsee howhMxifor a given spin initialised at its maximum\nvalue after the \u0019=2 pulse, decays with a Gaussian pro\fle\nMx(t) = (Mx(0)\u0000Mx(1))e\u0000(t\nT2)2\n+Mx(1):(6)\nThe calculated time constant is in the nanosecond range.\nThe short-time spin dynamics converges very rapidly\nwith respect to the inclusion of multiple spin-\rips in the\nbasis set. In fact, already for the minimal basis set, in-\ncluding only double spin-\rips, the decay part of the dy-\nnamics is well converged and only long-time features need\nto be perfected. This is an important aspect as it allows\nus to use the minimal basis-set to study the dephasing\nprocess, being the short-time dynamics the important\none.\nFIG. 1. Spin Echo dependence on the number of spin\nexcitations. The spin-echo pro\fle for the dynamics of 16\nspins, arranged on a 2 \u00022\u00022 supercell of the VO(acac) 2crys-\ntal. The time evolution is computed when 2 (green curve), 3\n(red) and 4 (black) spin excitations are included in the basis\nset. Note that the minimal basis set, containing 2 spin ex-\ncitations only, is already su\u000ecient to describe the short-time\ndynamics.\nStrong of these results we use the two spin-\rips basis-\nset to study the dependence of the hMxidynamics on the\ntotal number of spins included in the simulation. This is\nreported in Fig. 2. The spin dynamics features converge\nfast with respect to the number of interacting spins, and\nvirtually no di\u000berence is observed on the short-time pro-4\n\fle for 3\u00023\u00023 and 4\u00024\u00024 VO(acac) 2super-cells, con-\ntaining 54 and 128 spins, respectively. It is also interest-\ning to note how the long-time \ructuations get damped\nwith the inclusion of more spins in the simulation. This\nhappens as a result of an increased size of the Hilbert\nspace, which makes the chance of the system to visit\nstates with a large magnetisation less probable, in anal-\nogy to the classical recurrent time in the phase space.\nFIG. 2. Spin-echo dependence on the number of spins.\nThe echo pro\fles obtained from the simulation of di\u000berent\nsizes of the VO(acac) 2supercell. These are calculated with\na two spin-\rips basis sets. The green curve corresponds to a\n2\u00022\u00022 supercell, the red one to a 3 \u00023\u00023 and the black one\nto a 4\u00024\u00024.\nBeing the limits of our method understood, we are now\nready to study the e\u000bects of the magnetic dilution on\nthe transverse magnetisation decaying pro\fle. A simple\nmodel for addressing this problem consists in designing\na lattice of spins with the same symmetry of the crystal\nunit-cell and rescale the size of the lattice vectors in order\nto modulate the spin-spin distance as in a dilution exper-\niment. By applying this methodology to 27 VO(acac) 2\nspins, we scan magnetic dilution values ranging from 50%\nto 0.78% and the results are reported in Fig. (3). The\ndilution percentage is calculated as the number of spins\nwith respect the total number of Vanadium sites avail-\nble in the volume considered. The echo signal presents\nthe same features regardless of the dilution, since the\nsymmetry of the interactions is preserved. However, the\ntime-scale changes, leading to larger T2values for diluted\nsystems. These reach values of 0.1 \u0016sfor\u00181% dilution.\nFig. 4 shows the T2, extracted by \ftting the echo pro-\n\fles to Eq. (6), as a function of the dilution, revealing\na linear dependence. This fact has also been previously\nnoted in the context of nuclear magnetic impurities in-\nteracting with an electronic spin [17] and in diluted rad-\nicals [27]. It originates from the fact that, even though\nthe dipolar interactions goes as R\u00003, the average distance\nbetween spins does not scale linearly with the dilution\nFIG. 3. Spin echo dependence on the magnetic dilu-\ntion. The echo pro\fle obtained for 27 spins arranged accord-\ning to the VO(acac) 2crystal symmetry is reported for dif-\nferent values of the dilution: 50% (red curve), 6.25% (green\ncurve), 1.85% (red curve) and 0.78% (black curve).\npercentage.\nFIG. 4. T2dependence on the magnetic dilution. The\ndependence of T2, extracted by \ftting to the expression in\nEq. (6), is shown as function of dilution percentage (purple\ndots and line). The green line serves as guide to explicit the\nx\u00001dependence.\nInterestingly, even though the predicted order of mag-\nnitude of the spin dephasing is approaching the correct\nvalue for the typical dilutions used in experiments, the\nshape of the spin-echo envelope is qualitatively di\u000berent.\nSpin-echo signals are usually interpreted with generalised\nexponentials as\nMx(t) = (Mx(0)\u0000Mx(1))e\u0000(t\nT2)\f\n+Mx(1):(7)\nFor this class of molecular complexes an experimental\nvalue\f <1 is found [26], while here the pro\fle is Gaus-\nsian and\f= 2.5\nIn order to investigate this discrepancy we turn our at-\ntention to the study of another molecule in the Vanadyl\nfamily, namely VO(dmit) 2. This compound has been ex-\ntensively investigated experimentally [4] and the spin-\necho decay pro\fle is available. Furthermore, an experi-\nmental gtensor has been extracted and it is used here:\ngxx=gyy= 1:9843 andgzz= 1:9735. In this second\nexample we overcome the limitations of the previous di-\nluted study by designing several super-cells, where the\nVanadium sites are randomly occupied by a number of\nspins consistent with the desired level of dilution. A \frst\nset of 80 supercells contains 48 spins located on the sites\nof a 6\u00028\u00026 VO(dmit) 2supercell, and a second set of 80\ncontains 70 spins located on the sites of a 7 \u00029\u00027 super-\ncell. All the supercells correspond to a dilution of 4%,\na value comparable with experimental conditions. The\nspin echoes for all the di\u000berent simulations performed\non the same number of spins are then averaged. This\nprotocol o\u000bers a more realistic representation of the ex-\nperimental conditions, where spins experience di\u000berent\nenvironments.\nOur results are summarised in Fig. 5, where we show\na non-Gaussian decaying pro\fle of Mxfor both the sets\nof supercells, with the only noticeable di\u000berence among\nthem being the value of Mx(1). The \ft to Eq. (7) of\nthe curve corresponding to the simulation with 70 spins\nreturns\f= 0:70 andT2= 0:20\u0016s. In comparison,\nVO(acac) 2relaxes with T2\u00180:10\u0016sfor the same level of\ndilution, as extrapolated from Fig.4. Notably, the stretch\nfactor\fis now in excellent agreement with the experi-\nmental phenomenology. At the same time we predict a\nT2slightly shorter than the experimental one, which is\nestimated to be \u00185\u0016s. This is an interesting result,\nsince usually an incomplete model overestimates the re-\nlaxation time, while our predictions surpass the experi-\nmental rate of decoherence. This suggests that there is an\nactive mechanism, not cover in our simulations, slowing\ndown the dephasing process. However, other sources of\nerror might may also come from the simulated pulses be-\ning instantaneous and from the fact that in experiments\nmultiple spins resonate at the same frequency and would\n\rip at once under micro-wave pulses. Our model instead\nassumes that one single electronic spin is a\u000bected by the\npulse.\nV. DISCUSSION\nThe study of spin dephasing in molecular magnets has\nbeen mainly driven by experimental investigation and\nover the years several relevant \fgures of merit for the op-\ntimisation of T2have been proposed. However, the main\nfocus of these investigations concern the role of the H1\nspins in the solid-state matrix. The physics of this inter-\naction is relatively well understood and the basic guide-\nline to enhance T2consists in synthesising organic ligands\nand crystal's matrices without elements with abundant\nspin-active nuclear isotopes. The role of the residual elec-\nFIG. 5. Spin echo dependence on structural disorder.\nThe time-decay pro\fle of the x-component of the magneti-\nsation is reported for two di\u000berent sets of simulations, both\ncorresponding to a 4% of dilution. The average of 80 sim-\nulations performed with 48 spins is reported in blue, while\nthe red curve describes the average of 80 simulations with 70\nspins. The solid black line corresponds to the best \ft of the\nred curve to Eq. (7).\ntronic spins interactions is usually less investigated and\ndilution is generally thought to be su\u000ecient to eliminate\nit.\nHere, however, we have shown that T2is only linearly\ndependent on the dilution and extremely low concentra-\ntions are needed to bring it beyond the \u0016sboundary. Ac-\ncording to Fig. 3, a 0.01% dilution (or 0.3 mM) is needed\nto obtainT2>10\u0016s. This estimate also sets the min-\nimal distance between qubits inside a device to 15 nm.\nFor levels of dilution lower than this threshold, the elec-\ntronic spin-spin contribution is comparable to that of H1.\nTherefore, the method proposed here, is expected to be of\nhelp in the discrimination processes originating from dif-\nferent interactions. A less investigated aspect, due to the\nsynthetic complexity in gauging such interaction, con-\ncerns the role of the hyper\fne coupling of the nuclear\nspin borne by the same atom hosting the electronic one.\nThe method presented in this work can also be applied to\nthis situation, as well as to the central electronic spin in-\nteracting with an H1spin bath, and it will be argument of\nfuture applications. Another important ingredient that\ncan be added to our model, is the inclusion of tempera-\nture e\u000bects and spin-phonon relaxation. As we approach\nambient temperature, spin-phonon coupling becomes the\nbottle-neck for the coherence time [4, 28]. In particular\nthe understanding of the correlation between the chem-\nical structure and the spin-lattice relaxation might lead\nto new, unforeseen synthetic designing rules [29{31] and\nit must be considered a research priority.6\nVI. CONCLUSIONS\nIn conclusion, we have here presented a general\nmethodology to generate a truncated Hilbert space,\ndesigned to reproduce the dynamics of interacting spins\nin external magnetic \felds. The method has been tested\nfor VO(acac) 2and VO(dmit) 2molecular qubits, showing\nthat even simple two-spin \rips and second neighbour\nshells are able to capture the relevant physics leading to\nspin dephasing. The study of VO(acac) 2has also been\nextended to reproduce magnetic dilution experiments\nand has shown a linear dependence of T2with respect\nto the dilution. Our results therefore highlight the\nimportant role played by the residual electronic spin\ndipolar interaction in crystals of molecular qubits in\nsetting a upper limit to T2, reachable at not-extreme\ndilutions. Finally, our simulations also shed some light\non the origin of the stretched exponential-type of decay\nusually encountered in EPR investigations of transition\nmetal complexes, pointing to the disorder of magneticcentres inside the solid-state host as its origin.\nData Availability\nAll the relevant data discussed in the present paper are\navailable from the authors upon request.\nAcknowledgments\nThis work has been sponsored by Science Foundation\nIreland (grant 14/IA/2624). We acknowledge the\nMOLSPIN COST action CA15128. Computational\nresources were provided by the Trinity Centre for High\nPerformance Computing (TCHPC) and the Irish Centre\nfor High-End Computing (ICHEC).\nAuthor contributions\nAll the authors discussed the results and contributed to\nthe manuscript.\nCompeting \fnancial interests\nThe authors declare no competing \fnancial interests.\n[1] Y Nakamura, Y A Pashkin, and J S Tsai, \\Coherent\nControl of Macroscopic quantum states in a single cooper\npair box,\" Nature 398, 786{788 (1999).\n[2] C. Godfrin, A. Ferhat, R. Ballou, S. Klyatskaya,\nM. Ruben, W. Wernsdorfer, and F. Balestro, \\Operating\nQuantum States in Single Magnetic Molecules: Imple-\nmentation of Grover's Quantum Algorithm,\" Phys. Rev.\nLett. 119, 187702 (2017).\n[3] David P. DiVincenzo, \\The Physical Implementation\nof Quantum Computing,\" Fortschr. 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Sci. 8, 6051{6059 (2017)."    },    {        "title": "0801.2466v2.Charge_and_Spin_Currents_Generated_by_Dynamical_Spins.pdf",        "content": "arXiv:0801.2466v2  [cond-mat.mes-hall]  25 Jun 2008Typeset with jpsj2.cls <ver.1.2> Full Paper\nCharge and Spin Currents Generated by Dynamical Spins\nAkihitoTakeuchi1∗and Gen Tatara1,2\n1Graduate School of Science and Engineering, Tokyo Metropol itan University, Hachioji, Tokyo\n192-0397, Japan\n2PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Jap an\nWe demonstrate theoretically that a charge current and a spin cur rent are generated\nby spin dynamics in the presence of spin-orbit interaction in the pert urbative regime. We\nconsider a general spin-orbit interaction including the spatially inhom ogeneous case. Spin\ncurrent due to spin damping is identiﬁed as one origin of generated ch arge current, but other\ncontributions exist, such as the one due to an induced conservativ e ﬁeld and the one arising\nfrom the inhomogeneity of spin-orbit interaction.\nKEYWORDS: spintronics, inverse spin Hall eﬀect, spin curre nt, spin Hall eﬀect, current gen-\neration, spin battery, Keldysh formalism\n1. Introduction\nSpin Hall eﬀect1–6is one of the most interesting phenomena in spintronics, whi ch enables\nthe control of magnetic properties by purely electrical mea ns. The idea is to induce a spin\ncurrent in a transverse direction to an applied electric ﬁel d by using spin-orbit interaction. As\nan inverse eﬀect, one can expect the conversion of spin curren t into charge current or electric\nvoltage by use of spin-orbit interaction. This eﬀect, namely the inverse spin Hall eﬀect, was\nproposed by Saitoh et al.7and indeed observed experimentally in metallic systems7–9and in\nsemiconductor (GaAs).10One should note, however, that detection of spin Hall eﬀect ha s so\nfar been done by observing magnetization as a result of ﬂow of spin current, and not the spin\ncurrent itself. In the inverse eﬀect, similarly, the electri c voltage is measured as a response to\nan time-dependent external ﬁeld which drives magnetizatio n dynamics.\nThe inverse of spin Hall eﬀect was theoretically pointed out b y Zhang and Niu11and\nHankiewicz et al.,12where they discussed a transverse charge current by a gradie nt of a spin-\ndependent chemical potential (they called this eﬀect the rec iprocal spin Hall eﬀect). The\nspin-dependent chemical potential was argued to be related by an optical method. In a junc-\ntion of ferromagnet attached to normal metal, generation of electric voltage by applying an\nalternating magnetic ﬁeld observed by Costache et al.13Theoretical explanation of dc voltage\nwas done by Wang et al.14as due to the spin accumulation at the interface arising from a\nbackﬂow of pumpedspin current into the ferromagnet. Voltag e generation from spin dynamics\n∗E-mail address: atake@phys.metro-u.ac.jp\n1/14J. Phys. Soc. Jpn. Full Paper\nwas predicted by Stern15in a slightly diﬀerent context of Faraday’s law for a ﬁctitiou s ﬁeld of\nBerry’s phase. The application to magnetic domain wall was d one by Barnes and Maekawa16\nand Duine.17In the Berry’s phase mechanism, spin-orbit interaction is n ot essential but con-\ntributes as correction.17\nDirect relation between the pumped charge current and the ma gnetization dynamics was\ninvestigated recently by Ohe et al.18They considered a disordered two-dimensional electron\ngas with the Rashba spin-orbit interaction and interacting with dynamical magnetization.\nThe charge current was calculated perturbatively. Diﬀusive electron motion as represented\nby diﬀusion pole (proportional to 1 /q2for small momentum transfer, q) was taken account\nof since it leads to logarithmical long-range correlation i n two-dimensions. Considering a\ncase of uniform Rashba system, they found that pumped charge current had a contribution\nproportional to∝an}bracketle{tS×˙S∝an}bracketri}ht, whereSis a local spin and ∝an}bracketle{t···∝an}bracketri}hthere denotes average taking account\nthe diﬀusive motion of electrons. The quantity ∝an}bracketle{tS×˙S∝an}bracketri}htrepresents a spin damping and is\nrelated phenomenologically to a spin current across the int erface in the case of junctions.19,20\nThe result thus supports the idea of inverse spin Hall eﬀect, w here charge current is converted\nfrom the spin current. Qualitatively this contribution to t he current was, however, found and\nthe inverse spin Hall mechanism turned out to be too naive. Oh eet al. also noted that the\nuniform Rashba system is peculiar, with many cancellations among the Feynman diagrams,\nsimilarly to the peculiarity known in the spin Hall eﬀect.21\nIn this paper we extend the study by Ohe et al.18to the cases of general spin-orbit\ninteraction, including the case of inhomogeneous spin-orb it interaction. The result can thus\nbe applied to a ﬁnite Rashba system attached to leads. It turn s out that such inhomogeneity\nalso contributes to a current with diﬀerent symmetry proport ional to∝an}bracketle{t˙S∝an}bracketri}ht. We consider a\nthree-dimensional case and so do not take account of diﬀusion ladders which give only small\ncontributions unlike in a two-dimensional case considered in ref. 18.\n2. System\nWe consider an electron system with spin-orbit interaction and exchange interaction with\nlocal spins. The local spins can have arbitrary structure an d thus we can discuss various\nsystems, such as those with one or two ferromagnets attached to a nonmagnet as shown in\nFig. 1. We consider a disordered system which would be the cas e of most experiments in\nmetallic systems. The total Hamiltonian is H(t) =H0+Hex(t)+Hso+Himp, where\nH0=−/planckover2pi12\n2m/summationdisplay\nxψ†\nx∇2ψx, (1a)\nHex(t) =−Jex/summationdisplay\nxψ†\nx[Sx(t)·σ]ψx, (1b)\nHso=−i/summationdisplay\nxψ†\nx{[(∇Ux)×∇]·σ}ψx, (1c)\n2/14J. Phys. Soc. Jpn. Full Paper\nFig. 1. (Color online). Two typical systems with spin-orbit interactio n and local spins. The arrow is\na local spin ( S) which may have spatial structure and dynamics, and a bottom laye r describes\nmaterial with spin-orbit interaction (s-o).\nHimp=uni/summationdisplay\ni=1ψ†\nriψri. (1d)\nHere the annihilation (and creation) operator of conductio n electrons in coordinate space is\nψx(andψ†\nx). Theﬁrstterm describes freeelectron. Thesecond term Hexdenotes theexchange\ninteraction, where Jexis a strength of the exchange coupling, Sx(t) represents the local spins\nwhich can have any spatial and slow temporal structure, and σrepresent Pauli matrices.\nThe spin-orbit interaction is represented by Hso, whereUxis a scalar potential (including a\nfactor/planckover2pi12/4m2c2). The last term Himpis the spin-independent impurity scattering which gives\nrise an elastic electron lifetime τ≡(2πNeniu2//planckover2pi1V)−1, whereuis a strength of the impurity\nscattering,niis a number of impurities, Neis the electron’s density of states at Fermi energy,\nandVis system volume.\n3. Charge Current\nTheelectron velocity operator is deﬁned as ˆv=i\n/planckover2pi1[H,x] ([A,B] represents the commutator\nAB−BA), which reads ˆ vµ=−i/planckover2pi1\nm∂\n∂xµ+1\n/planckover2pi1ǫµνη∂Ux\n∂xησν. The charge current density is deﬁned as\njc(x,t)≡−e∝an}bracketle{tψ†\nx(t)← →v\n2ψx(t)∝an}bracketri}ht, whereA†← →vB≡(ˆvA)†B+A†(ˆvB) and∝an}bracketle{t···∝an}bracketri}htis the expectation\nvalue estimated by the total Hamiltonian H. It is given by\njcµ(x,t) =eTr/braceleftBigg/bracketleftBigg\n/planckover2pi12\n2m/parenleftbigg∂\n∂x−∂\n∂x′/parenrightbigg\nµ+iǫµνη∂Ux\n∂xησν/bracketrightBigg\nG<(x,t;x′,t)/bracerightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx′=x,(2)\nwhere Tr{···}represents trace over spin indices and G<(x,t;x′,t) represents a lesser Green\nfunction deﬁned as G<(x,t;x′,t′)≡i\n/planckover2pi1∝an}bracketle{tψ†\nx′(t′)ψx(t)∝an}bracketri}ht. This charge current satisﬁes the charge\ncontinuity equation,\n∂ρc(x,t)\n∂t+∇·jc(x,t) = 0, (3)\nwhereρc(x,t) (≡−e∝an}bracketle{tψ†\nx(t)ψx(t)∝an}bracketri}ht) is the charge density.\n3/14J. Phys. Soc. Jpn. Full Paper\njS\ns-oc\nFig. 2. (Color online). Diagrammatic representations of the charge current at the ﬁrst order in Jex.\nDotted lines and wavy lines denote the exchange interaction with loca l spin (S) and the spin-orbit\ninteraction (s-o), respectively.\nAssuming a dirty case ( Jex≪/planckover2pi1/τandkFU≪/planckover2pi1/τ,kFbeing Fermi momentum), we\ncarry out a perturbation expansion to calculate the charge c urrent. We treat the exchange\ninteraction to the second order and the spin-orbit interact ion to the ﬁrst order. Path ordered\nGreen function22,23is deﬁned as G(x,t;x′,t′)≡−i\n/planckover2pi1∝an}bracketle{tTC{ψx(t)ψ†\nx′(t′)}∝an}bracketri}ht, where T C{···}is a\npath ordering operator deﬁned on Keldysh contour C. This Green function satisﬁes the Dyson\nequation on complex contour,\nG(x,t;x′,t′) =gx−x′(t−t′)−Jex/summationdisplay\nX/integraldisplay\nCdtexgx−X(t−tex)[SX(tex)·σ]G(X,tex;x′,t′)\n−i/summationdisplay\nR/integraldisplay\nCdtsogx−R(t−tso){[(∇RUR)×∇R]·σ}G(R,tso;x′,t′),(4)\nwheregx(t) denotes free Green function deﬁned as gx−x′(t−t′)≡−i\n/planckover2pi1∝an}bracketle{tTC{ψx(t)ψ†\nx′(t′)}∝an}bracketri}ht0,\nwhere∝an}bracketle{t···∝an}bracketri}ht0is the expectation value estimated by free Hamiltonian H0and averaged over\nimpurity scatterings. Dyson equation is solved by iteratio n. If non-interacting, the charge\ncurrent inµ-direction is simply proportional to ∝an}bracketle{tkµ∝an}bracketri}htin momentum space. This contribution\nvanishes, for a system is spatial symmetry (we assume this th roughout this paper). The\ncharge current ﬁrst order either in the exchange interactio n or the spin-orbit interaction also\nvanishes since Tr{σ}= 0. Therefore the charge current only arises if exchange and spin-orbit\ninteractions couple.\n3.1 First order in Jex\nBy use of eqs. (2) and (4), contribution from the left diagram in Fig. 2 is given by\nj(Fig.2−left)\ncµ (x,t) =ie/planckover2pi12Jex\nm/parenleftbigg∂\n∂x−∂\n∂x′/parenrightbigg\nµ/summationdisplay\nX,R/bracketleftbigg/integraldisplay\nCdt1/integraldisplay\nCdt2Tr/braceleftBig\ngx−X(t−t1)[SX(t1)·σ]\n×gX−R(t1−t2){[(∇RUR)×∇R]·σ}gR−x′(t2−t)/bracerightBig/bracketrightbigg<\nx′=x,(5)\n4/14J. Phys. Soc. Jpn. Full Paper\nwhere<denotes taking lesser component. Using Tr {σασβ}= 2δαβand calculating a lesser\ncomponent, we obtain\nj(Fig.2−left)\ncµ (x,t) =ie/planckover2pi12Jex\nm/parenleftbigg∂\n∂x−∂\n∂x′/parenrightbigg\nµǫγνη/summationdisplay\nX,R/summationdisplay\nω,ΩeiΩt∂UR\n∂RηSν\nX,Ω\n×/braceleftbigg\n[f(ω+Ω)−f(ω)]gr\nx−X,ωga\nX−R,ω+Ω/parenleftbigg∂\n∂Rγga\nR−x′,ω+Ω/parenrightbigg\n+f(ω)ga\nx−X,ωga\nX−R,ω+Ω/parenleftbigg∂\n∂Rγga\nR−x′,ω+Ω/parenrightbigg\n−f(ω+Ω)gr\nx−X,ωgr\nX−R,ω+Ω/parenleftbigg∂\n∂Rγgr\nR−x′,ω+Ω/parenrightbigg/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx′=x,(6)\nwheregr\nx,ω(= (ga\nx,ω)∗) =1\nV/summationtext\nkeik·xgr\nk,ω(gr\nk,ω= (ga\nk,ω)∗= (/planckover2pi1ω−εk+εF+i/planckover2pi1\n2τ)−1, where\nεk≡/planckover2pi12k2/2mandεFis Fermi energy), f(ω) is the Fermi distribution function which is given\nasf(ω)≡θ(−ω) at zero temperature ( θ(ω) is a step function), and SX,Ωis Fourier transform\nofSX(t). Assuming that local spins vary slowly (Ω ≪τ−1), we obtain\nj(Fig.2−left)\ncµ (x,t)≃e/planckover2pi12Jex\n2πm/parenleftbigg∂\n∂x′−∂\n∂x/parenrightbigg\nµǫγνη/summationdisplay\nX,R∂UR\n∂Rη˙Sν\nX(t)gr\nx−Xga\nX−R/parenleftbigg∂\n∂Rγga\nR−x′/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx′=x,\n(7)\nwheregx≡gx,ω=0. We neglected terms containing only gr’s orga’s since they are higher order\nof/planckover2pi1/εFτ≪1 as compared with mixed terms. Other contributions are simi larly calculated\nand the whole contribution in Fig. 2 is obtained as\nj(Fig.2)\ncµ(x,t) =eJex\nπǫµνη∂Ux\n∂xηRe/summationdisplay\nX˙Sν\nX(t)gr\nx−Xga\nX−x\n+e/planckover2pi12Jex\nπm∂\n∂xµǫγνηRe/summationdisplay\nX,R∂UR\n∂Rη˙Sν\nX(t)gr\nx−X/parenleftbigg∂\n∂Rγga\nX−R/parenrightbigg\nga\nR−x\n−2e/planckover2pi12Jex\nπmǫγνηRe/summationdisplay\nX,R∂UR\n∂Rη˙Sν\nX(t)gr\nx−X/parenleftbigg∂\n∂Rγga\nX−R/parenrightbigg/parenleftbigg∂\n∂xµga\nR−x/parenrightbigg\n.(8)\nThe ﬁrst term corresponds to the right diagram in Fig. 2, i.e. , correction of the current vertex\ndue to the spin-orbit interaction. In momentum space, eq. (8 ) reads\nj(Fig.2)\ncµ(x,t) =−i2e/planckover2pi12Jex\nπmVǫγνη/summationdisplay\nq,pe−i(q+p)·xpηUp˙Sν\nq(t)\n×Re/summationdisplay\nk/bracketleftbigg/parenleftBig\nk−q\n2/parenrightBig\nµkγgr\nk−qga\nk/parenleftbig\nga\nk+p−ga\nk/parenrightbig\n+pµ\n2kγgr\nk−qga\nkga\nk+p/bracketrightbigg\n,(9)\nwhereSq(t)≡1\nV/summationtext\nXeiq·XSX(t) andUp≡1\nV/summationtext\nReip·RUR.\n5/14J. Phys. Soc. Jpn. Full Paper\n(b) (a)\njc\nFig. 3. (Color online). Charge current at the second order in Jex. (a) and (b) are in the presence and\nthe absence of spin-orbit interaction, respectively.\n3.2 Second order in Jex\nContribution from the second order in Jexshown in Fig. 3(a) is calculated similarly by\nusing Tr{σασβσγ}= i2ǫαβγas\nj(Fig.3a)\ncµ(x,t) =2eJ2\nex\nπǫµνη∂Ux\n∂xηIm/summationdisplay\nX1,X2/bracketleftBig\nSX1(t)×˙SX2(t)/bracketrightBigν\ngr\nx−X1gr\nX1−X2ga\nX2−x\n+e/planckover2pi12J2\nex\nπmǫγνη∂\n∂xµIm/summationdisplay\nX1,X2,R∂UR\n∂Rη/bracketleftBig\nSX1(t)×˙SX2(t)/bracketrightBigν\n\n+gr\nx−R/parenleftbig∂\n∂Rγgr\nR−X1/parenrightbig\ngr\nX1−X2ga\nX2−x\n+gr\nx−X1gr\nX1−X2ga\nX2−R/parenleftbig∂\n∂Rγga\nR−x/parenrightbig\n+gr\nx−X2ga\nX2−R/parenleftbig∂\n∂Rγga\nR−X1/parenrightbig\nga\nX1−x\n−2e/planckover2pi12J2\nex\nπmǫγνηIm/summationdisplay\nX1,X2,R∂UR\n∂Rη/bracketleftBig\nSX1(t)×˙SX2(t)/bracketrightBigν\n\n+/parenleftBig\n∂\n∂xµgr\nx−R/parenrightBig/parenleftbig∂\n∂Rγgr\nR−X1/parenrightbig\ngr\nX1−X2ga\nX2−x\n+/parenleftBig\n∂\n∂xµgr\nx−X1/parenrightBig\ngr\nX1−X2ga\nX2−R/parenleftbig∂\n∂Rγga\nR−x/parenrightbig\n+/parenleftBig\n∂\n∂xµgr\nx−X2/parenrightBig\nga\nX2−R/parenleftbig∂\n∂Rγga\nR−X1/parenrightbig\nga\nX1−x.\n(10)\nIn momentum space, this is written as\nj(Fig.3a)\ncµ(x,t) =−i2e/planckover2pi12J2\nex\nπmVǫγνη/summationdisplay\nq,Q,pe−i(Q+p)·xpηUp/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\n×Im/summationdisplay\nk\n\n+/parenleftBig\nk+Q+p\n2/parenrightBig\nµ(k+q)γgr\nkga\nk+q/parenleftBig\nga\nk+q+pga\nk+Q+p+ga\nk+qga\nk+Q/parenrightBig\n−/parenleftBig\nk+Q+p\n2/parenrightBig\nµkγ/parenleftBig\ngr\nk−p−gr\nk/parenrightBig\ngr\nkga\nk+qga\nk+Q\n−/parenleftBig\nk+Q+p\n2/parenrightBig\nµ(k+Q)γgr\nkga\nk+qga\nk+Q/parenleftBig\nga\nk+Q+p−ga\nk+Q/parenrightBig\n+pµkγgr\nk−pgr\nkga\nk+qga\nk+Q.(11)\nThe charge current arises even without the spin-orbit inter action at the second order in\n6/14J. Phys. Soc. Jpn. Full Paper\nJex. This contribution is shown in Fig. 3(b) and given as\nj(Fig.3b)\ncµ(x,t) =e/planckover2pi12J2\nex\nπm∂\n∂xµIm/summationdisplay\nX1,X2/bracketleftBig\nSX1(t)·˙SX2(t)/bracketrightBig\ngr\nx−X1gr\nX1−X2ga\nX2−x\n−2e/planckover2pi12J2\nex\nπmIm/summationdisplay\nX1,X2/bracketleftBig\nSX1(t)·˙SX2(t)/bracketrightBig\ngr\nx−X1gr\nX1−X2/parenleftbigg∂\n∂xµga\nX2−x/parenrightbigg\n.(12)\nThis current is simply due to chemical potential shift by the exchange interaction. It contains\na scalar product, S·˙S, in contrast with the contribution of Fig. 3(a), and so is sma ll if local\nspins are spatially slowly varying. In momentum space, eq. ( 12) is written as\nj(Fig.3b)\ncµ(x,t) =−i2e/planckover2pi12J2\nex\nπmV/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)·˙Sq(t)/bracketrightBig\nIm/summationdisplay\nk/parenleftbigg\nk+Q\n2/parenrightbigg\nµgr\nkga\nk+qga\nk+Q.(13)\nThe total charge current is given as jc=j(Fig.2)\nc+j(Fig.3a)\nc+j(Fig.3b)\nc. This current contains\na component which originates from a conservative scalar pot ential (or electric ﬁeld) created\nby the spin dynamics and the spin-orbit interaction, such as the second terms of eqs. (8) and\n(10) and the ﬁrst term of eq. (12). This scalar potential (or e lectric ﬁeld) is ﬁctitious, which\nonly acts if the charge current has spin degrees of freedom. T he existence of this conservative\npart is in contrast to the adiabatic case in ref. 16.\nThe above results are quite similar to those in two-dimensio ns.18The diﬀerence is that\naveraging over long-range diﬀusion motion in two-dimension s is replaced in three-dimensions\nby a short-ranged average within the scale of electron mean f ree path.\n4. Spin Current\nWe consider the spin current arising from exchange interact ion. We neglect here spin-orbit\ninteraction because we are interested in conversion of spin current into charge current by spin-\norbit interaction. Spin current density is deﬁned as jν\nsµ(x,t)≡/planckover2pi1\n2∝an}bracketle{tψ†\nx(t){ˆvµ,σν}\n2ψx(t)∝an}bracketri}ht({A,B}\nrepresents the anticommutator AB+BAand ˆvµ=−i/planckover2pi1\nm∂\n∂xµ), i.e.,\njν\nsµ(x,t) =/planckover2pi13\n4m/parenleftbigg∂\n∂x′−∂\n∂x/parenrightbigg\nµTr/braceleftbig\nσνG<(x,t;x′,t)/bracerightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx′=x. (14)\nIn contrast to the charge continuity equation (eq. (3)), thi s spin current satisﬁes the equation,\n∂ρν\ns(x,t)\n∂t+∇·jν\ns(x,t) =Tν\ns(x,t), (15)\nwhereρν\ns(x,t)(≡/planckover2pi1\n2∝an}bracketle{tψ†\nx(t)σνψx(t)∝an}bracketri}ht)isthespindensityand Tν\ns(x,t)(≡i\n2∝an}bracketle{tψ†\nx(t)[Hex,σν]ψx(t)∝an}bracketri}ht)\nis the torque density (or spin source) due to exchange intera ction. Spin is thus not conserved.\n(Deﬁnition of spin current has ambiguity when spin-orbit in teraction is taken account,24,25\nbut the continuity equation, eq. (15), is always satisﬁed wi thTsredeﬁned properly.)\nDiagrammatic representations of the spin current at the ﬁrs t and the second order in Jex\n7/14J. Phys. Soc. Jpn. Full Paper\n(a) (b)\njs\nFig. 4. (Color online). Diagrams of the spin current. (a) and (b) den ote contributions from the ﬁrst\nand the second order in Jex, respectively.\nare shown in Fig. 4. These contributions are given by\njν(Fig.4a)\nsµ(x,t) =/planckover2pi13Jex\n2πmIm/summationdisplay\nX˙Sν\nX(t)gr\nx−X/parenleftbigg∂\n∂xµga\nX−x/parenrightbigg\n, (16)\njν(Fig.4b)\nsµ(x,t) =/planckover2pi13J2\nex\n2πm∂\n∂xµRe/summationdisplay\nX1,X2/bracketleftBig\nSX1(t)×˙SX2(t)/bracketrightBigν\ngr\nx−X1gr\nX1−X2ga\nX2−x\n−/planckover2pi13J2\nex\nπmRe/summationdisplay\nX1,X2/bracketleftBig\nSX1(t)×˙SX2(t)/bracketrightBigν\ngr\nx−X1gr\nX1−X2/parenleftbigg∂\n∂xµga\nX2−x/parenrightbigg\n.(17)\nThey are written in momentum space as\njν(Fig.4a)\nsµ(x,t) =−i/planckover2pi13Jex\n2πmV/summationdisplay\nqe−iq·x˙Sν\nq(t)Im/summationdisplay\nkkµgr\nk−q\n2ga\nk+q\n2, (18)\njν(Fig.4b)\nsµ(x,t) =−i/planckover2pi13J2\nex\nπmV/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\nRe/summationdisplay\nk/parenleftbigg\nk+Q\n2/parenrightbigg\nµgr\nkga\nk+qga\nk+Q.\n(19)\nThetotal spincurrentcontains termslike ∝an}bracketle{t˙SX1∝an}bracketri}htand∝an}bracketle{tSX1×˙SX2∝an}bracketri}ht,where∝an}bracketle{t···∝an}bracketri}htdenotesaverage\nover position X1andX2within the length scale of electron mean free path. Comparin g these\nexpressions with those of charge currents (eqs. (8) and (10) ), we see that they are related,\nsuggesting the inverse spin Hall eﬀect. In fact, the ﬁrst term in eq. (8) (eq. (10)) arises from\n∝an}bracketle{t˙S∝an}bracketri}ht(∝an}bracketle{tS×˙S∝an}bracketri}ht), similarly to the spin current of eq. (16) (eq. (17)). These charge currents have\nextra factor of ∇Uxwhich justiﬁes the idea of inverse spin Hall eﬀect, conversio n of spin\ncurrent into charge current by spin-orbit ﬁeld, Eso≡−∇Ux. However, other (second and\nthird) terms of eqs. (8) and (10) do not have direct correspon dence to spin current.\n5. Smooth Spin Structures\nLet us ﬁrst consider a case Upis spatially smooth, i.e., p≪ℓ−1, whereℓis the electron\nmean free path. We obtain the charge currents as\nj(Fig.2)\ncµ(x,t) =ie/planckover2pi12Jex\n2πmVǫγνη∂2Ux\n∂xη∂xλ/summationdisplay\nqe−iq·x˙Sν\nq(t)Re/summationdisplay\nk/parenleftBig\nδµλkγ−δµγkλ/parenrightBig\ngr\nk−q(ga\nk)2+O(p3),\n(20)\n8/14J. Phys. Soc. Jpn. Full Paper\nj(Fig.3a)\ncµ(x,t) =4e/planckover2pi12J2\nex\nπmVǫγνη∂Ux\n∂xη/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\n×Im/summationdisplay\nk/parenleftbigg\nk+Q\n2/parenrightbigg\nµ(k+q)γgr\nk(ga\nk+q)2ga\nk+Q\n+ie/planckover2pi14J2\nex\nπm2Vǫγνη∂2Ux\n∂xη∂xλ/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\n×Im/summationdisplay\nk\n\n−/parenleftBig\nk+Q\n2/parenrightBig\nµ/bracketleftBig\nkγ(k+q)λ+(k+q)γkλ/bracketrightBig\n(gr\nk)2(ga\nk+q)2ga\nk+Q\n−/parenleftBig\nk+Q\n2/parenrightBig\nµ/bracketleftbig\nkγQλ−kλQγ/bracketrightbig\n(gr\nk)2ga\nk+q(ga\nk+Q)2\n+/parenleftBig\nk+Q\n2/parenrightBig\nµ/bracketleftBig\n(k+q)γ(k+Q)λ+(k+q)γ(k+Q)λ/bracketrightBig\ngr\nk(ga\nk+q)2(ga\nk+Q)2\n+O(p3).(21)\nWe also assume local spins vary slowly in space, and explore b ehavior of the currents in detail.\nCarrying out qandQexpansion assuming as q,Q≪ℓ−1, the currents are given as\njcµ(x,t)∼ie/planckover2pi12Jex\n3πmVǫγνη∂2Ux\n∂xη∂xλ/summationdisplay\nq/parenleftBig\nδµγqλ−δµλqγ/parenrightBig\ne−iq·x˙Sν\nq(t)Re/summationdisplay\nkεk(gr\nk)2(ga\nk)2\n+8eJ2\nex\n3πVǫµνη∂Ux\n∂xη/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\nIm/summationdisplay\nkεkgr\nk(ga\nk)3\n+i2e/planckover2pi12J2\nex\n3πmVǫγνη∂2Ux\n∂xη∂xλ/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\nImAµγλ\nq,Q\n−i2e/planckover2pi12J2\nex\nπmV/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)·˙Sq(t)/bracketrightBig\nImBµ\nq,Q,(22)\njν\nsµ(x,t)∼−i/planckover2pi13Jex\n3πmV/summationdisplay\nqqµe−iq·x˙Sν\nq(t)Im/summationdisplay\nkεkgr\nk(ga\nk)2\n−i/planckover2pi13J2\nex\nπmV/summationdisplay\nq,Qe−iQ·x/bracketleftBig\nSQ−q(t)×˙Sq(t)/bracketrightBigν\nReBµ\nq,Q,(23)\nwhere\nAµγλ\nq,Q≡/bracketleftBig\nδµγ(q+Q)λ+δµλ(q+Q)γ/bracketrightBig/summationdisplay\nkεkgr\nk(ga\nk)4\n−/bracketleftBig\nδµγ(q+Q)λ+δµλ(q−Q)γ/bracketrightBig/summationdisplay\nkεk(gr\nk)2(ga\nk)3\n+8\n3/bracketleftBig\nδµγ(q+Q)λ+δµλ(q+Q)γ/bracketrightBig/summationdisplay\nkε2\nkgr\nk(ga\nk)5\n−4\n3/bracketleftBig\nδµγ(2q+Q)λ+δµλ(2q+Q)γ/bracketrightBig/summationdisplay\nkε2\nk(gr\nk)2(ga\nk)4,\n9/14J. Phys. Soc. Jpn. Full Paper\nBµ\nq,Q≡/summationdisplay\nk/bracketleftbiggQµ\n2gr\nk(ga\nk)2+2(q+Q)µ\n3εkgr\nk(ga\nk)3/bracketrightbigg\n.\nHere we used∝an}bracketle{tkαkβ∝an}bracketri}ht=δαβk2/3 where the average is over direction. Sums over kare carried\nout as\n/summationdisplay\nkεk(gr\nk)2(ga\nk)2∼4πNeεFτ3\n/planckover2pi13,\n/summationdisplay\nkεkgr\nk(ga\nk)3∼−i3πNeτ2\n2/planckover2pi12−2πNeεFτ3\n/planckover2pi13,\n/summationdisplay\nkεkgr\nk(ga\nk)4∼−i2πNeεFτ4\n/planckover2pi14,\n/summationdisplay\nkεk(gr\nk)2(ga\nk)3∼i6πNeεFτ4\n/planckover2pi14,\n/summationdisplay\nkε2\nkgr\nk(ga\nk)5∼i5πNeεFτ4\n2/planckover2pi14,\n/summationdisplay\nkε2\nk(gr\nk)2(ga\nk)4∼−i5πNeεFτ4\n/planckover2pi14,\n/summationdisplay\nkgr\nk(ga\nk)2∼i2πNeτ2\n/planckover2pi12,\n/summationdisplay\nkεkgr\nk(ga\nk)2∼−3πNeτ\n2/planckover2pi1+i2πNeεFτ2\n/planckover2pi12,\nwhereNeis the density of states given as mVkF/2π2/planckover2pi12. We ﬁnally obtain\njcµ(x,t) =4eNeJexεFτ3\n3/planckover2pi1mV/bracketleftBigg\nǫγνη∂2Ux\n∂xµ∂xη∂˙Sν\nx(t)\n∂xγ−ǫµνη∂2Ux\n∂xη∂xλ∂˙Sν\nx(t)\n∂xλ/bracketrightBigg\n−4eNeJ2\nexτ2\n/planckover2pi12Vǫµνη∂Ux\n∂xη/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n−8eNeJ2\nexεFτ4\n9/planckover2pi12mV\n\n+ǫµνη∂2Ux\n∂xη∂xλ/braceleftBig\n4∂\n∂xλ/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n+9/bracketleftBig\nSx(t)×∂˙Sx(t)\n∂xλ/bracketrightBigν/bracerightBig\n+ǫγνη∂2Ux\n∂xµ∂xη/braceleftBig\n13∂\n∂xγ/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n+9/bracketleftBig\nSx(t)×∂˙Sx(t)\n∂xγ/bracketrightBigν/bracerightBig\n−2eNeJ2\nexτ2\nmV/bracketleftBigg\nSx(t)·∂˙Sx(t)\n∂xµ/bracketrightBigg\n,(24)\njν\nsµ(x,t) =2/planckover2pi1NeJexεFτ2\n3mV∂˙Sν\nx(t)\n∂xµ\n−4NeJ2\nexεFτ3\n3mV/braceleftBigg\n∂\n∂xµ/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n+/bracketleftBigg\nSx(t)×∂˙Sx(t)\n∂xµ/bracketrightBiggν/bracerightBigg\n.(25)\nWe see in eq. (24) that charge current driven by uniform spin- orbit ﬁeld ( Eso=−∇Ux) is\nperpendicular to spin damping, S×˙S, while discontinuity of Esoinduces new ˙Scomponents.\n10/14J. Phys. Soc. Jpn. Full Paper\nIn a case that local spins are spatially uniform, Sx(t) =S(t), the spin current totally\nvanishes and the charge current only arises as\njuniform\ncµ(x,t) =−4eNeJ2\nexτ2\n/planckover2pi12Vǫµνη∂Ux\n∂xη/bracketleftBig\nS(t)×˙S(t)/bracketrightBigν\n. (26)\n5.1 Rashba system\nWe apply above result to the case of Rashba spin-orbit intera ction. Rashba spin-orbit\ninteraction can be realized in various three-dimensional s ystems without inversion symme-\ntry.26,27Assuming uniform spin-orbit ﬁeld in z-direction,−∇Ux=Esoez, we obtain the\ncharge current as\njRashba\ncµ(x,t) =4eNeEsoJ2\nexτ2\n/planckover2pi12Vǫµνz/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n−2eNeJ2\nexτ2\nmV/bracketleftBigg\nSx(t)·∂˙Sx(t)\n∂xµ/bracketrightBigg\n.(27)\nItisinterestingtonotethatinthecaseofinhomogeneoussp in-orbitﬁeld,anadditional current\narises as\njadd\ncµ(x,t) =4eNeJexεFτ3\n3/planckover2pi1mV/bracketleftBigg\nǫµνz∂Eso\n∂xλ∂˙Sν\nx(t)\n∂xλ−ǫγνz∂Eso\n∂xµ∂˙Sν\nx(t)\n∂xγ/bracketrightBigg\n+8eNeJ2\nexεFτ4\n9/planckover2pi12mV\n\n+ǫµνz∂Eso\n∂xλ/braceleftBig\n4∂\n∂xλ/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n+9/bracketleftBig\nSx(t)×∂˙Sx(t)\n∂xλ/bracketrightBigν/bracerightBig\n+ǫγνz∂Eso\n∂xµ/braceleftBig\n13∂\n∂xγ/bracketleftBig\nSx(t)×˙Sx(t)/bracketrightBigν\n+9/bracketleftBig\nSx(t)×∂˙Sx(t)\n∂xγ/bracketrightBigν/bracerightBig.(28)\nThis additional current appears, for instance, at the inter face between Rashba region and\nleads when normal leads are attached.\n6. Equilibrium Components\nIn the above discussion on pumped currents, terms like ( gr)nand (ga)nwere negligibly\nsmall. These contributions in fact correspond to the equili brium contributions which exist\neven if local spins have no dynamics, Sx(t) =Sx. The currents are then given by\nj(eq)\ncµ(x) = 4eJexǫµνη∂Ux\n∂xηIm/summationdisplay\nXSν\nX/summationdisplay\nωf(ω)ga\nx−X,ωga\nX−x,ω\n+4e/planckover2pi12Jex\nmǫγνη∂\n∂xµIm/summationdisplay\nX,R∂UR\n∂RηSν\nX/summationdisplay\nωf(ω)ga\nx−X,ω/parenleftbigg∂\n∂Rγga\nX−R,ω/parenrightbigg\nga\nR−x,ω\n−8e/planckover2pi12Jex\nmǫγνηIm/summationdisplay\nX,R∂UR\n∂RηSν\nX/summationdisplay\nωf(ω)ga\nx−X,ω/parenleftbigg∂\n∂Rγga\nX−R,ω/parenrightbigg/parenleftbigg∂\n∂xµga\nR−x,ω/parenrightbigg\n,(29)\njν(eq)\nsµ(x) =/planckover2pi13J2\nex\nm∂\n∂xµIm/summationdisplay\nX1,X2[SX1×SX2]ν/summationdisplay\nωf(ω)ga\nx−X1,ωga\nX1−X2,ωga\nX2−x,ω\n−2/planckover2pi13J2\nex\nmIm/summationdisplay\nX1,X2[SX1×SX2]ν/summationdisplay\nωf(ω)ga\nx−X1,ωga\nX1−X2,ω/parenleftbigg∂\n∂xµga\nX2−x,ω/parenrightbigg\n.(30)\n11/14J. Phys. Soc. Jpn. Full Paper\nTheequilibrium charge current thus does not contain ∝an}bracketle{tSX1×SX2∝an}bracketri}htterm (j(eq)\nc∝∝an}bracketle{tSX∝an}bracketri}ht), while\nspin current does not have ∝an}bracketle{tSX∝an}bracketri}htterm (j(eq)\ns∝∝an}bracketle{tSX1×SX2∝an}bracketri}ht). In momentum space they are\nwritten as\nj(eq)\ncµ(x) =−i8e/planckover2pi12Jex\nmVǫγνη/summationdisplay\nq,pe−i(q+p)·xpηUpSν\nq/summationdisplay\nωf(ω)\n×Im/summationdisplay\nk/bracketleftbigg/parenleftBig\nk−q\n2/parenrightBig\nµkγga\nk−q,ωga\nk,ω(ga\nk+p,ω−ga\nk,ω)+pµ\n2kγga\nk−q,ωga\nk,ωga\nk+p,ω/bracketrightbigg\n,(31)\njν(eq)\nsµ(x) =−i2/planckover2pi13J2\nex\nmV/summationdisplay\nq,Qe−iQ·x[SQ−q×Sq]ν/summationdisplay\nωf(ω)Im/summationdisplay\nk/parenleftbigg\nk+Q\n2/parenrightbigg\nµga\nk,ωga\nk+q,ωga\nk+Q,ω.\n(32)\nThe equilibrium currents calculated above correspond to an initial response to local spins\nsuddenlyattached tothesystem.Inreality, as soonas theeq uilibriumchargeandspincurrents\nappear, the spin conﬁguration starts to evolve and dynamica l currents appear.\nIn a spatially smooth local spins and spin-orbit interactio n, the equilibrium components\nof the charge current contain at least three derivatives, ( ∇Ux)(∇2Sx) or (∇2Ux)(∇Sx) or\n(∇3Ux)Sx, and are negligibly small. The equilibrium spin current, in contrast, is large and\ngiven as\njν(eq)\nsµ(x)∼−/planckover2pi12NeJ2\nex\n12mVεF/bracketleftbigg\nSx×∂Sx\n∂xµ/bracketrightbiggν\n. (33)\nThis result indicates that the equilibrium component of spi n current cannot be converted into\ncharge one. This is physically reasonable since equilibriu m spin current can be dissipationless\nwhile charge current always has dissipation. Dynamical spi ns are thus essential in pumping of\ncharge current.\n7. Conclusion\nWe have discussed theoretically that charge and spin curren ts are induced by spin dynam-\nics in the presence of spin-orbit interaction. The spin curr ent pumped from dynamical spins\nwas found to contain two contributions, ∝an}bracketle{t˙S∝an}bracketri}htand∝an}bracketle{tS×˙S∝an}bracketri}ht, consistent with phenomenological\nargument.7,19We found that pumped charge current has a component ﬂowing pe rpendicu-\nlar to both spin-orbit ﬁeld ( Eso) and average spin polarization ∝an}bracketle{tS×˙S∝an}bracketri}ht. This contribution\nis thus explained as due to conversion of spin current ( ∝∝an}bracketle{tS×˙S∝an}bracketri}ht) into charge current, i.e.,\nthe inverse spin Hall eﬀect. For a pumping mechanism by use of l ocal spins, dynamical spins\nare essential because the static spin current is not convert ed into charge current. The charge\ncurrent has another component originating from a ﬁctitious conservative ﬁeld which only acts\nif the charge current has spin degrees of freedom. This curre nt is not associated directly with\nspin current. We also found a novel charge current proportio nal to∂Eso, arising from inho-\nmogeneity of spin-orbit ﬁeld. This contribution would be es sential in actual experiments on\nﬁnite size spin-orbit system attached to leads.\n12/14J. Phys. Soc. Jpn. Full Paper\nAcknowledgments\nWe are grateful to J.-I. Ohe and E. Saitoh for helpful discuss ions and comments.\n13/14J. Phys. Soc. Jpn. Full Paper\nReferences\n1) J.E. Hirsch: Phys. Rev. Lett. 83(1999) 1834.\n2) S. Zhang: Phys. Rev. Lett. 85(2000) 393.\n3) S. Murakami, N. Nagaosa, and S.C. Zhang: Science 301(2003) 1348.\n4) J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth, and A.H. M acDonald: Phys. Rev. Lett.\n92(2004) 126603.\n5) Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awschalom: Science 306(2004) 1910.\n6) J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth: Phy. R ev. Lett. 94(2005) 047204.\n7) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara: Appl. Phys. Lett .88(2006) 182509.\n8) S.O. Valenzuela and M. 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Mod. Phys. 77(2005) 1375.\n21) J. Inoue, G.E.W. Bauer, and L.W. Molenkamp: Phys. Rev. B 70(2004) 041303(R).\n22) H.HaugandA.-P.Jauho: Quantum Kinetics in Transport and Optics of Semiconductors (Springer-\nVerlag, 1998).\n23) J. Rammer and H. Smith: Rev. Mod. Phys. 58(1986) 323.\n24) Q.-F. Sun and X.C. Xie: Phys. Rev. B 72(2005) 245305.\n25) J. Shi, P. Zhang, D. Xiao, and Q. Niu: Phys. Rev. Lett. 96(2006) 076604.\n26) S. Fujimoto: J. Phys. Soc. Jpn. 76(2007) 051008.\n27) E.Bauer, H.Kaldarar, A.Prokoﬁev, E.Royanian, A.Amato, J .Sereni, W.Br¨ amer-Escamilla,and\nI. Bonalde: J. Phys. Soc. Jpn. 76(2007) 051009.\n14/14"    },    {        "title": "0707.2529v1.Decoherence_of_coupled_electron_spins_via_nuclear_spin_dynamics_in_quantum_dots.pdf",        "content": "arXiv:0707.2529v1  [cond-mat.mes-hall]  17 Jul 2007Decoherence of coupled electron spins via nuclear spin dyna mics in quantum dots\nW. Yang and R. B. Liu∗\nDepartment of Physics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China\nIn double quantum dots, the exchange interaction between tw o electron spins renormalizes the\nexcitation energy of pair-ﬂips in the nuclear spin bath, whi ch in turn modiﬁes the non-Markovian\nbath dynamics. As the energy renormalization varies with th e Overhauser ﬁeld mismatch between\nthe quantumdots, the electron singlet-triplet decoherenc e resulting from the bath dynamics depends\non sampling of nuclear spin states from an ensemble, leading to the transition from exponential\ndecoherence in single-sample dynamics to power-law decay u nder ensemble averaging. In contrast,\nthe decoherence of a single electron spin in one dot is essent ially the same for diﬀerent choices of\nthe nuclear spin conﬁguration.\nPACS numbers: 03.65.Yz, 03.67.Pp, 71.70.Gm, 71.70.Jp\nDecoherence draws a boundary between the micro-\nscopic quantum world and the macroscopic classical\nworld. It is also a main obstacle in quantum technolo-\ngies such as quantum computation. Thus, both for un-\nderstanding crossover from the quantum to the classical\nworld [1, 2, 3], and for exploiting quantum coherence of\nlarge systems [4], it is desirable to comprehend how de-\ncoherence develops with scaling up the size of a system.\nThe very initial step toward such a purpose is to examine\nthe diﬀerence between a two-level system (the simplest\nquantum object, called a qubit in quantum computation)\nand two coupled ones. For a system in a Markovian bath\n(which has broad-band ﬂuctuation), the decoherence is\ndescribed by the Lindblad formalism. For a system in a\nnon-Markovian bath, there are indications of nontrivial\nscaling behavior of decoherence [5, 6, 7, 8], such as the\nnon-additive decoherence in multiple baths [5]. In this\npaper, we study the decoherenceof a composite quantum\nobject in a non-Markovian mesoscopic bath, based on a\nparadigmatic system in mesoscopic physics and quantum\ninformationscience [9, 10, 11], namely, twoelectron spins\nin double quantum dots (QDs) where the nuclear spins\nserve as the bath.\nThe dynamics of a mesoscopic nuclear spin bath in\na QD is conditioned on the state of the electron spin in\ncontact with the bath [12, 13, 14]. The conditional evolu-\ntion of the nuclear spins establishes the electron-nuclear\nentanglement that causes the electron spin decoherence.\nWhen the electron spin is disturbed, the nuclear bath\ndynamics is altered. For example, the nuclear spin evo-\nlution can be shepherded by a structured sequence of\nelectron spin ﬂips so that the electron is disentangled\nfrom the bath and as a result the lost coherence is recov-\nered [13]. Besides external control, the disturbance may\nalso be due to interaction with another quantum object\nin proximity, such as the exchange interaction JexˆS1·ˆS2\nbetween two electron spins in coupled QDs. Intuitively\nspeaking,thedisturbanceduetotheexchangeinteraction\nmay be viewed as precessing of the electron spins about\neach other. For a more rigorous treatment, we should\nﬁrst diagonalize the electron spin Hamiltonian includingthe exchange interaction, and then study the bath dy-\nnamics and the decoherence in the electron eigenstate\nbasis. To demonstrate the most essential physics, we\nconsider only the decoherence between the singlet state\n|S∝angbracketrightand the unpolarized triplet state |T0∝angbracketright(by assuming\nthat the polarized triplet states |T±∝angbracketrightare well split oﬀ by\na large external magnetic ﬁeld). The singlet-triplet (S-\nT) dephasing due to the Overhauser ﬁeld distribution in\nan ensemble or a superposition state of nuclear spins has\nbeen studied in Refs. [15, 16], but the bath considered\nthere has no dynamical ﬂuctuation since the interaction\nbetween nuclear spins was neglected.\nTheroleofthe exchangeinteractioninthe decoherence\nmay be disclosed in a semiclassical spectral diﬀusion pic-\nture [17]. Let us denote the local Overhauser ﬁelds for\nthe nuclear spin conﬁguration |I∝angbracketrightby the electron Zee-\nman energies ΩI\n1and ΩI\n2in dot 1 and dot 2, respectively.\nThe pairwise nuclear spin ﬂip-ﬂops cause the dynamical\nﬂuctuation of the Overhauser ﬁelds and therefore a ran-\ndom phase of the electron spins. For a single electron\nspin in one QD, the Zeeman energy change due to the\nkth pair-ﬂip is 2 Zk. With exchange interaction, the S-T\nsplitting is ES-T=/radicalbig\nJ2ex+∆2\nI, varying with the Over-\nhauser ﬁeld mismatch ∆ I≡ΩI\n1−ΩI\n2[see Fig. 1(a)], So,\nthe S-T splitting change due to the kth pair-ﬂip is\n2Zk≈(2Zk)∂ES-T\n∂∆I≈2Zk∆I\nES-T. (1)\nThe exchange interaction modiﬁes the energy ﬂuctuation\n(or spectral diﬀusion) responsible for the decoherence.\nFor a full quantum mechanical description of the deco-\nherence [12, 13, 14, 18, 19], we focus on the nuclear spin\ndynamics which is driven by pair-ﬂips (as elementary ex-\ncitations in the bath). The pair-ﬂips are characterized\nby the ﬂip transition strength and the energy cost. For\nuncorrelated electrons in the double dots ( Jex= 0), the\nhyperﬁne-energy cost of a nuclear pair-ﬂip is ±Zkfor\nelectron spin up and down states, respectively. In pres-\nence of exchange interaction, the energy cost for the pair\nﬂip is renormalized to be ±Zkfor the triplet and singlet\nstates, respectively. Thus the bath dynamics itself is al-2\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s48/s45/s49/s50/s49/s48/s45/s57/s49/s48/s45/s54/s49/s48/s45/s51/s49/s48/s48/s49/s48/s51\n/s49/s48/s45/s55\n/s49/s48/s45/s53\n/s49/s48/s45/s51\n/s49/s48/s45/s49/s49/s48/s49/s49/s48/s50/s49/s48/s51\n/s45/s108/s110/s91/s124\n/s73/s40τ/s41/s124/s93/s32\nτ/s32/s40µ /s115/s41         ∆\n/s73/s32/s40µ /s101/s86/s41\n/s32/s48/s46/s48/s49\n/s32/s48/s46/s48/s52\n/s32/s48/s46/s49/s48\n/s32/s48/s46/s52/s48/s40/s98/s41\n∝τ/s52/s45\n/s107/s107\n/s124/s83〉/s124/s84\n/s48〉\n∆\n/s73/s32/s74\n/s101/s120/s32\n/s48/s90\n/s107/s73\n/s50/s32/s40µ /s115/s41\n∆\n/s73/s32/s40µ /s101/s86/s41/s40/s97/s41\n/s40/s99/s41\n∝ /s40∆\n/s73/s41/s45 /s49/s47/s50\nFIG. 1: (Color online) (a) The singlet and triplet energies\nas functions of the Overhauser ﬁeld mismatch ∆ I, and the\n(renormalized) excitation energy of a nuclear spin pair-ﬂi pZk\n(Zk). (b) FID of the S-T coherence for various initial nuclear\nspin states, indicated by ∆ I. (c) FID decoherence time TI\n2as\na function of ∆ I.\ntered when non-interacting quantum objects in the bath\nare replaced by interacting ones. In particular, through\nthedependenceofthebathﬂuctuationontheOverhauser\nﬁeld mismatch ∆ I, the dynamics of nuclear spins in one\ndot is aﬀected by the state of the nuclear spins in the\nother dot. Therefore, the resultant S-T decoherence time\nvaries with sampling of the nuclear spin conﬁguration |I∝angbracketright\nfrom an ensemble ˆ ρN=/summationtext\nIPI|I∝angbracketright∝angbracketleftI|. This, as will be\nshown later, leads to a transition from exponential deco-\nherence to power-law decay upon ensemble average. On\nthe contrary, the decoherence of a single electron spin in\na QD is essentially independent of the static Overhauser\nﬁeld [12, 14].\nWe consider a gate-deﬁned double-dot structure sim-\nilar to those used in Ref. [10, 11, 15, 16]. We assume\na large on-site Coulomb energy and gate voltages sup-\nporting one electron in each dot. The virtual inter-dot\ntunneling mediates the exchange interaction. Under a\nlarge magnetic ﬁeld (such that the electron Zeeman en-\nergy Ω e≫Jex), the unpolarized states |S∝angbracketrightand|T0∝angbracketrightare\nwell separated in energy from the polarized triplet states\n|T±∝angbracketright. The electrons and nuclei are coupled by the con-\ntact hyperﬁne interaction ˆS1·ˆh1+ˆS2·ˆh2, whereˆhj=/summationtext\nnaj,nˆIj,n, withaj,nbeing the hyperﬁne coeﬃcient for\nthenth nuclear spin at the jth dot. The oﬀ-diagonal hy-\nper��ne interaction (terms with ˆSx\nj,ˆSy\nj) couples |S∝angbracketrightand\n|T0∝angbracketrightto|T±∝angbracketright, which, however, has negligible eﬀect under\na strong external magnetic ﬁeld (Ω e≫aj,n) [20]. The\nOverhauser ﬁeld mismatch ˆ∆ =ˆhz\n1−ˆhz\n2couples|T0∝angbracketrightand\n|S∝angbracketright, which causes longitudinal T1relaxation. For a rela-\ntively large exchange splitting (e.g., Jex/greater≫rsimilar10∆I), theT1\nprocessis suppressed[15, 16], but virtual S-T ﬂips inducea self-energy correction\nˆHS-T≡/parenleftbigg\nES-T+∆I\nES-TˆδI/parenrightbigg\n⊗|T0∝angbracketright∝angbracketleftT0|−|S∝angbracketright∝angbracketleftS|\n2,(2)\nwhere the mean-ﬁeld part (∆ I≡ ∝angbracketleftI|ˆ∆|I∝angbracketright) is singled\nout from the quantum ﬂuctuation ( ˆδI≡ˆ∆−∆I) of\nthe Overhauser ﬁeld, and is included non-perturbatively.\nThis self-energy correction is responsible for the renor-\nmalized energy cost ( Zk) of a nuclear spin pair-ﬂip\ndriven by the intrinsic nuclear-nuclear spin interaction\nˆHN. Now the Hamiltonian of the electron-nuclear spin\nsystem is ˆH=ˆH+⊗ |T0∝angbracketright∝angbracketleftT0|+ˆH−⊗ |S∝angbracketright∝angbracketleftS|,where\nˆH±=ˆHN±ES-T/2±ˆHZwithˆHZ=ˆδI∆I/(2ES-T).\nSuch a block-diagonal Hamiltonian induces no T1relax-\nation but only pure S-T dephasing. We emphasize that\nthe intrinsic nuclear spin interaction ˆHNis essential to\nthe decoherence. Otherwise, without ˆHN, the pure de-\nphasing caused by the hyperﬁne interaction is totally\neliminated from spin echo signals [12, 13, 14, 21].\nThe S-T decoherence is caused by the electron-nuclear\nentanglement, established during the evolution of the nu-\nclear spin state predicated on the electron states: Sup-\npose the electrons are initially in a superposition state\nα|S∝angbracketright+β|T0∝angbracketrightand the nuclear state |I∝angbracketrightis one randomly\nchosenfrom athermalensemble. Theconditionalnuclear\nspin evolution |I∝angbracketright → |I±(τ)∝angbracketright ≡e−iˆH±τ|I∝angbracketrightestablishes\nan entangled state α|S∝angbracketright⊗|I−∝angbracketright+β|T0∝angbracketright⊗|I+∝angbracketright. The S-T\ncoherence is given by LI(τ) =∝angbracketleftI−(τ)|I+(τ)∝angbracketright. To cal-\nculate the nuclear spin evolution |I±(τ)∝angbracketright, we employ the\npair-correlationapproximation in which all possible pair-\nﬂips from the initial conﬁguration |I∝angbracketrightare taken as inde-\npendent of each other. The pair-correlation approxima-\ntion is justiﬁed for a mesoscopic nuclear spin bath with a\nsuﬃciently random conﬁguration where within the deco-\nherence timescale the occurred pair-ﬂips are much fewer\nthan the available pairs to be ﬂipped and therefore have\nlittle probability to be in neighborhood of each other or\nto be correlated [12, 13, 14, 18, 19, 22, 23]. A pair-ﬂip\nis characterized by a transition strength due to the oﬀ-\ndiagonal nuclear spin interaction Bk=∝angbracketleftI|ˆHN|I,k∝angbracketright, an\nenergy cost due to the diagonal nuclear spin interaction\nDk=∝angbracketleftI,k|ˆHN|I,k∝angbracketright − ∝angbracketleftI|ˆHN|I∝angbracketright, and a hyperﬁne en-\nergy cost ±Zk=±/parenleftBig\n∝angbracketleftI,k|ˆHZ|I,k∝angbracketright−∝angbracketleftI|ˆHZ|I∝angbracketright/parenrightBig\nfor the\ntriplet and the singlet state, respectively, where |I,k∝angbracketrightde-\nnotes the nuclear spin state after the kth pair-ﬂip. An\nindependent pair-ﬂip is mapped to be a pseudo-spin ˆsk\nprecessing about a pseudo-ﬁeld χ±\nk≡(2Bk,0,Dk±Zk),\ninitially pointing to the down direction [12, 13, 14]. The\nnuclear Hamiltonian in the pseudo-spin representation is\nˆH±≈ ±ES-T/2+/summationtext\nkχ±\nk·ˆsk.\nWithin the pair-correlationapproximation,the S-Tco-\nherence in free-induction decay (FID) is\nLI(τ) =e−iES-Tτ/productdisplay\nk/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n↓/vextendsingle/vextendsingle/vextendsinglee+iχ−\nk·ˆskτe−iχ+\nk·ˆskτ/vextendsingle/vextendsingle/vextendsingle↓/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle,(3)3\nThe short-time behaviour (for τ≪Z−1\nk) is a\nquartic exponential decay, LI(τ)≈exp(−iES-Tτ−/summationtext\nkB2\nkZ2\nkτ4/2)≡e−iES-Tτ−(τ/TI\n2)4. The decoherence\ntimeTI\n2∝(Zk)−1/2∝(∆I/ES-T)−1/2, varying with\nsampling of the nuclear spin conﬁguration from the en-\nsemble/summationtext\nIPI|I∝angbracketright∝angbracketleftI|. For comparison, the decoherence\nof a single electron spin in a QD [24], except for a trivial\nglobal phase factor related to the inhomogeneous broad-\nening, is essentially independent of choice of the initial\nstate [12, 13, 14], since the excitation energy of a nuclear\npair-ﬂip there Zk=/angbracketleftBig\nI,k/vextendsingle/vextendsingle/vextendsingleˆhz\nj/2/vextendsingle/vextendsingle/vextendsingleI,k/angbracketrightBig\n−/angbracketleftBig\nI/vextendsingle/vextendsingle/vextendsingleˆhz\nj/2/vextendsingle/vextendsingle/vextendsingleI/angbracketrightBig\nis\nindependent of |I∝angbracketright. Also, the S-T decoherence time is\nlonger than the single spin decoherence time by a factor\nof/radicalbig\nES-T/∆Ibecause of the reduction of the excitation\nenergy.\nIn numerical evaluation, we take a symmetric GaAs\ndouble-dot structure with height 6 nm, Fock-Darwin ra-\ndius 70 nm for a parabolic conﬁnement potential, and\ncenter-to-center separation 137 nm, under a perpendic-\nular magnetic ﬁeld B= 1 T at temperature T= 1 K.\nThe calculated varianceofthe localOverhauserﬁeld mis-\nmatch is Γ ≈0.12µeV, corresponding to an inhomo-\ngeneous dephasing time T∗\n2=√\n2/Γ≈8 ns. The ex-\nchange energy Jex≈ −1µeV is determined with the\nHund-Mulliken method [25], consistent with experimen-\ntal values [10].\nFigure 1 (b) shows the FID decoherence for several\nnuclear spin initial states |I∝angbracketrightrandomly chosen from the\nthermal ensemble. The suppression of the decoherence\nwith decreasing the Overhauser ﬁeld mismatch (∆ I) is\nevident. The dependence of the decoherence time on the\nOverhauser ﬁeld mismatch TI\n2∝(∆I/ES-T)−1/2is ver-\niﬁed in Fig. 1 (c). Note that for vanishing mismatch\n∆I→0, the energy cost Zkvanishes in the leading order\nofZkand the second-order correction Zk=Z2\nk/Jex, re-\nsulting in a saturation of the decoherence time at a large\nvalue. In ensemble dynamics, the signal is to be averaged\nbyL(τ) =/summationtext\nIPILI(τ). The inhomogeneous broaden-\ning of the Overhauser ﬁeld results in an eﬀective dephas-\ning/summationtext\nIPIe−iES-Tτwith a nanosecond timescale [15, 16],\nmuch faster than the entanglement-induced decoherence\nLI(τ) in single-sample dynamics. So below we study the\nensemble-averaged coherence in spin-echo conﬁgurations\nwheretheinhomogeneousbroadeningeﬀectiseliminated.\nIn the single-pulse Hahn echo ( τ-π-τ-echo), the S-T\ncoherence at the echo time 2 τfor a nuclear spin state |I∝angbracketright\nis\nL(1)\nI(2τ) =/productdisplay\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\n↓/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig\nˆU(1),−\nk/parenrightBig†ˆU(1),+\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle↓/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(4)\nwhereˆU(1),±\nk≡e−iχ∓\nk·ˆskτe−iχ±\nk·ˆskτ. The short-\ntime behavior (for τ≪Z−1\nk) isL(1)\nI(2τ)≈\nexp/parenleftbig\n−2/summationtext\nkB2\nkZ2\nkτ4/parenrightbig\n≡e−(2τ/TI\nH)4\nwith the single-\nsample decoherence time TI\nH=/parenleftbig/summationtext\nkB2\nkZ2\nk/8/parenrightbig−1/4=/s53 /s53/s48/s49/s48/s45/s54/s49/s48/s45/s51/s49/s48/s48/s49/s48/s51\n/s32/s45/s108/s110/s91/s40/s49/s41\n/s40/s50τ/s41/s93\n/s32/s45/s108/s110/s91/s76/s40/s49/s41\n/s40/s50τ/s41/s93∝τ4\n/s32/s32/s72/s97/s104/s110/s32/s101/s99/s104/s111/s32/s40/s50 τ/s41\nτ/s32/s40µ /s115/s41/s32/s48/s46/s48/s49\n/s32/s48/s46/s48/s52\n/s32/s48/s46/s49/s48\n/s32/s48/s46/s52/s48∆\n/s73/s40µ /s101/s86/s41\nFIG. 2: (Color online) Hahn echo signal ( −ln[L(1)\nI(2τ)]) for\nvarious nuclear spin conﬁguration |I/angbracketrightas indicated by ∆ I.\nThe solid black line shows the ensemble-averaged coherence\n(−ln[L(1)(2τ)]), compared to the echo signal ( −ln[L(1)(2τ)])\nof a single spin in a QD of the same size (solid gray line).\n√\n2TI\n2. The ensemble dynamics is studied by averaging\nover a large number of samples from a Gaussian distri-\nbution of the Overhauser ﬁeld. With the approximation\nES-T≈Jex, the ensemble-averaged result is analytically\nobtained for τ≪Z−1\nk,\nL(1)(2τ)≈/bracketleftBig\n1+(2τ/TH)4/bracketrightBig−1/2\n, (5)\nwith a power-law decay proﬁle, where the ensemble de-\ncoherence time TH=TI\nH/vextendsingle/vextendsingle\n∆I=√\n2Γ, i.e., the decoherence\ntime for anuclearspin conﬁgurationwith the Overhauser\nﬁeld mismatch equal to√\n2 times the standard variance.\nSuch a transition from an exponential decay in single-\nsample dynamics to a power-law decay in ensemble dy-\nnamics is shown in Fig. 2. In contrast, the echo signal of\na single electron spin in a QD is unchanged by ensemble\naveraging, i.e., L(1)=L(1)\nI.\nWe now study the S-T decoherence under concate-\nnated control which is designed to preserve the coher-\nence [13, 14, 26]. The coherence preserved after the\nmth order concatenated pulse sequence L(m)\nI(τm) is ob-\ntained by substituting ˆU(m),±\nkforˆU(1),±\nkin Eq. (4),\nwhereτm≡2mτ, andˆU(m),±\nkis recursively deﬁned as\nˆU(m−1),∓\nkˆU(m−1),±\nkform >1. The short-time proﬁle (for\nτ≪Z−1\nk) isL(m)\nI(τm)≈e−(τm/TI\n2,m)2m+2\n,with the deco-\nherence time TI\n2,m∼2m(m+3)\n2(m+1)[ZkBm\nk(∆I/ES-T)]−1\nm+1.\nAgain, the ensemble average leads to a power-law decay\nL(m)(τm) =/bracketleftbig\n1+(τm/T2,m)2m+2/bracketrightbig−1/2,(6)\nwithT2,m=TI\n2,m/vextendsingle/vextendsingle\n∆I=√\n2Γ. In contrast, for a single elec-\ntron spin, the ensemble-averaging has negligible eﬀect,4\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s49 /s49/s48 /s49/s48/s48 /s50/s48/s48/s49/s48/s45/s49/s50/s49/s48/s45/s56/s49/s48/s45/s52/s49/s48/s48/s49/s48/s52/s32/s32/s109/s61/s49\n/s32/s32/s109/s61/s50\n/s32/s32/s109/s61/s51\n/s32/s32/s109/s61/s52\n/s32/s32/s40/s109/s41\n/s40τ\n/s109/s41/s32/s38 /s32/s76/s40/s109/s41\n/s40τ\n/s109/s41/s32\nτ/s32/s40µ /s115/s41/s40/s97/s41/s45/s108/s110/s91/s40/s109/s41\n/s40τ\n/s109/s41/s93/s32/s38/s32/s45/s108/s110/s91 /s76/s40/s109/s41\n/s40τ\n/s109/s41/s93/s32\nτ/s32/s40µ /s115/s41/s32/s32/s109/s61/s49\n/s32/s32/s109/s61/s50\n/s32/s32/s109/s61/s51\n/s32/s32/s109/s61/s52/s40/s98/s41\nFIG. 3: (Color online) (a) Ensemble-averaged coherence un-\nder concatenated control, for the S-T decoherence in two cou -\npled dots ( L(m)(τm), as thick lines), and for the single spin\ndecoherence in one dot ( L(m)(τm), as thin lines), where m\nindicates the concatenation level. (b) Logarithmic plot of (a).\ni.e.,L(m)(τm) =L(m)\nI(τm)≈e−(τm/T2,m)2m+2,where the\ndecoherence time T2,mis shorter than the S-T decoher-\nence time T2,mby a factor ∼(Jex/Γ)1/(m+1). Fig. 3\ncompares the S-T decoherence to the single spin deco-\nherence, showing the suppression of the decoherence and\nthe crossover to a power-law decay due to the coupling\nbetween the electron spins.\nIn conclusion, the exchange interaction between two\nelectron spins in double QDs modiﬁes the nuclear spin\nbath dynamics through renormalizing the pair-ﬂip ex-\ncitation energy. As the renormalized excitation energy\nvaries with the Overhauser ﬁeld mismatch between the\ntwo dots, the nuclear spin dynamics in one dot becomes\ndependent on the nuclear spin state in the other dot,\nregardless of nonexistence of inter-dot nuclear-spin in-\nteraction in the considered situation. Consequently, the\nsinglet-tripletdecoherenceduetotheelectron-nuclearen-\ntanglement depends on choice of the nuclear spin conﬁg-\nuration from an ensemble, leading to a power-law decay\nof ensemble-averaged coherence, in contrast with the ex-\nponential decoherence of a single electron spin which is\ninsensitive to sampling of the nuclear spin ensemble. The\ndependence of the S-T decoherence on the Overhauser\nﬁeld mismatch may be observed by tuning the mismatch\nwith an inhomogeneous external ﬁeld. The exchange in-\nteraction alsoenhances the S-T decoherencetime by sup-\npressing the ﬂuctuation in the nuclear spin bath.\nThis work was supported by Hong Kong RGC Project\n2160285.\n∗Email: rbliu@phy.cuhk.edu.hk[1] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).\n[2] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and\nI.-O. Stamatescu, Decoherence and the appearance of a\nclassical world in quantum theory (Springer, New York,\n2003), 2nd ed.\n[3] M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2004).\n[4] D. Aharonov, A. Kitaev, and J. Preskill, Phys. Rev. Lett.\n96, 050504 (2006).\n[5] G.BurkardandF.Brito, Phys.Rev.B 72, 054528(2005).\n[6] J. Q. You, X. D. Hu, and F. Nori, Phys. Rev. B 72,\n144529 (2005).\n[7] H. G. Krojanski and D. Suter, Phys. Rev. Lett. 97,\n150503 (2006).\n[8] X. D. Hu and S. Das Sarma, Phys. Rev. Lett. 96, 100501\n(2006).\n[9] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n[10] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n[11] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,\nK. C. Nowack, T. Meunier, L. P. Kouwenhoven, and\nL. M. K. Vandersypen, Nature 442, 766 (2006).\n[12] W. Yao, R. B. Liu, and L. J. Sham, Phys. Rev. B 74,\n195301 (2006).\n[13] W. Yao, R. B. Liu, and L. J. Sham, Phys. Rev. Lett. 98,\n077602 (2007).\n[14] R. B. Liu, W. Yao, and L. J. Sham, New. J. Phys. 9, 226\n(2007).\n[15] W. A. Coish and D. Loss, Phys. Rev. B 72, 125337\n(2005).\n[16] J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby,\nC. M. Marcus, and M. D. Lukin, Cond-mat/0602470\n(2006).\n[17] R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 115322\n(2003).\n[18] W. M. Witzel, R. de Sousa, and S. Das Sarma, Phys.\nRev. B72, 161306(R) (2005).\n[19] W. M. Witzel and S. Das Sarma, Phys. Rev. Lett. 98,\n077601 (2007).\n[20] In principle, the virtual ﬂip-ﬂops ( |S/angbracketrightor|T0/angbracketright ↔ |T±/angbracketright)\nvia the oﬀ-diagonal hyperﬁne interaction mediate the in-\ndirect interaction between nuclear spins (being or not in\nthe same dot), with strength ∼aj,naj′,n′Jex/Ω2\ne, which,\nhowever, is negligible as Ω e≫Jex,aj,n.\n[21] N. Shenvi, R. de Sousa, and K. B. Whaley, Phys. Rev. B\n71, 224411 (2005).\n[22] S. K. Saikin, W. Yao, and L. J. Sham, Phys. Rev. B 75,\n125314 (2007).\n[23] W.M.Witzel andS.DasSarma, Phys.Rev.B 74, 035322\n(2006).\n[24] For two uncorrelated electrons in double QDs ( Jex= 0),\ntheT2pure dephasing of the two independent electron\nspins is a T1longitudinal relaxation in the S-T basis.\n[25] G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev.\nB59, 2070 (1999).\n[26] K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95,\n180501 (2005)."    },    {        "title": "2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf",        "content": "Dynamical Majorana Ising spin response in a topological superconductor-magnet\nhybrid by microwave irradiation\nYuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan.\n3Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: March 20, 2024)\nWe study a dynamical spin response of surface Majorana modes in a topological superconductor-\nmagnet hybrid under microwave irradiation. We find a method to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and\nthe microwave frequency. This reflects the topological nature of the Majorana modes, enhancing\nthe Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising\nspins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative\nspin devices.\nIntroduction.— The quest for Majoranas within matter\nstands as one of the principal challenges in the study of\ncondensed matter physics, more so in the field of quan-\ntum many-body systems [1]. The self-conjugate nature\nof Majoranas leads to peculiar electrical characteristics\nthat have been the subject of intensive research, both\ntheoretical and experimental [2]. In contrast, the focus of\nthis paper lies on the magnetic properties of Majoranas,\nspecifically the Majorana Ising spin [3–8]. A distinctive\ncharacteristic of Majorana modes, appearing as a surface\nstate in topological superconductors (TSC), is its exceed-\ningly strong anisotropy, which makes it behave as an Ising\nspin. In particular, this paper proposes a method to ex-\nplore the dynamical response of the Majorana Ising spin\nthrough the exchange interaction at the magnetic inter-\nface, achieved by coupling the TSC to a ferromagnet with\nferromagnetic resonance (FMR) (as shown in Fig.1 (a)).\nFMR modulation in a magnetic hybrid system has at-\ntracted much attention as a method to analyze spin ex-\ncitations in thin-film materials attached to magnetic ma-\nterials [9, 10]. Irradiating a magnetic material with mi-\ncrowaves induces dynamics of localized spin in magnetic\nmaterials, which can excite spins in adjacent thin-film\nmaterials via the magnetic proximity effect. This setup\nis called spin pumping, and has been studied intensively\nin the field of spintronics as a method of injecting spins\nthrough interfaces [11, 12]. Recent studies have theoret-\nically proposed that spin excitation can be characterized\nby FMR in hybrid systems of superconducting thin films\nand magnetic materials [13–18]. Therefore, it is expected\nto be possible to analyze the dynamics of surface Majo-\nrana Ising spins using FMR in hybrid systems.\nIn this work, we consider a TSC-ferromagnetic insula-\ntor (FI) hybrid system as shown in Fig. 1 (a). The FMR\nis induced by microwave irradiation on the FI. At the\ninterface between the TSC and the FI, the surface Ma-\n(b)\n(c)(a)\nFI~~\n~~Microwave\nϑS\nY, yX\nxZhdcHex\nTSC\n(d)\nhdchdc+δhα+δα\nHz\nFIG. 1. (a) The TSC-FI hybrid schematic reveals how,\nunder resonance frequency microwave irradiation, localized\nspins commence precessional motion, consequently initiating\nthe dynamical Majorana Ising spin response at the TSC inter-\nface. (b) In the TSC context, the liaison between a spin-up\nelectron and a spin-down hole with the surrounding sea of\nspin-triplet Cooper pairs drastically modulate their proper-\nties; notably, a spin-down hole can engage with a spin-triplet\nCooper pair, thereby inheriting a negative charge. (c) No-\ntably, spin-triplet Cooper pairs amass around holes and scat-\nter around electrons, thereby eroding the rigid distinction be-\ntween the two. (d) The interplay between the Majorana mode\nand the localized spin manipulates the FMR spectrum, trig-\ngering a frequency shift and linewidth broadening.\njorana modes interact with the localized spins in the FI.\nAs a result, the localized spin dynamics leads to the dy-\nnamical Majorana Ising spin response (DMISR), which\nmeans the Majorana Ising spin density is dynamically in-\nduced, and it is possible to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjust-\ning the external magnetic field angle and the microwave\nfrequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2  [cond-mat.mes-hall]  19 Mar 20242\nspin dynamics due to the interface interaction leads to a\nfrequency shift and a linewidth broadening, which reflect\nthe properties of the Majorana Ising spin dynamics. This\nwork proposes a setup for detecting Majorana modes and\npaves the way for the development of quantum comput-\ning and spin devices using Majoranas.\nModel.— We introduce a model Hamiltonian Hconsist-\ning of three terms\nH=HM+HFI+Hex. (1)\nThe first, second, and third terms respectively describe\nthe surface Majorana modes on the TSC surface, the bulk\nFI, and the proximity-induced exchange coupling. Our\nfocus is on energy regions significantly smaller than the\nbulk superconducting gap. This focus allows the spin ex-\ncitation in the TSC to be well described using the surface\nMajorana modes. The subsequent paragraphs provide\ndetailed explanations of each of these three terms.\nThe first terms HMdescribes the surface Majorana\nmodes,\nHM=1\n2Z\ndrψT(r)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r),(2)\nwhere r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant\nvelocity, and σ= (σx, σy, σz) are the Pauli matrices.\nThe two component Majorana field operator is given by\nψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization\naxis along the xaxis. The Majorana field operators sat-\nisfy the Majorana condition ψσ(r) =ψ†\nσ(r) and the an-\nticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′)\nwhere σ, σ′=→,←. We can derive HMby using surface-\nlocalized solutions of the BdG equation based on the bulk\nTSC Hamiltonian. The details of the derivation of HM\nare provided in the Supplemental Material [19].\nA notable feature of the surface Majorana modes is\nthat the spin density is Ising like, which we call the Majo-\nrana Ising spin [3–8]. The feature follows naturally from\nthe Majorana condition and the anticommutation rela-\ntion. The Majorana Ising spin density operator is given\nbys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r))\n(See the Supplemental Material for details [19]). The\nanisotropy of the Majorana Ising spin is the hallmark of\nthe surface Majorana modes on the TSC surface.\nThe second term HFIdescries the bulk FI and is given\nby the ferromagnetic Heisenberg model,\nHFI=− JX\n⟨n,m⟩Sn·Sm−ℏγhdcX\nnSZ\nn, (3)\nwhere J>0 is the exchange coupling constant, Snis the\nlocalized spin at site n,⟨n, m⟩means summation for near-\nest neighbors, γis the electron gyromagnetic ratio, and\nhdcis the static external magnetic field. We consider the\nspin dynamics of the localized spin under microwave irra-\ndiation, applying the spin-wave approximation. This al-\nlows the spin excitation to be described by a free bosonic\noperator, known as a magnon [20].The third term Hexrepresents the proximity exchange\ncoupling at the interface between the TSC and the FI,\nHex=−Z\ndrX\nnJ(r,rn)s(r)·Sn=HZ+HT,(4)\nHZ=−cosϑZ\ndrX\nnJ(r,rn)sz(r)SZ\nn, (5)\nHT=−sinϑZ\ndrX\nnJ(r,rn)sz(r)SX\nn, (6)\nwhere the angle ϑis shown in Fig. 1 (a). HZis the\ncoupling along the precession axis and HTis the coupling\nperpendicular to the precession axis. In our setup, HZ\nleads to gap opening of the energy spectrum of the surface\nMajorana modes and HTgives the DMISR under the\nmicrowave irradiation.\nDynamical Majorana Ising spin response.— We con-\nsider the microwave irradiation on the FI. The coupling\nbetween the localized spins and the microwave is given\nby\nV(t) =−ℏγhacX\nn\u0000\nSX\nncosωt−SY\nnsinωt\u0001\n,(7)\nwhere hacis the microwave amplitude, and ωis the mi-\ncrowave frequency. The microwave irradiation leads to\nthe precessional motion of the localized spin. When the\nfrequency of the precessional motion and the microwave\ncoincide, the FMR occurs. The FMR leads to the DMISR\nvia the exchange interaction. The DMISR is character-\nized by the dynamic spin susceptibility of the Majorana\nmodes, ˜ χzz(q, ω), defined as\n˜χzz(q, ω) :=Z\ndre−iq·rZ\ndtei(ω+i0)tχzz(r, t),(8)\nwhere χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩\nwith the interface area L2and the spin den-\nsity operator in the interaction picture, sz(r, t) =\nei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou-\npling, we consider configuration average and assume\n⟨P\nnJ(r,rn)⟩ave=J1, which means that HZis treated\nas a uniform Zeeman like interaction and the interface\nis specular [21]. Using eigenstates of Eq. (2) and after a\nstraightforward calculation, the uniform spin susceptibil-\nity is given by\n˜χzz(0, ω)\n=−X\nk,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ)\n2Ek,λ+ℏω+i0,\n→ −Z\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω+i0, (9)\nwhere |k, λ⟩is an eigenstate of HMwith eigenenergy\nEk,λ=λp\n(ℏvk)2+M2, (λ=±).M=J1Scosϑis\nthe Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3\ndistribution function, and D(E) is the density of states\ngiven by\nD(E) =L2\n2π(ℏv)2|E|θ(|E| − |M|), (10)\nwith the Heaviside step function θ(x). It is important to\nnote that the behavior of the uniform spin susceptibil-\nity is determined by the interband contribution, which is\nproportional to the Fermi distribution function, i.e., the\ncontribution of the occupied states. This mechanism is\nsimilar to the Van Vleck paramagnetism [22]. The con-\ntribution of the occupied states often plays a crucial role\nin topological responses [23].\nReplacing the localized spin operators with their statis-\ntical average values, we find the induced Majorana Ising\nspin density, to the first order of J1S, is given by\nZ\ndr⟨sz(r, t)⟩= ˜χzz\n0(0,0)J1Scosϑ\n+ Re[˜ χzz\n0(0, ω)]hac\nαhdcJ1Ssinϑsinωt, (11)\nwhere ˜ χzz\n0(0,0) is the spin susceptibility for M= 0. The\nfirst term originates from HZand gives a static spin den-\nsity, while the second term originates from HTand gives\na dynamic spin density. Figure 2 shows the induced Ising\nspin density as a function of time at several angles. As\nshown in Eq. (11), the Ising spin density consists of the\nstatic and dynamic components. The dynamic compo-\nnent is induced by the precessional motion of the local-\nized spin, which means one can induce the DMISR using\nthe dynamics of the localized spin.\nThe inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of\nϑat a fixed frequency. When the frequency ℏωis smaller\nthan the Majorana gap, Im˜ χzz(0, ω) is zero. Once the\nfrequency overcomes the Majorana gap, Im˜ χzz(0, ω) be-\ncomes finite. The implications of these behaviors are that\nif the magnon energy is smaller than the Majorana gap,\nthere is no energy dissipation due to the DMISR. How-\never, once the magnon energy exceeds the Majorana gap,\nfinite energy dissipation associated with the DMISR oc-\ncurs at the surface of the TSC. Therefore, one can toggle\nbetween dissipative and non-dissipative Majorana Ising\nspin dynamics by adjusting the precession axis angle and\nthe microwave frequency.\nFMR modulation.— The retarded component of the\nmagnon Green’s function is given by GR(rn, t) =\n−(i/ℏ)θ(t)⟨[S+\nn(t), S−\n0(0)]⟩with the interaction picture\nS±\nn(t) =eiHFIt/ℏS±\nne−iHFIt/ℏ. The FMR signal is char-\nacterized by the spectral function defined as\nA(q, ω) :=−1\nπIm\"X\nne−iq·rnZ\ndtei(ω+i0)tGR(rn, t)#\n.\n(12)\nSSImχzz(0, ω) ˜⟨s   z⟩\n2\n1ωtϑ\nFInon-dissipativenon-dissipativedissipativedissipativeTSC\nFITSC000.00.51.0\nπ/4\nπ/2\n0 π/4 π/20\nϑ2π\nπFIG. 2. The induced Ising spin density, with a unit\n˜χzz\n0(0,0)J1S, is presented as a function of ωtandϑ. The\nfrequency and temperature are set to ℏω/J1S= 1.5 and\nkBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is\nset to 0 .3. The static Majorana Ising spin density arises\nfrom HZ. When the precession axis deviates from the di-\nrection perpendicular to the interface, the precessional mo-\ntion of the localized spins results in the dynamical Majorana\nIsing spin response (DMISR). Energy dissipation due to the\nDMISR is zero for small angles ϑas the Majorana gap ex-\nceeds the magnon energy. However, once the magnon energy\novercomes the Majorana gap, the energy dissipation becomes\nfinite. Therefore, one can toggle between dissipative and non-\ndissipative DMISR by adjusting ϑ.\nFor uniform external force, the spectral function is given\nby\nA(0, ω) =2S\nℏ1\nπ(α+δα)ω\n[ω−γ(hdc+δh)]2+ [(α+δα)ω]2.\n(13)\nThe peak position and width of the FMR signal is given\nbyhdc+δhandα+δα, respectively. hdcandαcorre-\nspond to the peak position and the linewidth of the FMR\nsignal of the FI alone. δhandδαare the FMR modu-\nlations due to the exchange interaction HT. We treat\nHM+HFI+HZas an unperturbed Hamiltonian and HT\nas a perturbation. In this work, we assume the specular\ninterface, where the coupling J(r,rn) is approximated\nasDP\nn,n′J(r,rn)J(r′,rn′)E\nave=J2\n1. The dynamics\nof the localized spins in the FI is modulated due to the\ninteraction between the localized spins and the Majo-\nrana Ising spins. In our setup, the peak position and the\nlinewidth of the FMR signal are modulated and the FMR4\nmodulation is given by\nδh= sin2ϑSJ2\n1\n2NγℏRe˜χzz(0, ω), (14)\nδα= sin2ϑSJ2\n1\n2NℏωIm˜χzz(0, ω), (15)\nwhere Nis the total number of sites in the FI. These for-\nmulas were derived in the study of the FMR in magnetic\nmultilayer systems including superconductors. One can\nextract the spin property of the Majorana mode from the\ndata on δhandδα. Because of the Ising spin anisotropy,\nthe FMR modulation exhibits strong anisotropy, where\nthe FMR modulation is proportional to sin2ϑ.\nFigure 3 shows the FMR modulations (a) δαand (b)\nδh. The FMR modulation at a fixed frequency increases\nwith angle ϑand reaches a maximum at π/2, as can be\nread from Eqs. (14) and (15). When the angle ϑis fixed\nand the frequency ωis increased, δαbecomes finite above\na certain frequency at which the energy of the magnon\ncoincides with the Majorana gap. When ϑ < π/ 2 and\nℏω≈2M,δαlinearly increases as a function of ωjust\nabove the Majorana gap. The localized spin damping is\nenhanced when the magnon energy exceeds the Majorana\ngap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes\nandδαis proportional to ω/T. In the high frequency\nregion ℏω/J 1S≫1,δαconverges to its upper threshold.\nThe frequency shift δhis almost independent of ωand\nhas a finite value even in the Majorana gap. This behav-\nior is analogous to the interband contribution to the spin\nsusceptibility in strongly spin-orbit coupled band insula-\ntors, and is due to the fact that the effective Hamiltonian\nof the Majorana modes includes spin operators. It is im-\nportant to emphasize that although the Majorana modes\nhave spin degrees of freedom, only the zcomponent of the\nspin density operator is well defined. This is a hallmark\nof Majorana modes, which differs significantly from elec-\ntrons in ordinary solids. Note that δhis proportional to\nthe energy cutoff, which is introduced to converge energy\nintegral for Re˜ χzz(0, ω). The energy cutoff corresponds\nto the bulk superconducting gap, which is estimated as\n∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap-\nplicable in the frequency region below ℏω∼0.1[meV]\n(∼30[GHz]). In addition, we assume that Majorana gap\nis estimated to be J1S∼0.01[meV] ( ∼0.1[K]).\nDiscussion.— Comparing the present results with spin\npumping (SP) in a conventional metal-ferromagnet hy-\nbrid, the qualitative behaviors are quite different. In con-\nventional metals, spin accumulation occurs due to FMR.\nIn contrast, in the present system, no corresponding spin\naccumulation occurs due to the Ising anisotropy. Also, in\nthe present calculations, the proximity-induced exchange\ncoupling is assumed to be an isotropic Heisenberg-like\ncoupling. However, in general, the interface interaction\ncan also be anisotropic. Even in such a case, it is no qual-\nitative change in the case of ordinary metals, although a\n0.00.5\n(a) (b)\nϑℏω/J1S 0\nπ/4\nπ/2024\nϑℏω/J1S 0\nπ/4\nπ/2024δ  α δ  h10\n0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a)\nThe damping modulation δαonly becomes finite when the\nmagnon energy exceeds the Majorana gap; otherwise, it van-\nishes. This behavior corresponds to the energy dissipation of\nthe Majorana Ising spin. (b) The peak shift is finite, except\nforϑ= 0, and is almost independent of ω. This behavior\nresembles the spin response observed in strongly spin-orbit\ncoupled band insulators, where the interband contribution to\nspin susceptibility results in a finite spin response, even within\nthe energy gap.\ncorrection term due to anisotropy is added [24]. There-\nfore, the Ising anisotropy discussed in the present work\nis a property unique to the Majorana modes and can\ncharacterize the Majorana excitations.\nLet us comment on the universal nature of the toggling\nbetween non-dissipative and dissipative dynamical spin\nresponses observed in our study. Indeed, such toggling\nbecomes universally feasible when the microwave fre-\nquency and the energy gap are comparable, and when the\nHamiltonian and spin operators are non-commutative,\nindicating that spin is not a conserved quantity. The\nnon-commutativity can be attributed to the presence of\nspin-orbit couplings [25–27], and spin-triplet pair corre-\nlations [28].\nMicrowave irradiation leads to heating within the FI,\nso that thermally excited magnons due to the heating\ncould influence the DMISR. Phenomena resulting from\nthe heating, which can affect interface spin dynamics, in-\nclude the spin Seebeck effect (SSE) [29], where a spin\ncurrent is generated at the interface due to a tempera-\nture difference. In hybrid systems of normal metal and\nFI, methods to separate the inverse spin Hall voltage due\nto SP from other signals caused by heating have been\nwell studied [30]. Especially, it has been theoretically\nproposed that SP and SSE signals can be separated us-\ning a spin current noise measurement [24]. Moreover, SP\ncoherently excites specific modes, which qualitatively dif-\nfers from SSE induced by thermally excited magnons [14].\nTherefore, even if heating occurs in the FI in our setup,\nthe properties of Majorana Ising spins are expected to\nbe captured. Details of the heating effect on the DMISR\nwill be examined in the near future.\nWe also mention the experimental feasibility of our the-\noretical proposals. As we have already explained, the\nFMR modulation is a very sensitive spin probe. Indeed,\nthe FMR modulation by surface states of 3D topological5\ninsulators [31] and graphene [32–36] has been reported\nexperimentally. Therefore, we expect that the enhanced\nGilbert damping due to Majorana Ising spin can be ob-\nservable in our setup when the thickness of the ferromag-\nnetic insulator is sufficiently thin.\nFinally, it is pertinent to mention the potential candi-\ndate materials where surface Majorana Ising spins could\nbe detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39],\nSrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p-\nwave superconducting state and theoretically can host\nsurface Majorana Ising spins. Recent NMR measure-\nments indicate that UTe 2could be a bulk p-wave su-\nperconductor in the Balian-Werthamer state [41], which\nhosts the surface Majorana Ising spins with the per-\npendicular Ising anisotropy, as considered in this work.\nAxBi2Se3(A= Cu, Sr, Nb) is considered to possess in-\nplane Ising anisotropy [8], differing from the perpendic-\nular Ising anisotropy explored in this work. Therefore,\nwe expect that it exhibits anisotropy different from that\ndemonstrated in this work.\nConclusion.— We present herein a study of the spin\ndynamics in a topological superconductor (TSC)-magnet\nhybrid. Ferromagnetic resonance under microwave irra-\ndiation leads to the dynamically induced Majorana Ising\nspin density on the TSC surface. One can toggle between\ndissipative and non-dissipative Majorana Ising spin dy-\nnamics by adjusting the external magnetic field angle and\nthe microwave frequency. Therefore, our setup provides\na platform to detect and control Majorana excitations.\nWe expect that our results provide insights toward the\ndevelopment of future quantum computing and spintron-\nics devices using Majorana excitations.\nAcknowledgments.— The authors are grateful to R.\nShindou for valuable discussions. This work is partially\nsupported by the Priority Program of Chinese Academy\nof Sciences, Grant No. XDB28000000. We acknowl-\nedge JSPS KAKENHI for Grants (Nos. JP20K03835,\nJP21H01800, JP21H04565, and JP23H01839).\nSUPPLEMENTAL MATERIAL\nSurface Majorana modes\nIn this section, we describe the procedure for deriv-\ning the effective Hamiltonian of the surface Majorana\nmodes. We start with the bulk Hamiltonian of a three-\ndimensional topological superconductor. Based on the\nbulk Hamiltonian, we solve the BdG equation to demon-\nstrate the existence of a surface-localized solution. Us-\ning this solution, we expand the field operator and show\nthat it satisfies the Majorana condition when the bulk\nexcitations are neglected. As a result, on energy scales\nmuch smaller than the bulk superconducting gap, the\nlow-energy excitations are described by surface-localized\nMajorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three-\ndimensional coordinates and r∥for two-dimensional ones\nin the Supplemental Material.\nWe start with the mean-field Hamiltonian given by\nHSC=1\n2Z\ndrΨ†\nBdG(r)HBdGΨBdG(r), (16)\nwithr= (x, y, z ). We consider the Balian-Werthamer\n(BW) state, in which the pair potential is given by\n∆ˆk=∆\nkF\u0010\nˆk·σ\u0011\niσywith the bulk superconducting gap\n∆. Here, we do not discuss the microscopic origin of the\npair correlation leading to the BW state. As a result, the\nBdG Hamiltonian HBdGis given by\nHBdG=\nεˆk−EF 0 −∆\nkFˆk−∆\nkFˆkx\n0 εˆk−EF∆\nkFˆkx∆\nkFˆk+\n−∆\nkFˆk+∆\nkFˆkx−εˆk+EF 0\n∆\nkFˆkx∆\nkFˆk− 0 −εˆk+EF\n,\n(17)\nwith ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2\n2m. The four\ncomponent Nambu spinor ΨBdG(r) is given by\nΨBdG(r) :=\nΨ→(r)\nΨ←(r)\nΨ†\n→(r)\nΨ†\n←(r)\n, (18)\nwith the spin quantization axis along the xaxis. The\nmatrices of the spin operators are represented as\nσx=\u00121 0\n0−1\u0013\n, (19)\nσy=\u0012\n0 1\n1 0\u0013\n, (20)\nσz=\u00120−i\ni0\u0013\n. (21)\nThe fermion field operators satisfy the anticommutation\nrelations\n{Ψσ(r),Ψσ′(r′)}= 0, (22)\n{Ψσ(r),Ψ†\nσ′(r′)}=δσσ′δ(r−r′), (23)\nwith the spin indices σ, σ′=→,←.\nTo diagonalize the BdG Hamiltonian, we solve the BdG\nequation given by\nHBdGΦ(r) =EΦ(r). (24)\nWe assume that a solution is written as\nΦ(r) =eik∥·r∥f(z)\nu→\nu←\nv→\nv←\n, (25)6\nwithk∥= (kx, ky) and r∥= (x, y). If we set the four\ncomponents vector to satisfy the following equation (Ma-\njorana condition)\n\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n\nu→\nu←\nv→\nv←\n=±\nu→\nu←\nv→\nv←\n, (26)\nwe can obtain a surface-localized solution. If we take a\npositive (negative) sign, we obtain a solution localized\non the top surface (bottom surface). As we will consider\nsolutions localized on the bottom surface below, we take\na negative sign. Finally, we obtain the normalized eigen-\nvectors of the BdG equation given by\nΦλ,k∥(r) =eik∥·r∥\n√\nL2fk∥(z)uλ,k∥, (27)\nwith\nfk∥(z) =Nk∥sin(k⊥z)e−κz, (28)\nNk∥=s\n4κ(k2\n⊥+κ2)\nk2\n⊥, (29)\nκ=m∆\nℏ2kF, (30)\nk⊥=q\nk2\nF−k2\n∥−κ2, (31)\nand\nu+,k∥=\nu+,→k∥\nu+,←k∥\nv+,→k∥\nv+,←k∥\n=1√\n2\nsinϕk∥+π/2\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\n,(32)\nu−,k∥=\nu−,→k∥\nu−,←k∥\nv−,→k∥\nv−,←k∥\n=1√\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\nsinϕk∥+π/2\n2\n.(33)\nThe eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can\nshow that the eigenvectors satisfy\nu−,−k∥=u+,k∥. (34)\nConsequently, the field operator is expanded as\nΨBdG(r) =X\nk∥\u0012\nγk∥eik∥·r∥\n√\nL2+γ†\nk∥e−ik∥·r∥\n√\nL2\u0013\n×fk∥(z)u+,k∥+ (bulk modes) ,(35)\nwhere γk∥(γ†\nk∥) is the quasiparticle creation (annihila-\ntion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk\nmodes and performing the integration in the z-direction,\nwe obtain the effective Hamiltonian for the surface states\nHM=1\n2Z\ndr∥ψT(r∥)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r∥),(36)\nwhere v= ∆/ℏkFand we introduced the two component\nMajorana field operator\nψ(r∥) =\u0012ψ→(r∥)\nψ←(r∥)\u0013\n, (37)\nsatisfying the Majorana condition\nψσ(r∥) =ψ†\nσ(r∥), (38)\nand the anticommutation relation\nn\nψσ(r∥), ψσ′(r′\n∥)o\n=δσσ′δ(r∥−r′\n∥). (39)\nThe spin density operator of the Majorana mode is\ngiven by\ns(r∥) =ψ†(r∥)σ\n2ψ(r∥). (40)\nThexcomponent is given by\nsx(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00121/2 0\n0−1/2\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ→(r∥)−ψ†\n←(r∥)ψ←(r∥)\u0003\n=1\n2\u0002\nψ2\n→(r∥)−ψ2\n←(r∥)\u0003\n= 0. (41)\nIn a similar manner, the yandzcomponents are given\nby\nsy(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120 1/2\n1/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ←(r∥) +ψ†\n←(r∥)ψ→(r∥)\u0003\n=1\n2\b\nψ→(r∥), ψ←(r∥)\t\n= 0, (42)\nand\nsz(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120−i/2\ni/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=−i\n2\u0000\nψ†\n→(r∥)ψ←(r∥)−ψ†\n←(r∥)ψ→(r∥)\u0001\n=−iψ→(r∥)ψ←(r∥), (43)\nrespectively. As a result, the spin density operator is\ngiven by\ns(r∥) =\u0000\n0,0,−iψ→(r∥)ψ←(r∥)\u0001\n. (44)\nOne can see that the spin density of the Majorana mode\nis Ising like.7\nMajorana Ising spin dynamics\nIn this section, we calculate the Ising spin density in-\nduced on the TSC surface by the proximity coupling Hex.\nHexconsists of two terms, HZandHT.HZleads to the\nstatic spin density and HTleads to the dynamic spin\ndensity. First, we calculate the static spin density. Next,\nwe calculate the dynamic spin density.\nThe total spin density operator is given by\nsz\ntot=Z\ndr∥sz(r∥). (45)\nThe statistical average of the static spin density is calcu-\nlated as\n⟨sz\ntot⟩=−X\nk∥M\n2Ek∥\u0002\nf(Ek∥)−f(−Ek∥)\u0003\n→ −\u0012L\n2πℏv\u00132Z∆\nMEdEZ2π\n0dϕM\n2E[f(E)−f(−E)]\n=−Z∆\n0dED (E)f(E)−f(−E)\n2EM. (46)\nAt the zero temperature limit T→0, the static spin\ndensity is given by\n⟨sz\ntot⟩=1\n2L2\n2π(ℏv)2(∆−M)M≈˜χzz\n0(0,0)M, (47)\nwhere ˜ χzz\n0(0,0) = D(∆)/2 and we used ∆ ≫M.\nThe dynamic spin density is given by the perturbative\nforce\nHT(t) =Z\ndr∥sz(r∥)F(r∥, t), (48)\nwhere F(r∥, t) is given by\nF(r∥, t) =−sinϑX\nnJ(r∥,rn)\nSX\nn(t)\u000b\n≈ −sinϑJ1Sγhacp\n(ω−γhdc)2+α2ω2cosωt\n=:Fcosωt. (49)\nThe time dependent statistical average of the Ising spin\ndensity, to the first order of J1S, is given by\nZ\ndr∥\nsz(r∥, t)\u000b\n=Z\ndr∥Z\ndr′\n∥Z\ndt′χzz(r∥−r′\n∥, t′)F(r′\n∥, t−t′)\n= Re\u0002\n˜χzz(0, ω)Fe−iωt\u0003\n≈Re[˜χzz\n0(0, ω)]Fcosωt, (50)\nwhere we used Re˜ χzz\n0(0, ω)≫Im˜χzz\n0(0, ω). The real part\nof ˜χzz(0, ω) is given by\nRe˜χzz(0, ω) =−PZ\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω,\n(51)where Pmeans the principal value. When the integrand\nis expanded with respect to ω, the lowest order correc-\ntion term becomes quadratic in ω. In the frequency range\nconsidered in this work, this correction term is signifi-\ncantly smaller compared to the static spin susceptibility\nRe˜χzz(0,0). Therefore, the spin susceptibility exhibits\nalmost no frequency dependence and remains constant\nas a function of ω. The imaginary part of ˜ χzz(0, ω) is\ngiven by\nIm˜χzz(0, ω)\n=πD(ℏω/2)(ℏω/2)2−M2\n2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)].\n(52)\nFMR modulation due to the proximity exchange\ncoupling\nIn this section, we provide a brief explanation for the\nderivation of the FMR modulations δhandδα. The FMR\nmodulations can be determined from the retarded com-\nponent of the magnon Green’s function, which is given\nby\n˜GR(k, ω) =2S/ℏ\nω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53)\nwhere we introduce the Gilbert damping constant αphe-\nnomenologically. In the second-order perturbation calcu-\nlation with respect to HT, the self-energy is given by\nΣR(k, ω) =−\u0012sinϑ\n2\u00132X\nq∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54)\nwhere ˜J(q∥,0) is given by\n˜J(q∥,k) =1\nL2√\nNZ\ndr∥X\nnJ(r∥,rn)ei(q∥·r∥+k·rn)\n(55)\nThe pole of ˜GR(k, ω) signifies the FMR modulations,\nincluding both the frequency shift and the enhanced\nGilbert damping. These are given by\nδh=2S\nγℏReΣR(0, ω), δα =−2S\nℏωImΣR(0, ω).(56)\nFrom the above equations and Eq. (54), it is apparent\nthat FMR modulations provide information regarding\nboth the properties of the interface coupling and the dy-\nnamic spin susceptibility of the Majorana modes.\nThe form of matrix element ˜J(q∥,0) depends on the\ndetails of the interface. In this work, we assume the\nspecular interface. |˜J(q∥,0)|2is given by\n|˜J(q∥,0)|2=J2\n1\nNδq∥,0. (57)8\nUsing Eq. (57), the self-energy for the uniform magnon\nmode is given by\nΣR(0, ω) =−\u0012sinϑ\n2\u00132J2\n1\nN˜χzz(0, ω). (58)\n[1] F. Wilczek, Nat. 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Jpn. 92, 063701 (2023)."    },    {        "title": "0808.1377v1.Quantum_frustration_of_dissipation_by_a_spin_bath.pdf",        "content": "arXiv:0808.1377v1  [cond-mat.mes-hall]  9 Aug 2008Quantum frustration of dissipation by a spin bath\nD D Bhaktavatsala Rao,1Heiner Kohler,2and Fernando Sols3\n1Department of Physics, Indian Institute of Technology Kanpur, K anpur 208016,\nIndia.\n2Department of Physics, University of Duisburg-Essen, D-47057 D uisburg, Germany\n3Departamento de F´ ısica de Materiales, Universidad Complutense de Madrid,\nE-28040 Madrid, Spain\nPACS numbers: 03.65.Yz,42.50.Lc,75.10.Dg\nE-mail:ddbrao@iitk.ac.in\nAbstract. We investigate the evolution of a central spin coupled to a spin bath\nwithout internal dynamics. We compare the cases where the bath c ouples to one or\ntwo components of the spin. It is found that the central spin dyna mics is enhanced in\nthe latter case, which may be interpreted as a frustration of dissip ation. However, the\nquantum purity of the spin decays fast in both scenarios. We conclu de that symmetric\ncoupling of the bath to two orthogonal components of the spin inhib its dissipation but\nnot decoherence.Quantum frustration of dissipation by a spin bath 2\n1. Introduction\nThe study of the role of a dissipative environment is of central impor tance to the ﬁeld\nof quantum computation and for the fundamental understanding of the transition from\nquantum to classical behavior. In that context, the dissipative tw o-level system (TLS)\nis a well-studied paradigm [1, 2]. The generic TLS, which in particular can be a spin1\n2\nparticle, experiences dissipation due to its coupling to a bath of harm onic oscillators. In\ntheresultingspin-bathproblem, anexternalmagneticﬁeldinterac tswithonecomponent\nofthespinoperator(e.g. Sz)whileasecond component (e.g. Sx)couples toanoscillator\nbath. Depending on the relative strength of the two interactions, one can switch over\nbetween underdamped and overdamped behaviour, as may become manifest through\nquantities such as the average spin energy or various correlation f unctions. Recently\nCastro Neto et al.[3, 4] have shown that if two components ( SxandSy) of the\neﬀective spin are coupled to two diﬀerent baths, then the competin g eﬀects of those\nbathscanreducetheeﬀect ofdissipation. Inparticular, theyhav e foundthat, foragiven\ncoupling strength, symmetric coupling to the two spin components is less decoherent\nthan coupling to a single component. Moreover, in the case of symme tric coupling to\nSxandSy, coherent behaviour is preserved for arbitrarily strong coupling. These two\nproperties are remarkable because one would naively expect that c oupling to a higher\nnumber of bath oscillators would increase the eﬀect of dissipation. R ather, they have\nshown that the competition between the baths contributes to pro tect the TLS energy\ngap. The reduction of dissipation stems from the non-commutative character of the\nspin operators coupled to the diﬀerent baths. Logically, if the two b aths interact with\nthe same spin component, then no dissipation reduction is observed . In Refs. [3, 4], it\nis argued that this feature arises from the lack of a preferred bas is to which the TLS\nmay relax at long times. This new phenomenon has been coined quantu m frustration of\ndecoherence, since it is interpreted as the frustrated attempt o f the two environments\nto “measure” simultaneously two non-commuting observables. The study of Refs. [3, 4]\nhas been restricted to equilibrium properties such as the transver se susceptibility, which\nwas evaluated using the numerical renormalization group.\nFeatures of quantum frustration were also reported in Ref. [5], wh ere it was noted\nthat the phase-number variable of a superconducting Josephson junction is coupled\nsimultaneously to two diﬀerent dissipative environments: through t he phase to the\nquasiparticle ﬁeld and through the Cooper pair number to the quant um electromagnetic\nﬁeld. The result is that the uncertainty in the macroscopic phase ha s contributions from\nboth baths which tend to cancel each other. They never cancel c ompletely because, in\nthat particular case, the sources of dissipation diﬀer widely both in n ature and strength.\nFrustration of decoherence in Josephson networks has also been investigated [6]\nIn Refs. [7, 8] the dynamics of an oscillator coupled through its posit ion and\nmomentum to two diﬀerent oscillator baths was studied. The problem can be shown\nto be equivalent to that of a large (quasiclassical) spin impurity in a fer romagnetic\nenvironment. Itwasnotedthatthedissipativeoscillatormaybedriv enfromoverdampedQuantum frustration of dissipation by a spin bath 3\nto underdamped behaviour by the symmetrical addition of a second bath. This\nsurprising eﬀect was found only for the case where the momentum o perator ( p) and\nthe position operator ( q) are coupled to two diﬀerent baths. Like in Refs. [3, 4], these\neﬀects were investigated through equilibrium properties such as th e position-position\nresponse function. However, some dynamical aspects were also a nalyzed, noting that\nthe purity of the quantum oscillator decays faster in the presence of two baths than\nin the presence of a single bath. This last result is important because it reveals that\nthe competition between two baths coupling to non-commuting obse rvables is not a\nuniversal panacea to suppress decoherence. Our present stud y is motivated by the need\nfor a more detailed understanding of the equilibrium and dynamical pr operties of a\nquantum system in the presence of competing environments.\nIn the studies of quantum frustration made earlier [3, 4, 5, 7, 8], t he bath has been\nmodeled by a set of non-interacting oscillators. However, if the inte rpretation is correct\nthat the essence of quantum frustration stems from the canonic ally conjugate character\nof the two observables which couple to separate baths, then one s hould expect a similar\nbehaviour to appear when the dissipative environment is formed by a bath of spins\nacting on a central spin impurity. We note that, because of the vec tor nature of the\nspin, the spin bath can by itself be viewed as formed by several bath s. Thus a single\nspin bath may exhibit features of quantum frustration. Spin baths have been studied\n[9, 10, 11, 12, 13, 14, 16, 17] as an alternative to the conventiona l oscillator models of\nquantum dissipation [1, 2]. They are known to give rise to non-Markov ian evolution,\nwith the system evolution showing a strong dependence on the polar ization of the initial\nstate [15].\nIn this work we shall evaluate the dynamical properties of a TLS cou pled to a bath\nof spins. Similar to the work of Novais et.al.[3, 4], we shall consider the situation where\nthe components of the spin are coupled to two diﬀerent baths and s tudy the eﬀects of\nfrustration arising due to the non-commuting nature of the spin co mponents. The work\nof Refs. [3, 4] was based on the method of the numerical renormaliz ation group. By\nassuming that the bath has no internal dynamics of its own, we are a ble to perform\nan analytical study. Comparing the case of coupling to a single compo nent to that of\nsymmetric coupling to both components, we ﬁnd that the spectral function behaves\nsimilarly to the study of Refs. [3, 4]. Namely, the spectral function d evelops a peak in\nthe symmetric case which is absent in the case of single-component c oupling. In Refs.\n[3, 4], this was interpreted as the preservation of decoherence ar ising from the frustrated\nattempt of the two environments to measure two non-commuting o bservables. However,\nwe ﬁnd that the emergence of the peak as the second bath interve nes is compatible\nwith a fast decay of the quantum purity of the central spin. This me ans that the\nfrustration induced by the two competing environments has more t o do with dissipation\nthanwithdecoherence. Energyrelaxationisindeed inhibitedbythep resence ofasecond\nbath coupled to the other spin component, while the quantum purity decays fast in the\npresence ofasecond bath, withonlyaminor formoffrustrationre vealed by ashort-lived\nrevival which will be discussed.Quantum frustration of dissipation by a spin bath 4\nSection 2 is devoted to the presentation of the model of a central spin coupled to\na spin bath. In Sec. 3 we study the time evolution analytically, deriving expressions\nfor the expectation values and the quantum purity of the central spin. Sections 4 and\n5 focus on the density of states and the response function of the central spin. Section 6\ndeals with the tailoring of the spin bath properties which mimics the beh avior of a spin\ncoupled to a conventional bath of harmonic oscillators. Finally, the m ain conclusions of\nthis work are summarized and discussed in Sec. 7.\n2. Central spin model\nWe consider the dynamics of a two-level-system which is linearly couple d through two\nnon-commuting observables to two independent environments of t wo-level-systems. We\nwill refer indistinctively to both the central impurity and the constit uents of the bath as\nparticles of spin1\n2or two-level-systems. If the spins of the environment carry their own\ndynamics, in general there areno conserved quantities other tha nenergy and the system\ncan not be treated analytically without approximations. However, in most of the solid-\nstatespinsystems wherethespinbathinteractionisadominant sou rceofmechanism for\nthe dissipation of a TLS, the internal bath dynamics is generally very slow (for example\nin quantum dot systems, where the bath spins are nuclear spin half p articles and the\nTLS is the electronic spin, [18]). We therefore assume that both en vironments carry no\ndynamics of its own, i. e. that their Hamiltonians are zero. Thus the b ath dynamics is\nexclusively due to its interaction with the central spin [16]. The total Hamiltonian of\nsystem and bath is then given by\nH=HS+HSB (1)\nwhere\nHS=ω0Sz, (2)\nHSB=g1SxN/summationdisplay\nk=1Ix\nk+g2SyN/summationdisplay\nl=1Jy\nl, (3)\nwhereSi,i=x,y,zare the components of the spin operator of the central spin and\nIx\nkandJy\nkare spin operators of the bath spins. We assume homogeneous inte raction\nbetween the central spin and the baths. Moreover we assume tha t the number of spins\nof each environment to be the same. The strength of the coupling t o each environment\nis thus described by one parameter gi(i= 1,2) only. For a single bath–environment\nthe case of non–homogeneous coupling ( g1,g2dependent on index k,l, respectively) was\nsolved explicitly by in Ref. [10]. However, it was shown in Refs. [10, 16] t hat the\nonly eﬀect of inhomogeneous interaction is that of destroying cert ain revival eﬀects.\nAll other features can be captured within the homogeneous intera ction between system\nand bath. The main advantage of the homogeneous interaction app roximation is that\nexact, closed-form expressions for the expectation values and c orrelation functions can\nbe obtained.Quantum frustration of dissipation by a spin bath 5\nThe Hamiltonian (1) is similar to that employed in Refs. [3, 4] in that two d iﬀerent\nenvironments couple to the two perpendicular components of the c entral spin. The\nmain diﬀerence with the model of Refs. [3, 4] is the non-dynamic char acter of the\nbath, which we consider, which contrasts with the oscillator bath th ere considered. We\nshall examine whether our simpler model (1) can yield frustration eﬀ ects similar to\nthose obtained from the more complex model of Refs. [3, 4], which wa s solved with the\nnumerical renormalization group method.\nIn the particular case where the spin bath couples to only one compo nent of the\ncentral spin, a number of non-trivial eﬀects are known to appear , despite its apparent\nsimplicity. One instance is the crossover from overdamped to under damped behaviour\nas the coupling strength increases, similar to the spin-boson model where the TLS is\ncoupled to an oscillator bath [1, 2]. For a more complete account on t he dynamics of a\nspin coupled to a single bath we refer to Refs. [9, 16].\n3. Time evolution\nIn the absence of bath dynamics the x– andy–components of the total spin of the\nrespective environments, Ix\ntot=/summationtext\nkIx\nkandJy\ntot=/summationtext\nkJy\nk, are conserved quantities. We\nwrite the total Hilbert space as a tensor product HS⊗HB. The total 22Ndimensional\nHilbert space of the baths HBdecomposes into invariant subspaces H(m1,m2)\nB. These\nare labeled by the eigenvalues m1,m2of the total spin operators Ix\ntotandJy\ntot. Each\nmi(i= 1,2) runs from −N/2 toN/2. The Hamiltonian (1) acts on the subspace\nHS⊗H(m1,m2)\nBas\nH|σ/an}bracketri}ht|α1,α2;m1,m2/an}bracketri}ht= (ω0Sz+Sxm1g1+Sym2g2)|σ/an}bracketri}ht|α1,α2;m1,m2/an}bracketri}ht,(4)\nfor|α1,α2;m1,m2/an}bracketri}ht ∈ H(m1,m2)\nBand|σ/an}bracketri}ht ∈ H S. Hereαi, which is not important in\nthe following, labels the irreducible representation. We therefore c an write Hmost\nconveniently as a direct sum H=⊕N/2\nm1,m2=−N/2Hm1m2where\nHm1m2= (ω0Sz+g1m1Sx+g2m2Sy)⊗Iλm1⊗Iλm2\n=/vectorS·/vectorΩm1m2⊗Iλm1⊗Iλm2. (5)\nHereIλis theλ×λunit matrix and the parameters λmi,\nλmi=/parenleftBigg\nN\nN/2−mi/parenrightBigg\n, i= 1,2 (6)\nmeasure the dimension of the invariant subspace H(m1,m2)\nB. From Eq. (5) it is clear that\nthe eﬀect of the environment is to give rise to an eﬀective magnetic ﬁ eld/vectorΩm1m2=\n(m1g1,m2g2,ω0). However, this eﬀective magnetic ﬁeld is diﬀerent from the static\nmagnetic ﬁeld pointing in the z–direction, since it does not take a single value but\nrather is a distribution characterized by the degeneracy coeﬃcien tsλmi.\nSinceHm1m2acts on the subspace of the environment, we will often write Hm1m2=\n/vectorS·/vectorΩm1m2for short. The eigenvalues of Hm1m2are±Ωm1m2/2, where we introduced theQuantum frustration of dissipation by a spin bath 6\nfrequency Ω m1m2= (ω2\n0+g2\n1m2\n1+g2\n2m2\n2)1/2. We denote the eigenstates of Hm1m2by\n|±,m1,m2/an}bracketri}ht. They are related to the eigenstates | ↑/an}bracketri}ht,| ↓/an}bracketri}htof the non–interacting system\nHamiltonian HSby the unitary transformation/parenleftBigg\n|+,m1,m2/an}bracketri}ht\n|−,m1,m2/an}bracketri}ht/parenrightBigg\n=/parenleftBigg\ncosθm1m2sinθm1m2eiφm1m2\n−sinθm1m2e−iφm1m2cosθm1m2/parenrightBigg/parenleftBigg\n| ↑/an}bracketri}ht\n| ↓/an}bracketri}ht/parenrightBigg\n(7)\nwhere the angles φm1m2andθm1m2are given by\nφm1m2= arctan/parenleftbiggm2g2\nm1g1/parenrightbigg\n(8)\ncos2θm1m2=Ωm1m2+ω0\n2Ωm1m2. (9)\nThe case of a single environment is recovered by setting g2or, equivalently, φm1m2equal\nto zero.\nThe time evolution operator is straightforwardly derived from Eq. ( 5). We obtain\nU=⊕N/2\nm1,m2=−N/2Um1m2with\nUm1m2(t) = cos/parenleftbiggt\n2Ωm1m2/parenrightbigg\nI2+2isin/parenleftbigt\n2Ωm1m2/parenrightbig\nΩm1m2/vectorS·/vectorΩm1m2, (10)\nWe can decompose an arbitrary system (central spin) operator OasO=\n⊕N/2\nm1,m2=−N/2Om1m2. In particular we are interested in the Heisenberg spin operator\n/vectorS(t) and its commutators and anticommutators at diﬀerent times. Usin g (10), we ﬁnd\n/vectorSm1m2(t) = cos(Ω m1m2t)/vectorS(0)−sin(Ωm1m2t)(/vectorS(0)×/vector nm1m2)\n+[1−cos(Ωm1m2t)](/vectorS(0)·/vector nm1m2)/vector nm1m2(11)\n−i[Si\nm1m2(t),Sj\nm1m2(0)] = cos(Ω m1m2t)ǫijkSk(0) (12)\n+sin(Ω m1m2t)/parenleftBig\nδij/vector nm1m2/vectorS(0)−Si(0)nj\nm1m2/parenrightBig\n+[1−cos(Ωm1m2t)]ni\nm1m2/parenleftBig\n/vectorS(0)×/vector nm1m2/parenrightBig\nj\n2{Si\nm1m2(t),Sj\nm1m2(0)}= cos(Ω m1m2t)δij−sin(Ωm1m2t)ǫijknk\nm1m2\n+[1−cos(Ωm1m2t)]ni\nm1m2nj\nm1m2.(13)\nThe vector /vector nm1m2is a unit vector pointing in the direction of the eﬀective magnetic ﬁeld\n/vectorΩm1m2.\nIf the density matrix ρ(t) of the total system is initially invariant under rotations\nwithin a subspace H(m1,m2)\nBdue to the trivial action of the Hamiltonian in this subspace,\nthis invariance will persist at all times. In particular, the density mat rix can be written\nfor all times as ρ(t) =⊕N/2\nm1,m2=−N/2ρm1m2(t). This means that ρ(t) shares for all times\nthe block structure of the Hamiltonian. If ρ(0) fulﬁlls this condition, the expectation\nvalue of an arbitrary system operator O(t) with respect to ρ(0) can then be written as\n/an}bracketle{tO(t)/an}bracketri}ht ≡tr[ρ(0)O(t)]\n=N/2/summationdisplay\nm1,m2=−N/2λm1λm2Om1m2(t)ρm1m2(0) (14)Quantum frustration of dissipation by a spin bath 7\n00.2 0.4 0.6 0.8 10.20.10   −0.1 −0.2 −0.3 −0.4 −0.5 \n/radicalbig\ng2\n1+g2\n2t/π/angbracketleftSz(t)/angbracketrightg1= 1,g2= 0\ng1= 1/√\n2,g2= 1/√\n2\ng1= 1/2,g2=√\n3/2\nFigure 1. The magnetization of the TLS is plotted with time for various values of the\nsystem bath interaction, in zero-ﬁeld. In the presence of two diﬀe rent baths a change\nof sign arises during the time evolution. The sign reversal reaches it s maximum value\nwhen the two baths are identical. The total number of spins in each b ath is taken\nto beN= 100. The system bath couplings given in the inset are dimensionless\n(gi//radicalbig\ng2\n1+g2\n2).\nIn the following we will analyse the expectation values of the operato rs (11) to (13)\nwith respect to an initially unpolarized bath. Since the magnetic ﬁeld ap plied along the\nz–direction only aﬀects the central spin, we take this to be initially in th e ground state\ndetermined by HSand consequently choose the initial density matrix as\nρ(0) =1\n22N| ↑/an}bracketri}ht/an}bracketle{t↑ |⊗ I2N⊗I2N. (15)\nWe immediately see that, in this state, /an}bracketle{tSi(0)/an}bracketri}ht=−δi3/2.\nUsing the above formalism for evaluating the dynamical properties o f the TLS\nwe shall now calculate various quantities for the TLS operators and study the eﬀects\nbrought about by the coupling to two diﬀerent baths.\n3.1. Expectation values\nThe expectation values /an}bracketle{tSi(t)/an}bracketri}htcan now be calculated using Eqs. (11) and (15). We ﬁnd\n/an}bracketle{tSz(t)/an}bracketri}ht=−1\n22N+1N/2/summationdisplay\nm1,m2=−N/2λm1λm2\n/parenleftbiggg2\n1m2\n1+g2\n2m2\n2\nΩ2m1m2cos(Ωm1m2t)+ω2\n0\nΩ2m1m2/parenrightbigg\n(16)\n/an}bracketle{tSx(t)/an}bracketri}ht=/an}bracketle{tSy(t)/an}bracketri}ht= 0 (17)Quantum frustration of dissipation by a spin bath 8\n00.2 0.4 0.6 0.8 10.10   −0.1 −0.2 −0.3 −0.4 −0.5 \n/radicalbig\ng2\n1+g2\n2t/π/angbracketleftSz(t)/angbracketrightg1= 1,g2= 0\ng1= 1/√\n2,g2= 1/√\n2\ng1= 1/2,g2=√\n3/2\nFigure 2. Same as Fig. 1 but for a nonzero ﬁeld such that ω0//radicalbig\ng2\n1+g2\n2= 2. In\nthe presence of the external ﬁeld the magnetization for the single bath case decays\nnon-monotonically and saturates to a non-zero value.\nIn the absence of external ﬁeld ω0= 0, we would expect that the initial polarization of\ntheTLSwould decay faster incomparison tothesingle bathcase, sin ce thetotal number\nof spins with which the TLS is interacting is doubled. In Fig. 1 we have plo tted the\ntime variation of the /an}bracketle{tSz(t)/an}bracketri}htfor various values of g1,g2keepingg=/radicalbig\ng2\n1+g2\n2constant.\nAs one can see, for a single bath the polarization decays to zero ver y fast, whereas in\nthe presence of the second bath, the decay is comparatively slow. In contrast to the\ncase of single bath, one observes a change of sign in the time-depen dent behaviour of\nthe polarization of the TLS, indicating the presence of a nonzero ﬁe ld.\nIn Fig. 2 the time variation of the /an}bracketle{tSz(t)/an}bracketri}htis plotted for non–vanishing external ﬁeld\nω0/ne}ationslash= 0. We ﬁnd that the polarization saturates to a ﬁnite value at long tim es with\nfaster oscillations in the case of a symmetric double bath, which is con sistent with the\nspectral properties discussed later in the text. Inspection of Fig s. 1 and 2 reveals that\nthe change of sign occurs in the single bath case only if there is a nonz ero ﬁeld, while it\nis observed for both zero and nonzero ﬁeld in the case of a symmetr ic double bath. Since\nthe baths are completely unpolarized (peaked at m= 0), it is clear that the eﬀective\nﬁeld responsible for this change of sign can only stem from the compe ting eﬀect of two\nbaths coupled to non-commuting components of the central spin.\n3.2. Quantum purity\nFor an arbitrary density matrix ρ, purity is deﬁned as P= Trρ2. Purity is a convenient,\nbasis-independent measure of the degree of coherence, if ρis the reduced density matrix\nof the central spin (that which results from tracing out the bath d egrees of freedom\nin the total density matrix). In our case the decay of purity is direc tly related to theQuantum frustration of dissipation by a spin bath 9\n0 0.1 0.2 0.3 0.4 0.50.50.60.70.80.91\n/radicalbig\ng2\n1+g2\n2t/πP(t)g1= 1,g2= 0\ng1= 1/√\n2,g2= 1/√\n2\ng1= 1/2,g2=√\n3/2\nFigure 3. The purity of TLS is plotted against time for various values of the sys tem\nbath interaction. Though the initial decay rate strongly depends o n the combined\ninteraction strength/radicalbig\ng2\n1+g2\n2(which is same for all the three cases), the TLS looses\nits initial polarization faster in the presence of a single bath in compar ison to the case\nof two baths. The external ﬁeld is set to zero and the number of sp ins in each bath\nare taken to be equal with N= 100. The system bath couplings given in the inset are\ndimensionless ( gi//radicalbig\ng2\n1+g2\n2).\nrelaxation of the spin expectation values to equilibrium.\nP(t) =1\n2+3/summationdisplay\ni=1/an}bracketle{tSi(t)/an}bracketri}ht2. (18)\nThe result is plotted in Fig. 3. We notice that, both in the single and dou ble\nsymmetric bathcases, thepuritydecaysfasttoitsminimumvalue1/ 2. Inthesymmetric\ncase, we notice a small, short-lived revival that may be interpreted as a weak form of\ndecoherence frustration which however does not aﬀect the long t ime behavior of the\ncentral spin.\n4. Density of states\nTo get further analytical insight we introduce the function\nD(ω) =1\n22N+1−N/2/summationdisplay\nm1,m2=−N/2λm1λm2[δ(ω−Ωm1m2)+δ(ω+Ωm1m2)] (19)\nwhichisessentially thedensityofstates, normalisedtofulﬁllthesum rule/integraltext\ndωD(ω) = 1.\nThe expectation value /an}bracketle{tSz(t)/an}bracketri}htis related to D(ω) by\n/an}bracketle{tSz(t)/an}bracketri}ht=−1\n2/integraldisplay\ndωD(ω)/parenleftbigg\ncos(ωt)ω2−ω2\n0\nω2+ω2\n0\nω2/parenrightbigg\n. (20)Quantum frustration of dissipation by a spin bath 10\n0 5 10 15 20 2500.511.522.533.54\n(ω−ω0)/ω0D(ω)/D(ω0)θg=π/50\nθg=π/10\nθg= 3π/20\nθg=π/4\nFigure 4. Density of states as a function of frequency in normalized units.\nD(ω) can further be evaluated by using the approximation\n1\n2N/summationdisplay\nmλm≈/radicalbigg\n2\nπN/integraldisplay∞\n−∞dme−2m2/N, (21)\nfor the binomials λmi, which is known as Laplace–de Moivre formula in probability\ntheory [19] and which is valid only for large N. We ﬁnd\nD(ω) =4|ω|θ(ω−ω0)\nNg2sin(2θg)I0/parenleftbigg4cot(2θg)(ω2−ω2\n0)\nNg2sin(2θg)/parenrightbigg\nexp/parenleftbigg\n−4(ω2−ω2\n0)\nNg2sin2(2θg)/parenrightbigg\n.(22)\nwhere the total coupling strength gand the angle θgare deﬁned by\ng1=gcosθg\ng2=gsinθg, (23)\nandI0is the modiﬁed Bessel function. The case θg= 0 corresponds to the single bath\nandthecase θg=π/4correspondstotwo identical baths. InEq.(22)thelimit ofasingle\nbath can be taken by using the asymptotic expansion of the modiﬁed Bessel function\nlimz→∞I0(z) =ez/√\n2πz+...The results are plotted in Fig. 4. As the coupling of the\nTLS with the baths becomes symmetric, i.e. θg→π/4, the density of states peaks at\na frequency away from ω0. Asθg→0, this peaks shifts towards ω0, which is expected\nfor the case of TLS coupling to a single bath. Thus the density of sta tes can by itself\nreveal the frustrating eﬀects of decoherence more elegantly.\nFor the particular case of HS= 0 (ω0= 0), one can obtain simpliﬁed expressions\nfor the density of states. We obtain\nD(ω)≃1\ng√\nNe−ω2/g2N, g1=g, g2= 0\nD(ω)≃ω\ng2Ne−ω2/g2N, g1=g2=g (24)Quantum frustration of dissipation by a spin bath 11\nIn the case of a single bath the peak is at ω= 0, where as for two baths, the peak is\nshifted to ω=/radicalbig\nN/2g. We note in this respect that, if the two baths were coupled to\nsame spin component, then the behavior would be similar to that of an eﬀective single-\nbath coupled to one spin component. In such a case the peak in D(ω) would remain\natω= 0. Thus, the emergency of a peak at ω/ne}ationslash= 0 may be viewed as a frustration of\ndissipationduetothecompetitionbetweentwoenvironments couple dtonon-commuting\nspin components.\nWe end by noting that D(ω) is the Fourier transform of /an}bracketle{tSz(t)Sz(0)/an}bracketri}htand is a\nmeasure of the strength of the transitions induced by a periodic pe rturbation ∼cos(ωt).\n5. Correlation functions\nWe now investigate the spin–spin correlation functions deﬁned by Cij(t) =/an}bracketle{tSi(t)Sj(0)/an}bracketri}ht.\nSpeciﬁcally, we focus on its symmetrized and antisymmetrized versio ns,Sij(t)≡\n1\n2/an}bracketle{t{Si(t),Sj(0)}/an}bracketri}htandAij(t)≡ −i/an}bracketle{t[Si(t),Sj(0)]/an}bracketri}ht.\nSince the system is initially in an eigenstate of Sz, the symmetrized autocorrelation\nfunction in z–direction is simply Szz(t) =−1\n2/an}bracketle{tSz(t)/an}bracketri}ht. Using the general formulae of\nSec. 3, we ﬁnd for the transversal symmetrized auto–correlatio n functions\nSxx(t) =−1\n22N+2N/2/summationdisplay\nm1,m2=−N/2λm1λm2/parenleftbiggω2\n0+g2\n2m2\n2\nΩ2m1m2cos(Ωm1m2t)+g2\n1m2\n1\nΩ2m1m2/parenrightbigg\n(25)\nand a similar result is obtained for Syy(t) by exchanging indices m1andm2. All\nsymmetrized cross–correlation functions Sij(t), (i/ne}ationslash=j) are zero.\nWe look at the autocorrelation function in z–direction in the case that there is no\nexternal magnetic ﬁeld applied ω0= 0. Since Szz(t) is proportional /an}bracketle{tSz(t)/an}bracketri}htwe can use\nthe integral representation (20). In general, i. e. for intermedia te values of θgand for\nnon–zero frequency ω0, the integral (20) becomes quite diﬃcult and cannot be solved\nanalytically. However in some limits closed expressions can be derived. Forω0= 0 we\nobtain\nSzz(t) = exp(−Ng2t2/8), θg= 0\nSzz(t) = 1−/radicalbigg\nπNg2\n8texp(−Ng2t2/8)Erﬁ/parenleftBigg√\nNgt\n2√\n2/parenrightBigg\n, θg=π/4 (26)\nThe antisymmetrized correlation functions are related to the dyna mical\nsusceptibilities, deﬁned as\nχij(ω) =/integraldisplay∞\n0dt\n2πeiωtAij(t). (27)\nIn particular, the imaginary part of the susceptibility can be used as a measure of\nthe energy dissipated from the system to the bath. Using Eqs. (12 ) and (14) we ﬁnd\nAzz(t) = 0,\nAxx(t) =−1\n22N+1N/2/summationdisplay\nm1,m2=−N/2λm1λm2ω0\nΩm1m2sin(Ωm1m2t)Quantum frustration of dissipation by a spin bath 12\n=/integraldisplay\ndωD(ω)ω0\nωsin(ωt) (28)\nandAyy(t) =Axx(t), where D(ω) has been analyzed in the previous section. All anti–\nsymmetrized cross–correlation function but Axy(t) are zero. For Axy(t) we ﬁnd\nAxy(t) =−1\n22N+1N/2/summationdisplay\nm1,m2=−N/2λm1λm2cos(Ωm1m2t)\n=/integraldisplay\ndωD(ω)cos(ωt) (29)\nIn the second lines of Eqs. (28) and (29) we used an integral repre sentation in terms of\nD(ω). In this form the dynamical susceptibilities are readily evaluated\nχxx(ω) =ω0\n2π/integraldisplay\ndω′D(ω′)\nω2−ω′2+sgn(ω)i0+(30)\nSplitting χij(ω) in its real and its imaginary part, χij(ω) =χ′\nij(ω)+iχ′′\nij(ω), one obtains\nthe relation (22)\nχ′′\nxx(ω) =ω0\n2ωD(ω), (31)\nwhich holds for ω≥ω0. Moreover χ′′\nxx(ω) = 0 for |ω|< ω0, andχ′′\nxx(−ω) =−χ′′\nxx(ω).\nWe can use the approximation (21) and obtain\nχ′′\nxx(ω) =2ω0\nNg2exp/parenleftbigg\n−4(ω2−ω2\n0)\nNg2/parenrightbigg\n, θg=π\n4(32)\nχ′′\nxx(ω) =ω0/radicalbig\n2πNg2(ω2−ω2\n0)exp/parenleftbigg\n−2(ω2−ω2\n0)\nNg2/parenrightbigg\n, θg= 0 (33)\nFrom the above equations it can be seen that there is a strong singu larity at ω=ω0, in\naddition to the Gaussian spread arising due to the interaction with th e bath. For the\nsymmetric coupling, this singularity is removed and only a Gaussian spr ead peaked at\nω=ω0remains. Both functions are peaked at ω=ω0, and hence one can say that, since\nthere is no peak shifting there is no frustration. If we try to remov e the singularity for\nthe single bath case by multiplying/radicalbig\nω2−ω2\n0withχ′′\nxx(ω) then one can immediately see\nthat the peak for χ′′\nxx(ω) is shifted away from ω=ω0for the symmetric case. Scalings of\nsuch kind can be avoided by considering other kinds of distributions f or the bath spins.\nIn the next section we consider bath spin distributions λmwith a Gaussian cutoﬀ.\n6. Tailoring the density of states\nIn Refs. [3, 4] respectively [5, 8] similar expressions were obtained f orχ′′\nxx(ω) in the ﬁrst\ncase and for the Fourier transform of the antisymmetrized positio n–position correlation\nfunction in the second case. In both cases the function under con sideration is essentially\na Lorentzian\nχ′′\nxx(ω) =Zω\n(ω2−/tildewideω2\n0)2+geﬀω2, (34)Quantum frustration of dissipation by a spin bath 13\nwhere/tildewideω0is the renormalized frequency of the system (Larmor frequency, respectively\noscillatorfrequency), geﬀistheeﬀectivedampingcoeﬃcient. Forthedetailedexpressions\nof/tildewideω0,geﬀandZsee Eq. (26) of Ref. [4] and Eq. (20) of Ref. [8]. The Lorentzian for m,\nand in particular the linear behavior for small values of ωis typical for Ohmic type of\ndissipation.\nWe model our system to best mimic Ohmic behavior. Since χ′′\nxx(ω) is related to\nthe density of states D(ω) by the simple relation (31), we can focus directly on D(ω) as\ndeﬁned in Eq. (19). We now use a general λmof the form\nλ(α)\nm=1\n2/parenleftbigg2\nN/parenrightbigg(α+1)/2|m|α\nΓ((1+α)/2)exp/parenleftbigg\n−2m2\nN/parenrightbigg\n. (35)\nFor small values of mthe behavior of λmis dominated by the power mαwith a\ncharacteristic exponent α. We note that the case α= 1 cannot directly be identiﬁed\nwith an Ohmic bath. In Eq. (35) we took a Gaussian cutoﬀ for large va lues ofωwith\na cutoﬀ frequency chosen as/radicalbig\nN/2 in order to make contact with the former results.\nIt is a well known fact in the theory of open quantum systems that t he speciﬁc form\nof the cutoﬀ function is not relevant [2]. The function λmis normalized such that/integraltext\ndmλ(m) = 1 holds. We see that the form of λmdescribed in Eq. (6) is just a special\ncase of Eq. (35) corresponding to α= 0 [see also Eq. (21)].\nIn a calculation which is similar to that performed in Sec. 4, we obtain fo r the\ndensity of states\nD(ω) =4|ω|√π\nΓ((α+1)/2)Ng2sin(2θg)/parenleftbigg2\nNg2cos(2θg)/parenrightbiggα/2\n(ω2−ω2\n0)α/2θ(ω−ω0)\nIα/2/parenleftbigg4cot2θg(ω2−ω2\n0)\nNg2sin(2θg)/parenrightbigg\nexp/parenleftbigg\n−4(ω2−ω2\n0)\nNg2sin2(2θg)/parenrightbigg\n, (36)\nwhere, as before, g2=g2\n1+g2\n2and the angle θgis deﬁned in Eq. (23). Moreover we\nhave introduced the modiﬁed Bessel function Iα/2of order α/2. For the transverse\nsusceptibility we ﬁnd in the two limiting cases θg= 0 and θg=π/4 the expressions\nχ′′\nxx(ω) =1\n2(ω2−ω2\n0)αω0\nΓ(α+1)/parenleftbigg4\nNg2/parenrightbiggα+1\nexp/parenleftbigg\n−4(ω2−ω2\n0)\nNg2/parenrightbigg\n, θg=π\n4(37)\nχ′′\nxx(ω) =1\n2Ng2(ω2−ω2\n0)(α−1)/2ω0\nΓ((α+1)/2)/parenleftbigg2\nNg2/parenrightbigg(α−1)/2\nexp/parenleftbigg\n−2(ω2−ω2\n0)\nNg2/parenrightbigg\n, θg= 0. (38)\nwhich satisﬁes the general properties given after Eq. (31). The t ransverse susceptibility\nis zero for ω=ω0. This zero value is found because the distributions λmis centred at\nm= 0. If the distribution is shifted to be centred at a non-zero mvalue, then the value\nofχ′′(ω)12will be non-zero at ω=ω0for suﬃciently large α. In Fig. 5 we have plotted\nthe transverse susceptibility for the symmetric ( θg=π/4) and single-bath ( θg= 0)Quantum frustration of dissipation by a spin bath 14χ′′xx(ω−ω0−ǫ)/χ′′xx(ω0+ǫ)\n(ω−ω0−ǫ)/ω0\nFigure 5. Plot of the transverse suceptibility χ′′\nxx(ω) for an Ohmic type of density\nof states α= 1 and for three diﬀerent angles θg= 0,π/8,π/4. The other parameters\nareω0= 10,g= 1 and N= 1000. The oﬀset at which the function is normalized is\nǫ= 0.1\ncases, as well as for an intermediate situation. In order to compar e with other results,\nin particular with the curves obtained in Ref. [4], we have normalized χ′′\nxx(ω) so that\nχ′′\nxx(ω0+ǫ) = 1, where ǫis a convenient small oﬀset which is chosen for proper scaling\nand comparison. We note that making ǫtoo small shoots the peak to inﬁnity in those\ncases where lim ǫ→0χ′′\nxx(ω0+ǫ) = 0. Though the natural sum rule/integraltext\ndωωχ′′\nxx(ω) =ω0\nis spoiled by adding this epsilon the essential physics behind is unaﬀect ed. When there\nis no interaction with the bath, χ′′(ω) is a delta function peaked at ω=ω0. In the\npresence of one bath the peak broadens with the maximum still locat ed atω=ω0.\nSurprisingly, thepeak at ω=ω0disappears when we introduce a second bath which\ncouples to a diﬀerent component of the central spin. A similar shift w as reported in\nRefs. [3, 4]. It results from a pure frustration eﬀect due to the no n-commuting nature\nof the spin operators.\nIn Fig. 6 we have plotted the transverse susceptibility for three diﬀ erent types\nof infrared behavior α= 0,1,2, mimicking a subohmic, an Ohmic, respectively a\nsuperohmic bath. Dissipationdecreases asthepower αincreases, asexpected forgeneral\ndissipative quantum systems [1]. On the other hand we see that the f rustration eﬀect\nof an additional bath increases with increasing power α(not shown).\n7. Conclusions\nWe have analyzed the spectral properties of a two-level-system w hich is coupled to one\nor two dissipative baths through non-commuting observables. A pe ak in the spectrum\nat a nonzero frequency reveals the existence of an eﬀective magn etic ﬁeld experienced by\nthe central spin. We have seen seen that the coupling to a second b ath enhances rather\nthan diminishes that eﬀective ﬁeld and, with it, the dynamics of the ce ntral spin. In\nthe extreme case of a zero external ﬁeld, the nonzero ﬁeld is gene rated solely from theQuantum frustration of dissipation by a spin bath 15\n0.00.51.01.52.02.53.03.54.00.00.51.01.52.02.53.03.54.0\nΑ/EquΑl2Α/EquΑl1Α/EquΑl0χ′′xx(ω−ω0−ǫ)/χ′′xx(ω0+ǫ)\n(ω−ω0−ǫ)/ω0\nFigure 6. Plot of the transverse suceptibility χ′′\nxx(ω) for the symmetric case θg=π/4\nfor the values α= 0 (dotted), α= 1 (dashed) and α= 2 (full). The parameters are\nω0= 10,g= 1 andN= 1000. The oﬀset at which the function is normalized is ǫ= 0.5\ncompetition of two environments coupling to non-commuting spin com ponents. This\nfact is remarkable if one notes that the baths are assumed to be init ially unpolarized.\nThese physical eﬀects arising from the non-commuting nature of t he spin operators\nare a general feature of the dynamics which does not depend on de tails such as the\nMarkovian or non-Markovian character of the reduced system dy namics or the strength\nof the system-bath interactions. In fact, we explicitly prove that the emergence of a\npeak in the spectral function is compatible with a fast decay of the q uantum impurity.\nThis suggests that, while dissipation is inhibited by the competition bet ween the two\nbaths, decoherence is not, at least in the long time behavior. Our pr esent results are\nentirely consistent with the results of Refs. [7, 8] for a harmonic os cillator (equivalent\nto a large or quasiclassical spin), where a fast decay of the quantu m purity was found\nto coexist with a weak form of suppression of dissipation. Here we ha ve proved that the\nphenomenon of frustration of dissipation also exists for a dissipativ e two-level-system\nwhich, given its reduced dimensionality, is much more quantum in natur e than the\nharmonic oscillator.\nThe intuitive idea that two competing environments attempt to meas ure non-\ncommuting observables and thus fail to generate decoherence so unds appealing but\nmay be misleading. The statement would be true if the only possible res ult of\na quantum measurement were to select a narrow distribution of eige nstates of the\nmeasured observable, since two non-commuting observables cann ot be simultaneously\nwell deﬁned. However, a possible outcome of the coupling to a dissipa tive environment\nis that the reduced density matrix, while becoming diagonal in the rep resentation of the\neigenstates of the measured observable, may display a broad prob ability distribution in\nthat representation. In the limit in which that distribution is very bro ad, the reduced\ndensity approaches the identity matrix, which is invariant under a ch ange of basis. Thus\na reduced density matrix may be simultaneously diagonal in the repre sentations of twoQuantum frustration of dissipation by a spin bath 16\nnon-commuting observables, provided it is close to the identity matr ix. This is what\nactually happens to our central spin-1\n2, as is clearly revealed by the quantum purity\ntending to its minimum value 1/2 at long times, both for a single and a sym metric\ndouble bath. As we have seen, this feature is compatible with the rein forcement of the\ncentral spin dynamics resulting from the competition of the two env ironments. The\nupshot of the present study on the eﬀect of competing environme nts is that, at least for\ndissipative two-level-systems, it may be misleading to speak of quant um frustration of\ndecoherence and is more appropriate to introduce the concept of quantum frustration\nof dissipation.\nWe cannot rule out however the possibility that the coexistence of d ecoherence and\ndynamics enhancement by two competing environments is a particula r property of our\ndissipative model where the bath has no internal dynamics. A ﬁrmer conclusion on\nthe existence or absence of decoherence frustration will require an understanding of the\nbehavior of genuinely quantum properties such as purity or pair ent anglement in the\npresence of competing environments with internal dynamics.\nAcknowledgments\nThis work has been supported by the German Research Council (DF G) through grant\nNo. Ko 3538/1-1 and by MEC (Spain) through grants No. FIS2004- 05120and FIS2007-\n65723.\nReferences\n[1] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger. Rev.\nMod. Phys. , 59:1, 1987.\n[2] U. Weiss. Quantum Dissipative Systems . World Scientiﬁc, Singapore, 2nd edition, 1999.\n[3] A. H. Castro Neto, E. Novais, L. Borda, G. Zarand, and I. Aﬄec k.Phys. Rev. Lett. , 91:096401,\n2003.\n[4] E. Novais, A. H. Castro Neto, L. Borda, I. Aﬄeck, and G. Zaran d.Phys. Rev. B , 72:014417, 2005.\n[5] H. Kohler, F. Guinea and F. Sols. Ann. Phys. , 72:014417, 2004.\n[6] D. Giuliano and P. Sodano. arXiv:0710.5554.\n[7] H. Kohler and F. Sols. Phys. Rev. B , 72:014417, 2005.\n[8] H. Kohler and F. Sols. New J. Phys. , 8:149, 2006.\n[9] N. V. Prokof’ev and P. C. E. Stamp. Rep. Prog. Phys. , 63:669, 2000.\n[10] F. M. Cucchietti, J. P. Paz, and W. H. Zurek. Phys. Rev. B , 70:035311, 2005.\n[11] A. Relano, J. Dukelsky, and R. A. Molina. Phys. Rev. E , 76:046223, 2007.\n[12] W. H. Zurek, F. M. Cucchietti, and J. P. Paz. Acta Phys. Polonica , 38:1685, 2005.\n[13] D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, and R. F azio.Phys. Rev. A , 75:032333,\n2007.\n[14] D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, and R. F azio.J. Phys A: Math. Theor. ,\n40:8033, 2007.\n[15] D. D. Bhaktavatsala Rao, V. Ravishankar and V. Subrahmanya m.Phys. Rev. A , 74:22301, 2006.\n[16] D. D. Bhaktavatsala Rao. Phys. Rev. A , 76:042312, 2007.\n[17] C. Y. Lai, J. T. Hung, C. Y. Mou, P. Chen. Phys. Rev. B , 77:205419, 2008.\n[18] J. Schliemann, A. Khaetskii, and D. Loss. J. Phys.: Cond. Matter , 15:1809, 2003.Quantum frustration of dissipation by a spin bath 17\n[19] G. R. Grimmett and D. R. Stirzaker. Probability and Random Processes . Oxford University Press,\nOxford, 3nd edition, 2001."    },    {        "title": "1304.4980v1.Universal_spin_dynamics_in_two_dimensional_Fermi_gases.pdf",        "content": "Universal spin dynamics in two-dimensional Fermi gases\nMarco Koschorreck1, Daniel Pertot1, Enrico Vogt1, and Michael K ohl1;2\n1Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB30HE, United Kingdom\n2Physikalisches Institut, University of Bonn, Wegelerstrasse 8, 53115 Bonn, Germany\nHarnessing spins as carriers for information has\nemerged as an elegant extension to the transport\nof electrical charges1. The coherence of such spin\ntransport in spintronic circuits is determined by\nthe lifetime of spin excitations and by spin di\u000bu-\nsion. Fermionic quantum gases are a unique sys-\ntem to study the fundamentals of spin transport\nfrom \frst principles since interactions can be pre-\ncisely tailored and the dynamics is on time scales\nwhich are directly observable2{11. In particular at\nunitarity, spin transport is dictated by di\u000busion\nand is expected to reach a universal, quantum-\nlimited di\u000busivity on the order of ~=m. Here,\nwe study the non-equilibrium dynamics of a two-\ndimensional Fermi gas following a quench into\na metastable, transversely polarized spin state.\nUsing the spin-echo technique12, we measure the\nyet lowest transverse spin di\u000busion constant13,14\nof0:25(3) ~=m. For weak interactions, we observe\na coherent collective transverse spin-wave mode\nthat exhibits mode softening when approaching\nthe hydrodynamic regime.\nStudying transport in low-dimensional nanostructures\nhas a long and rich history because of its non-trivial fea-\ntures and its relevance for electronic devices. The most\ncommon case, charge transport, has great technological\nimplications and it determines the current-voltage char-\nacteristics of a device. With the development of the \feld\nof spintronics1, however, also spin transport has moved\ninto the focus of the research interest. Spin transport\nhas unique properties, setting it aside from charge trans-\nport: Firstly, the transport of spin polarization is not\nprotected by momentum conservation and is greatly af-\nfected by scattering5,15. Therefore, the question arises\nwhat is the limiting case of the spin transport coe\u000ecients\nwhen interactions reach the maximum value allowed by\nquantum mechanics? Secondly, unlike charge currents\n(which lead to charge separation and the buildup of an\nelectrical \feld, counteracting the current), spin accumu-\nlation does not induce a counteracting force.\nThe main mechanism to even out a non-equilibrium\nmagnetization ~M(~ r;t) =M(~ r;t)~ p(~ r;t) is spin di\u000busion,\nwhich is an overall spin-conserving process. Other, spin\nnon-conserving, processes are much slower in ultra cold\nFermi gas experiments and are neglected hereafter. The\ngradient of the non-equilibrium magnetization r~M=\n~ prM+Mr~ pdrives two distinct spin currents16: The\n\frst term produces a longitudinal spin current parallel to\n~Mand the second term induces a transverse spin current\northogonal to ~M. These spin currents are proportional\na\nb\nc\nxz\nyBz(x)FIG. 1. Quench of a two-dimensional Fermi gas in which all\natoms were initially prepared in the j#istate. ( a), A\u0019=2-\npulse prepares the Fermi gas polarized in the Sy-direction.\n(b) A magnetic \feld gradient @Bz=@xcauses the spins to ac-\nquire di\u000berent phase-angles \u001e(x) in the equatorial plane. ( c)\nCollisions cant the spins out of the equatorial plane due to the\nidentical spin rotation e\u000bect. The acquired projection along\nSztogether with the motion of the atoms in the harmonic\ntrap impedes rephasing of the spins when the magnetic \feld\ngradient is reversed. The spin states are illustrated in the\nrotating frame.\nto the longitudinal Dkand transverse spin di\u000busivity D?,\nrespectively.\nIn general, the spin di\u000busivity in Ddimensions behaves\nasD=vlmfp=D, wherevis the collision velocity and lmfp\nis the mean free path. The mean-free path is determined\nby the elastic scattering cross section \u001bbetween atoms.\nFor short-range s-wave interactions at unitarity, the cross\nsection attains its maximum value allowed by quantum\nmechanics \u001b\u00181=\u0015D\u00001\ndB, where\u0015dBis the de-Broglie\nwavelength of the colliding particles. At high temper-\natures in the classical gas regime, the spin di\u000busion con-\nstant hence scales as D/TD=2(up to logarithmic correc-\ntions in two dimensions owing to the energy dependence\nof the cross-section10). In the degenerate regime (of any\ndimensionality), the situation is quite di\u000berent: the de-\nBroglie wavelength \u0015dBis of order 1 =kFand hence the\nmean-free path is lmfp= 1=(n\u001b)\u00191=kF, wherekFis the\nFermi wave vector of the gas and n\u0018kD\nFis the density.\nHence, the spin di\u000busion constant is given by Planck's\nconstant ~divided by the mass mof the particle car-\nrying the spin. This quantum limit can also be viewedarXiv:1304.4980v1  [cond-mat.quant-gas]  17 Apr 20132\nas a result of the uncertainty principle by noticing that\nthe mean-free path is limited by the mean interparticle\nspacing11. The simple scaling argument, however, hides\nmuch of the rich underlying physics. In particular, it can-\nnot explain the Leggett-Rice e\u000bect13,14,17, the di\u000berence\nbetween longitudinal and transverse spin di\u000busivities18,\nand the transition to weak interactions where the physics\nchanges because the system evolves from hydrodynamic\nto collisionless. So far, lowest spin di\u000busion constant\nhas been measured for longitudinal spin currents to be\nDk= 6:3~=min three-dimensional degenerate Fermi\ngases at unitarity5, approximately two orders of mag-\nnitude smaller than in semiconductor nanostructures15.\nHere, we study the coherence properties of a trans-\nversely polarized two-dimensional Fermi gas under the\nin\ruence of magnetic \feld gradient. By employing the\nspin-echo technique12we gain access to the intriguing,\nbut little understood, transverse spin dynamics in two-\ndimensional Fermi systems16. The experiment starts\nwith a polarised Fermi gas, where all spins are in the\nsame statej#i. Three resonant rf-pulses are applied to\nrotate the spin state by respective angles \u0019=2 -\u0019-\u0019=2\nabout a \fxed axis in spin space, Sx. Consecutive pulses\nare separated by a time \u001cand we refer to the spin evolu-\ntion time as t= 2\u001c. The initial \u0019=2-pulse creates trans-\nverse spin polarisation in a coherent superposition be-\ntweenj#iandj\"i(see Fig. 1a). An applied magnetic\n\feld gradient B0\u0011@Bz=@x gives rise to a transverse\nspin wave (see Figure 1b) of wave vector Q=\u000e\rtB0,\nwhere\u000e\r= 152 kHz/G is the di\u000berential gyromagnetic\nratio betweenj\"iandj#iat 209 G and tis the evolution\ntime (see Methods). On time scales much shorter than\nthe trapping period, Qis independent of the interaction\nstrength19, and nearby atoms acquire a relative phase-\nangle of \u0001\u001e\u0019Q=kF. This lifts the spin-polarisation of\nthe Fermi gas and the spins can collide with each other\nas they move in the harmonic potential or di\u000buse. Triv-\nial dephasing induced by the magnetic \feld gradient is\nreversed by the \u0019-pulse and the spin state refocus after a\ntime\u001cif no decoherence has occured. The \fnal \u0019=2-pulse\nmaps the spin state from the SyontoSz, which is mea-\nsured by performing a Stern-Gerlach experiment in time\nof \right, and we record hMzi= (N\"\u0000N#)=(N\"+N#).\nEven though the interaction potential between atoms\nis not explicitly spin-dependent, an e\u000bective spin-\nexchange interaction is mediated by the required anti-\nsymmetrisation of the scattering wave function. In bi-\nnary collisions, this leads to the identical spin rotation\ne\u000bect20: in a collision both spins rotate about the axis\nde\fned by the sum of their spin orientations. In the\nFermi degenerate regime, the binary collision picture\nhas to be replaced by Landau's quasiparticle description.\nHere, quasiparticle excitations can be considered being\na\u000bected by a \\molecular \feld\" resulting from the e\u000bec-\ntive spin-exchange interaction13,14,21,22. As a result, the\nspin wave cants out of the Sx\u0000Sy{plane and forms a\nspin spiral which acquires a component along Sz(see\nFigure 1c). The magnitude of the identical spin rota-\n-1 0 1 30.000.501.00\n0 10 20 30Mz\n0.00.51.0\nSpin E volution Ti me        [ms]a\nbdc\nln(kFa2D)0.250.75 ln(kFa2D)=-0.2\nyk\nkx\nx x+dx\nFerm i surface mi xed state\nFerm i surface pol arized  compone ntFIG. 2. Spin-echo signals in the strongly interacting regime.\na, Spin echo-signal at ln( kFa2D) =\u00000:2. The blue line is a\n\ft/exp[\u0000(2\u0000\u001c)3].b, Transverse spin di\u000busion constant D?\nas a function of interaction strength deduced from the decay\nconstant \u0000 of the spin-echo signal. Illustration of the di\u000berent\nspatial variation of Fermi surfaces of a polarized Fermi gas for\nthe case of longitudinal ( c) and transverse ( d) spin di\u000busion.\ntion e\u000bect is proportional to the mean-\feld interaction\nstrengthg=\u00002\u0019~2=[mln(kFa2D)] of the gas. Here, kF\nis the Fermi wave vector for the initially spin-polarized\nsample, and a2Dis the two-dimensional scattering length,\nwhich is linked to the binding energy of the con\fnement-\ninduced dimer by23EB=~2=ma2\n2D.\nIn the strongly interacting regime, i.e., \u00001<\nln(kFa2D)<3, spin transport is dominated by di\u000bu-\nsion and the spin-echo signal attains a characteristic non-\nexponential decay of the form12{14/exp[\u0000(2\u0000\u001c)3]. Fit-\nting the envelope of the spin-echo signal24(see Figure\n2a) with this function we deduce the transverse spin\ndi\u000busion constant from the measured decay rate as12,25\nD?= 12 ~2\u00003=(\u000e\rB0)2. We extract the transverse spin\ndi\u000busion constant for various interaction strengths and\n\fndD?= 0:25(3) ~=mat a shallow minimum around\nln(kFa2D) = 0 (see Figure 2b). From the arguments\ngiven at the beginning, we expect the di\u000busivity to be\non the order of ~=m. Observing a smaller value than for\nlongitudinal spin di\u000busion in three dimensions5is prob-\nably less linked to the dimensionality rather than to the\nphase space available for collisions necessary to drive spin\ndi\u000busion18: In the case of longitudinal spin currents, a\ngradient of the Mz(x) polarization along the x-direction\ncan be considered as a spatial variation of the local Fermi\nsurfaceskF;\"(x) andkF;#(x) (see Fig. 2c). Only j\"ispins\nin a small region near the Fermi surface can di\u000buse from\nxtox+dx, invoking the typical T\u00002scaling for quasipar-\nticles in the deeply degenerate regime. In contrast, in the\ncase of transverse spin currents the Fermi surfaces at dif-\nferent positions are of the same size but have slightly dif-\nferent directions of magnetization. (see Fig. 2d). Hence,\na spin moving along xfrom anywhere between the Fermi\nsurfaces for spin-up and down must scatter to reach lo-\ncal equilibrium, which scales as n\"\u0000n#and provides a\nmuch larger phase space. The result is that for a degen-\nerate system the transverse di\u000busivity is smaller than the\nlongitudinal and becomes independent of temperature18.\nFor a repulsively interacting Fermi gas in three dimen-3\n−11 ky/kF\n−1 1−11 ky/kF\nkx/kF\n−11 ky/kF00.51\n00.10.20.3\n0 10 20 30 60a\nb\nd\nSpin E volutionT ime 2   [ms]40 50[kF]\n−1 1−11 ky/kF\nkx/kFc\nd\ncMz\nMz(x,y)\n|∆k|\nFIG. 3. Observation of spin wave for the non-interacting\ngas, i.e., ln( kFa2D)!+1:a, We plot the envelope of the\nspin-echo signal (dots) and show the \ft function (grey line).\nb, centre of mass separation of j\"iandj#iafter 11:8 ms of\ntime of \right. candd, momentum distribution of local spin\npolarisation for large and small spin polarisation, respectively.\nThe lower two panels in dshow the distribution of spin up\nand down, respectively.\nsions the decay of a spin-spiral state into a Stoner itin-\nerant ferromagnet was considered19. In this case, the\ngrowth rate of the instability towards the formation of\nthe ferromagnet order was found to scale /Q2and hence\nthe spin-polarisation signal would decay /exp[\u0000(\u000bt)3],\nsimilar to spin di\u000busion.\nIn the weakly interacting attractive Fermi liquid\nregime26, i.e., ln(kFa2D)>3, we observe a qualitatively\ndi\u000berent behaviour (see Figure 3a). The envelope of the\nspin-echo signal decays exponentially and is modulated\nby a slow oscillation and both the frequency and the\ndamping constant of the oscillation depend on the in-\nteraction strength. We attribute this slow periodic mod-\nulation to transverse spin waves which are excited by\nthe spin-echo sequence. The spin current induced by the\nmagnetic \feld gradient can only be inverted completely\nby the\u0019\u0000pulse when applied in phase with the harmonic\nmotion. This can be qualitatively (for zero interactions)\nunderstood in a phase-space picture by considering that\nthe magnetic \feld gradient causes a spatial o\u000bset between\nthe potentials for the j\"iand thej#ispin states and,\ncorrespondingly, the phase-space trajectories of the two\nspin-components are displaced9. Indeed, we observe that\nthe di\u000berence of the center-of-mass momenta hk\"i\u0000hk#i\nof the two spin densities n\"(k) andn#(k) oscillates out-\nof-phase with the contrast of the spin-echo signal (Figure\n3a and 3e). The total density pro\fle n(k) =n\"(k)+n#(k)\nis stationary, which indicates a pure spin mode.\nThe measured envelope of the spin-echo signal (see\nFig. 3a) is \ftted with an empirical function of the form:\nAexp(\u0000\rt)\u0000\n1\u0000Bjsin(!t=2)j3\u0001\n+C. Here,jAj;jBj \u0018\nSpin Echo Time (ms)\nPolarization M\n1 10 100 10001010002040\nb\nc\nln(kFa2D)0.000.250.500.751.00\n[Hz]\n[Hz]a\n900\n35.77.59.713182539\nln(k Fa2D)0\n10\n30\n40\n50\n6020\nSpin- Evolu tion\nTime 2   [ms]MzFIG. 4. Spin wave in the collisionless regime as function of in-\nteraction strength. a, Time evolution of global spin polariza-\ntion for di\u000berent interaction strengths. b, Frequency of oscil-\nlating spin-polarization. For the data point at ln( kFa2D= 3)\nwe were not able to \ft a frequency and set the value to zero.\ncDamping coe\u000ecient \rof the spin-polarization. The dashed\nline indicates the elastic scattering rate and the solid line is a\n\ft to the measured decay.\nO(1) andjCj \u001c 1 are amplitudes and a global o\u000bset\nof the spin echo signal, respectively. We \fnd that this\nperiodic function of frequency !is approximating the\nnon-sinusoidal signal best. The exponent is \fxed to the\nvalue 3 for all interactions, and we do not see a large in-\n\ruence on the deduced parameters. Both the frequency\n!and the exponential damping \rdepend on the interac-\ntion strength of the gas.4\nFor zero interaction strength the mode is at half the\ntrap frequency and we observe a softening of the spin\nmode when approaching the hydrodynamic regime (Fig-\nure 4a and b). Such a behaviour is reminiscent of\nthe collisionless-to-hydrodynamic transition of the spin-\ndipole oscillation5,27, whereas spin-symmetric modes\nhave non-zero frequencies even in the hydrodynamic\nregime28. The decay rate \rof the oscillation (see Figure\n4c) is found smallest for the non-interacting gas, where\nit is two orders of magnitude smaller than the dephasing\nrate 1=\u001cRamsey in the inhomogeneous magnetic \feld (see\nMethods). As the interaction strength is increased, the\nrate\ris growing (see Fig. 4c). We \ft this growth with\na power-law behaviour \r=\u000b=ln(kFa2D)\f+\r0and we\ndetermine the exponent to be \f= 1:3\u00060:15. Observ-\ning a scaling with an exponent close to unity signals that\nthe loss of global coherence is dominated by the identi-\ncal spin-rotation e\u000bect, which scales proportional to the\nmean-\feld interaction strength g. In contrast, the rate\nof elastic collisions scales as /1=ln(kFa2D)2, i.e.,\f= 2\n(see dashed line in Fig. 4c). This suggest the following\ndecoherence mechanism in a spin-1/2 Fermi gas: after\nbuilding up in the magnetic \feld gradient, the spin wave\ndecays when neighbouring atoms have acquired a su\u000e-\nciently large phase-angle \u0001 \u001ebetween their spins such\nthat in a spin-conserving collision both spins rotate out\nof theSx\u0000Sy-plane. Initially, this is a coherent pro-\ncess, which subsequently dephases due to the coupling of\nthe spin-rotation to the momentum of the atoms29. This\npicture contrasts to the previously conjectured mecha-\nnism of the decoherence of a spin mixture resulting from\na magnetic \feld curvature3,30, which has been probed,\nfor example, using the Ramsey technique30. Our spin-\necho measurements show that the dephasing time mea-\nsured by the Ramsey technique underestimates than the\nactual spin decoherence time in the weakly interacting\nregime by orders of magnitude, a\u000becting the question of\nthe equilibration of the Fermi gas.\nWe thank E. Altman, E. Demler, U. Ebling, A.\nEckardt, C. Kollath, M. Lewenstein, A. Recati, W. Zw-\nerger for discussions and B. Fr ohlich and M. Feld for early\nwork on the experimental apparatus. The work has been\nsupported by EPSRC (EP/J01494X/1, EP/K003615/1),\nthe Leverhulme Trust (M. K.), the Royal Society, the\nWolfson Foundation, and an Alexander-von-Humboldt\nProfessorship.\nCorrespondence and requests for materials should be\naddressed to M. K. (email: mk673@cam.ac.uk) or M.\nK ohl (email: michael.koehl@uni-bonn.de).\nI. METHODS\nA. Preparation of the two-dimensional gases\nWe prepare a quantum degenerate Fermi gas of40K\natoms in a one-dimensional optical lattice of wavelength\u0015= 1064 nm populating a stack of approximately 40 in-\ndividual two-dimensional quantum gases31,32. The radial\ncon\fnement of the two-dimensional gases is harmonic\nwith a trap frequency of !r= 2\u0019\u0002127 Hz and the ax-\nial trap frequency is !z= 2\u0019\u000275 kHz. Using a radio-\nfrequency cleaning pulse, we ensure preparation of a spin-\npolarized gas in the jF= 9=2;mF=\u00009=2i\u0011j#i ground\nstate with 1 :5\u0002105atoms, corresponding to the Fermi\nenergy of the spin-polarised gas EF= 12:3(2:0) kHz at\na temperature of kBT=EF= 0:24(3). The coupling\nstrength of a two-dimensional Fermi gas is given by the\nparameterg=\u00002\u0019~2=[mln(kFa2D)]23. Here,kFis the\nFermi wave vector for the initially spin-polarized sample,\nanda2Dis the two-dimensional scattering length which is\nlinked to the binding energy of the con\fnement-induced\ndimer byEB=~2=ma2\n2D.\nOur experiments on the spin dynamics start by ramp-\ning the magnetic \feld from B= 209:15 G to the desired\nvalueBnear the Feshbach resonance in 30 ms and wait\nfor 200 ms. Subsequently we apply the spin-echo pulse\nsequence (see main text) of three radio-frequency pulses\nwith frequencies of \u001850 MHz. The pulses have a square\nenvelope and duration of t\u0019=2\u001846\u0016s for a\u0019=2-pulse and\n2\u0002t\u0019=2for a\u0019-pulse. The pulses are separated by the\nsame time \u001cwhich gives the total spin evolutions time\nt= 2\u001c.\nB. Ramsey measurements: Dephasing in the\nmagnetic \feld gradient\nWe assess the time scale of simple dephasing of the\nspin in the magnetic \feld gradient by performing Ramsey\nspectroscopy on the non-interacting gas using the follow-\ning sequence: the initial \u0019=2-pulse is followed by a sec-\nond\u0019=2-pulse with a variable time delay \u001c. We observe a\nRamsey coherence time of \u001cRamsey = (600\u000630)\u0016s. Since\nthe Ramsey time is more than an order of magnitude\nshorter than the inverse trap frequency, we neglect the\nmotional contribution to the dephasing. Instead, we re-\nlate the Ramsey time to the dephasing time of the (quasi\nstationary) spins across the whole cloud, and we \fnd from\nthis the magnetic \feld gradient. We \ft the data with\nP(t) =8NJ2(\u001e)\n\u001e2Where,\u001e= 2\u0019\u0001t=d\rB0RFt. \u0001 is the\nZeeman shift due to the gradient at the edge of the cloud\ncompared to the center. The \ft function \fts \u001e=bt, i.e.,\nb= 2\u0019\u0001. The \ft results gives b= 8:1(1) kHz and we\nknowd\r= 152 kHz/G at 209 :15 G andRF= 17:7\u0016m.5\n1D. D. Awschalom, D. Loss, and N. Samarth, eds.,\nSemiconductor Spintronics and Quantum Computation ,\nNanoScience and Technology (Springer, 2007).\n2S. D. Gensemer and D. S. Jin, Phys. Rev. Lett. 87, 173201\n(2001).\n3X. Du, L. Luo, B. Clancy, and J. E. Thomas, Phys. Rev.\nLett. 101, 150401 (2008).\n4X. Du, Y. Zhang, J. Petricka, and J. E. Thomas, Phys.\nRev. Lett. 103, 010401 (2009).\n5A. Sommer, M. Ku, G. Roati, and M. W. Zwierlein, Na-\nture472, 201 (2011).\n6A. Sommer, M. Ku, and M. W. Zwierlein, New Journal of\nPhysics 13, 055009 (2011).\n7G. M. Bruun, New Journal of Physics 13, 035005 (2011).\n8D. Wulin, H. Guo, C.-C. Chien, and K. 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Ketterle, Science 300, 1723 (2003).\n31B. Fr ohlich, M. Feld, E. Vogt, M. Koschorreck, W. Zw-\nerger, and M. K ohl, Phys. Rev. Lett. 106, 105301 (2011).\n32M. Feld, B. Fr ohlich, E. Vogt, M. Koschorreck, and\nM. K ohl, Nature 480, 75 (2011)."    },    {        "title": "1109.2410v2.Spin_Excitation_Assisted_by_Non_Softening_Phonon_for_Spin_Peierls_Model.pdf",        "content": "arXiv:1109.2410v2  [cond-mat.str-el]  2 Dec 2011Spin Excitation Assisted by Non-Softening Phonon\nfor Spin-Peierls Model\nTakanori Sugimoto,∗Shigetoshi Sota, and Takami Tohyama\nYukawa Institute for Theoretical Physics, Kyoto Universit y, 606-8502, Kyoto, Japan\n(Dated: October 1, 2018)\nWe study spin dynamics of a spin-Peierls chain with nearest- neighbor and next-nearest-neighbor\nHeisenberg spin exchange interactions together with a gapp ed and dispersionless phonon. The dy-\nnamical spin correlation functionand phononexcitation sp ectrumare calculated atzerotemperature\nby using dynamical density-matrix renormalization-group method. We ﬁnd a new spin excitation\nassisted bynon-softening phonon. The excitation is locate d above phonon in energy and shows a dis-\npersive feature with strong intensity near the momentum π. The phonon excitation spectrum is also\ninﬂuenced by the spin-phonon interaction. We discuss the po ssibility of observing the spin-phonon\ncoupled features in inorganic spin-Peierls compound CuGeO 3.\nI. INTRODUCTION\nOne-dimensional (1D) quantum spin system coupled\nwith lattice degree of freedom has been extensively stud-\nied experimentally and theoretically, since the systems\nprovide a playground of spin-Peierls transition. Conven-\ntional spin-Peierls compounds like organic materials ex-\nhibit the transition with spin dimerization and lattice al-\nternationaccompaniedbysoft-phononmode. [1, 2] Theo-\nretically the spin-Peierls transition can be derived under\nthe presence of soft-mode phonon by using the random\nphase approximation approach in the adiabatic limit. [3]\nThediscoveryofCuGeO 3,[4]however,hascastedaprob-\nlem on the conventional mechanism of the spin-Peierls\ntransition, since the soft-phonon mode associated to lat-\ntice alternation has never been found in CuGeO 3so\nfar. [4–7]\nA recent study has shown a theoretical explanation of\nspin-Peierlsinstability in the antiadiabaticlimit. [8] The\nspin-Peierls Hamiltonian can be mapped approximately\ntospin-1\n2J1-J2HeisenbergHamiltonianbytheﬂowequa-\ntionintheantiadiabaticlimit, where J1andJ2isnearest-\nneighbor (NN) and next-nearest-neighbor (NNN) ex-\nchange interactions, respectively. The spin-Peierls tran-\nsition, therefore, corresponds to the Kosterlitz-Thouless\n(KT) transition in the J1-J2Heisenberg model obtained\nby changing the ratio of J1andJ2. The KT transi-\ntionisorder-disordertype quantumphasetransition, and\nhas a critical exchange ratio, αc∼=0.241167, between\nspin-liquid and dimer phases at the ground state. [9–11]\nNamely, for the exchange ratio J2/J1≡α < α c, the\nground state belongs to the spin-liquid phase, but for\nα > α c, it is in the dimer phase. The renormalization of\nphonon degree of freedom by the ﬂow equation changes\nthe ratio α:αincreases as spin-phonon coupling gin-\ncreases. Thus, there occurs the KT transition without\nsoft phonons in the antiadiabatic limit for large spin-\nphonon coupling g > gc∼=0.8. [12–18] This means that\nno soft phonons exist below the critical temperature. [8]\n∗takanori@yukawa.kyoto-u.ac.jpThe spin-Peierls transition was experimentally ob-\nserved for the organic compounds, tetrathiafulvalene\n(TTF), tetracyanoquinodimethane (TCNQ) series in\nthe 1970s, [1, 2, 19–22] and the inorganic compounds\nCuGeO 3in 1993. [4] Experimental studies have shown\nthat phonon frequency associated with lattice distor-\ntion in CuGeO 3is higher than that in the organic com-\npounds. In (TTF)CuS 4C4(CF3)4, a soft-phonon fre-\nquency is estimated to be ωph∼=1.4meV∼=16K, and\nspin-Peierls gap is ∆ ∼=21K. [1, 19] On the other hand,\nfor CuGeO 3, a dispersive phonon frequency related to\nlattice distortion is ωph∼=6.8THz∼=330K, a disper-\nsionless phonon is ωph∼=3.2THz∼=150K, and the gap\nis ∆∼=2meV∼=23K. [5, 23–25] A condition ωph<∆\nfor the conventional spin-Peierls mechanism is fulﬁlled in\nthe organic compounds, but not in CuGeO 3. Therefore,\nthe spin-Peierls transition in CuGeO 3can be explained\nby the theory in the antiadiabatic limit.\nCuGeO 3has an antiferromagnetic NN interaction J1\nand an antiferromagnetic NNN interaction J2that in-\nduces frustration. Experimental and theoretical studies\nhave shown that J1≈100K and α∼=0.36. [5–7] The\ninstability of the spin-Peierls phase due to J2has been\ndiscussed by Weiße [26].\nSpin excitations in CuGeO 3have been observed by in-\nelastic neutron scattering. [27–29] The separation of spin\nand phonon excitations in the experiments has not been\ncomplete, althoughsomeofphononexcitationshavebeen\nidentiﬁed by the same technique. [23–25] A recent devel-\nopment of polarized neutron scattering may resolve this\nproblem in the near future. To give theoretical supports\nfor inelastic polarized neutron scattering experiment, we\ninvestigate spin excitation of the spin-Peierls model with\nthe non-softening phonon. Our chief aim in the present\npaper is to clarify the eﬀect of non-softening phonon on\nspin excitations in CuGeO 3.\nWe use the dynamical density-matrix renormalization-\ngroup (D-DMRG) method to calculate dynamical\nspin correlation function and phonon excitation spec-\ntrum. This method is a dynamical version of DMRG\nmethod [30] presented by Jeckelmann [31].\nIn this paper, we perform D-DMRG calculation of the2\ndynamical spin correlation function for both the spin-\nPeierls model and its eﬀective spin model ( J1-J2model).\nTreating phonons in the spin-Peierls model as quantum\nobjects, we ﬁnd a new spin excitation assisted by non-\nsoftening phonon. There is no corresponding structure\nin the eﬀective J1-J2model. The new structure shows\na dispersive feature with strong intensity near the mo-\nmentum π, and it is located above the phonon energy.\nThe new structure is explained by using particle-hole ex-\ncitation assisted by phonon in the Su-Schrieﬀer-Heeger\nmodel onto which the spin-Peierls model with XY-type\nspin is mapped. The phonon excitation spectrum is also\ninﬂuenced by the spin-phonon interaction. Main peaks\nof phonon near the momentum πshow a new tail struc-\nture whose energy range is the same as the new spin\nexcitation. In addition, spin-assisted phonon structures\nappear at low-energy region. These features induced by\nspin-phonon interaction are expected to be observed in\ninelastic neutron scattering experiments in the near fu-\nture.\nThis paper is organized as follows. In Sec. II, we show\na model Hamiltonian of the frustrated spin-Peierls chain\nwith Einstein phonon and introduce a renormalized J1-\nJ2Hamiltonian. We explain the numerical method, D-\nDMRG, in Sec. III. In preparation for the spin-Peierls\nmodel, we demonstrate dynamical spin correlation func-\ntion in the J1-J2model in Sec. IVA. Dynamical spin\ncorrelation function and phonon excitation spectrum of\nthe spin-Peierls model are presented in §IVB. Summary\nof the present paper is given in Sec. V.\nII. MODEL\nWe consider the following model Hamiltonian that de-\nscribes 1D frustrated spin-Peierls chain.\nH=Hs+Hp+Hsp (1)\nwith\nHs=J/summationdisplay\njh(1)\nj(∆)+α0J/summationdisplay\njh(2)\nj(∆),(2)\nHp=/summationdisplay\njp2\nj\n2M+K\n2x2\nj, (3)\nHsp=g/summationdisplay\nj(xj−xj+1)h(1)\nj(∆), (4)\nand\nh(r)\nj(∆) =Sj·Sj+r−∆Sz\njSz\nj+r. (5)\nwhereSjis thej-site spin operator for S=1\n2,xjis\nthej-site displacement of the coordinate with respect to\nequilibrium position, and pjis the conjugate momentum\noperator. Jis bare NN exchange interaction, and α0is\nthe ratio of bare NN and NNN interactions. In this work,\nwe consider only antiferromagnetic coupling for the NNand NNN interactions. MandKare the eﬀective mass\nand the elastic coupling constant, respectively. gis the\nspin-phononcouplingconstant. ∆isthe Isinganisotropic\nparameter.\nAfter the second quantization of lattice degree of free-\ndom, the phonon part of the Hamiltonian ( 3) and the\nspin-phonon coupling term ( 4) are rewritten by\nHp=ω0/summationdisplay\njb†\njbj (6)\nand\nHsp=λJ\n2/summationdisplay\nj/parenleftBig\nbj+b†\nj−bj+1−b†\nj+1/parenrightBig\nh(1)\nj(∆),(7)\nrespectively, where b†\njandbjare the creation and annihi-\nlation operatorof j-site phonon, respectively. We employ\nthe Einstein phonon with frequency ω0=/radicalbig\nK/M. The\ndimensionless spin-phonon coupling constant λis deﬁned\nasλ= (g/J)/radicalbig\n2/Mω0. We use the natural unit, /planckover2pi1= 1.\nThespin-PeierlsmodelwithouttheNNNinteractionin\ntheXYlimit (α0= 0 and ∆ = 1), that we call XYspin-\nPeierlsmodel, canbe mappedtotheSu-Schrieﬀer-Heeger\n(SSH) model [40] by the Jordan-Wigner transformation,\nHe=−J/summationdisplay\nkcos(k)˜c†\nk˜ck (8)\nand\nHep=iλJ/summationdisplay\nk,l[sin(k)−sin(l)](˜bk−l+˜b†\nl−k)˜c†\nk˜cl,(9)\nwhere˜bqand˜b†\nqare the momentum representation of the\nphonon creation and annihilation operators. ˜ c†\nkand ˜ck\nare the momentum representation of the spinless-charge\ncreation and annihilation operators.\nAn eﬀective Hamiltonian with Heisenberg spin ex-\nchange interactions (∆ = 0) after renormalization of\nphonon degree of freedom by the ﬂow equation is given\nby [8]\nHJ1J2=J1N/summationdisplay\nj=1Sj·Sj+1+J2N/summationdisplay\nj=1Sj·Sj+2(10)\nwith\nJ1=J/bracketleftbigg\n1+λ2J\n4ω0−3(1−α0)λ2J2\n8ω2\n0/bracketrightbigg\n+O(λ3),(11)\nJ2=J/bracketleftbigg\nα0+λ2J\n8ω0+(3−5α0)λ2J2\n8ω2\n0/bracketrightbigg\n+O(λ3),(12)\nwhereλJ/ω0andJ/ω0are small parameters for expan-\nsion and O(λ3) is the third or higher order terms of λ.\nEquation ( 10) is aJ1-J2model with eﬀective NN and\nNNN interactions. In this model, with increasing the ra-\ntioα=J2/J1from zero, the KT transition from spin\nliquid phase to dimer phase occurs at αc. [9–11] In the3\nspin-liquid phase where α < α c, a gapless excitation ap-\npears. On the other hand, in the dimer phase where\nα > α c, there is a spin gap between the ground state\nand the lowest triplet excitation, accompanied by spon-\ntaneouslydimerization ofspin pairs. The quantum phase\ntransition does not require any soft phonon.\nIII. METHOD\nTo examine dynamical behavior for the spin-Peierls\nmodel, we calculate two quantities, dynamical spin cor-\nrelation function and phonon excitation spectrum.\nThe dynamical spin correlation function at zero tem-\nperature is given by\nχs(k,ω) =ℑ1\nπL/angbracketleft0|Sz(k)1\nω+ǫ0−H+iγSz(k)|0/angbracketright,(13)\nwhereLis the system size, γis the damping factor with\nsmall positive number, and |0/angbracketrightis the ground state. Sz(k)\nis thezcomponent of the spin operator at momentum k.\nWe deﬁne phonon excitation spectrum as\nχp(q,ω) =−ℑ1\nπL/angbracketleft0|b(q)1\nω+ǫ0−H+iγb†(q)|0/angbracketright,(14)\nwhereb(q) andb†(q) arethe phononannihilationandcre-\nationoperatorsatmomentum q, respectively. In eq. ( 14),\nwe consider a pair of b(q) andb†(q). In the phonon\nGreen’s function, there are additionally three pairs of\nb†(q) andb(q),b(q) andb(q), andb†(q) andb†(q). If the\nspin-phonon coupling is small, there are few phonons in\nthe ground state. In such a case, the contribution from\nthe three pairs is expected to be small. Therefore, we\nconsider only the case of eq. ( 14) to represent the phonon\nexcitation spectrum.\nFor strongly-correlated 1D systems, it is well-known\nthat the DMRG method can provide a good numerical\nsolution of the ground state. [30] The dynamically ex-\ntended version of DMRG, D-DMRG, is also suitable for\nobtaining dynamical properties at zero temperature. [31]\nSince the spin-phonon interaction ( 7) breaks the mir-\nror symmetry, we consider two reduced density matri-\nces for both the system and the environment blocks in\nthe DMRG process. We use the inﬁnite-size algorithm\nof DMRG [30] for systems with open boundary condition\n(OBC). In OBC, the Fourier transform of the spin and\nphonon operators read\nSz(k) =/radicalbigg\n2\nL+1L/summationdisplay\nj=1Sz\njsin(jk), (15)\nb(q) =/radicalbigg\n2\nL+1L/summationdisplay\nj=1bz\njsin(jq), (16)\nwith momenta kandqgiven by nπ/(L+ 1), (n=\n1,2,···,L).For the D-DMRG method, we use three target\nstates:|0/angbracketright,Sz(k)|0/angbracketright, and [ω+ǫ0−H+iγ]−1Sz(k)|0/angbracketright\nfor eq. ( 13), and |0/angbracketright,b†(q)|0/angbracketright, and\n[ω+ǫ0−H+iγ]−1b†(q)|0/angbracketrightfor eq. ( 14). In the D-\nDMRG procedure, we use a modiﬁed version of the\nconjugate gradient method to calculate the correction\nvector, [ω+ǫ0−H+iγ]−1ˆO|0/angbracketright.\nIV. RESULTS\nIn this section, we ﬁrst show calculated results of dy-\nnamical spin correlation function for the eﬀective J1-J2\nmodel (10) deduced from the spin-Peierls Hamiltonian\n(1). Secondly, treating phonons quantum-mechanically,\nwe calculate the dynamical spin correlation function of\nthe spin-Peierls model ( 1). New structures originated\nfrom the quantum phonons are identiﬁed by making a\ncomparison with the J1-J2model. Finally the eﬀect of\nspin-phonon coupling on phonon excitation spectrum is\nexamined.\nIn order to simulate CuGeO 3, we take the phonon en-\nergyω0= 3J, [5, 23–25] except in the case explicitly\nprovided. The value of spin-phonon coupling is not clear\nfor CuGeO 3. Therefore, we take λsatisfying λJ/ω0<1.\nA. Eﬀective J1-J2model\nThe dynamical spin correlation function for a 64-site\nJ1-J2Heisenberg chain is shown in Fig. 1. The damp-\ning factor γin eq. (13) is taken to be 0 .1J. We note\nthat a preliminary result has been reported in ref. 32).\nIn Fig.1(a), we take α0= 0 and λ= 1.12, resulting in\nJ2/J1= 0.1 from eqs. ( 11) and (12). The ratio is below\nαc. This means that Fig. 1(a) represents the dynamical\nspin correlation in a spin-liquid phase. The distribution\nof spectral weight is similar to the exact results of the\n1D Heisenberg model ( J2/J1= 0) where spectral weight\nconsistsofthe des Cloizeaux-Pearsonmode at the lowest-\nenergy branch and multi-spinon continuum. [33–35] The\ninsetin Fig. 1(a)showsthe system-sizedependence ofthe\nposition of the des Cloizeaux-Pearsonmode at the small-\nest momentum k=π/(L+ 1). A ﬁtting function gives\nnearly zero excitation energy at 1 /L→0 as expected in\nthe spin-liquid phase.\nThespincorrelationfunction inadimerphaseisshown\nin Fig.1(b), where α0= 0.36, [5–7] λ= 1.06, and thus\nJ2/J1= 0.4. We can ﬁnd two characteristics as com-\nparedwithFig. 1(a), i.e., strongintensityaround k=π/2\n(ω/J∼1) and a peak structure at the upper edge of the\nspinon continuum around k=π(ω/J∼2). These fea-\ntures in the dimer phase have been reported in a previ-\nous study for small systems up to 16 sites under periodic\nboundary condition. [36, 37] The inset shows the pres-\nence of spin gap in the thermodynamic limit. The gap\nmagnitude ( ∼0.08J) is similar to a previous report eval-\nuated from DMRG calculations. [38] These results also4\nFIG. 1. (Color online) Intensity map of dynamical spin cor-\nrelation function (SCF) in a 64-site J1-J2chain under two\nconditions: (a) J2/J1= 0.1 (ω0/J= 3,α0= 0,λ∼=1.12),\n(b)J2/J1= 0.4 (ω0/J= 3,α0= 0.36,λ∼=1.06). The inset\nin both panels shows system-size dependence of the energy of\npeak position at the smallest momentum k=π/(L+1). Blue\nsquares represent calculated peak positions, and red lines de-\nnote ﬁttingby a function with a power: f(1/L) =a(1/L)b+c.\na= 5.63,b= 0.90, and c=−0.01 for (a), and a= 7.43,\nb= 1.14, andc= 0.07 for (b). The gap in (b) remains ﬁnite\nin the thermodynamical limit 1 /L→0. The truncation num-\nber of DMRG and the broadening factor of Lorentzian are set\nto bem= 100 and γ= 0.1J, respectively.\nconﬁrm the validity of our D-DMRG calculation.\nThere is a breakpoint near k=π/2 in the lowest-\nenergy branch in Fig. 1(b). This is an artifact by ﬁnite-\nsize eﬀect under OBC, since momenta near k=π/2 have\na substantial contribution from the edges of the system\nas expected from eq. ( 15). This, however, does not occur\nfor momenta close to k= 0 and close to k=π, for\nwhich sin( jk) in eq. ( 15) goes to zero with approaching\nthe edges of the system.\nB. Spin-Peierls model\nWe calculate dynamical spin correlation function and\nphonon excitation spectrum for a 16-site frustrated spin-\nPeierls chain in this section. The system size is smaller\nthan that for the J1-J2model (64 sites). This is becausesingle-site dimension of spin-phonon coupled system is\nseveraltimes largerthan that of pure spin system, result-\ning in the requirement of huge computational resources.\nWe checkedthe convergenceof the calculation in terms\nof two parameters, i.e., the DMRG truncation number,\nm, and the maximal phonon number per site, np. We\nfound that good convergence not only for the ground\nstate but also for spectral weight at high-energy region\ncan be achieved for m= 100 and np= 2. In this paper,\nweusethe broadeningfactorforthe dynamicalquantities\nγ= 0.1J.\n1. Dynamical spin correlation function\nFigure2(a) shows the case without spin-phonon cou-\npling, i.e., only Hsterm in eq. ( 2) withα0= 0.36. This\nis nothing but a 16-site result of the J1-J2model, and\nthe spectral behavior is consistent with the 64-site result\nwith spin gap mentioned in §IVA.\nThe introduction of λchanges the spectrum as shown\nin Fig.2(b), where λJ/ω0= 0.5. The most striking\nchange is the appearance of high-energy spectral weight\naroundω∼4J(see red solid lines and intensity map\nshown in the inset). The intensity is strongest at k=π\nand widens toward low-energy side with decreasing kfol-\nlowed by an energy minimum at k=π/2. The energy\nposition is higher than the dispersionless phonon located\natω0= 3J. From this result, it is expected that addi-\ntional spin excitations originating from the spin-phonon\ncoupling exist above the energy of the phonons. Note\nthat the intensity of the new structure increases with in-\ncreasing λ.\nComparing red solid lines in Fig. 2(b) with those in\nFig.2(a), we ﬁnd a change of spectral distribution along\nthe lowest energy branch: the spectral intensity at k=π\nis suppressed and the weight transfers toward k=π/2.\nThischangeis similartothat causedby increasing J2/J1.\nThis is reasonable, since the eﬀective value of J2/J1eval-\nuated from the second-order expression of λin eqs. (11)\nand (12) isα=J2/J1= 0.44, which is larger than the\nbare value α0= 0.36. For comparison, the spin cor-\nrelation function for the J1-J2model with α= 0.44 is\nplotted in Fig. 2(b) as blue dotted lines. The lowest-\nenergy branch follows that of the spin-Peierls model.\nThis tempts us to justify the use of eﬀective J1-J2model.\nHowever, we can ﬁnd a qualitative diﬀerence between\ntheJ1-J2model and the spin-Peierls model: the upper\nedge of multi-spinon excitation ( ∼2.5Jneark=π) in-\ncreases in the spin-Peierls model while decreases in the\nJ1-J2model ascomparedwiththat ofFig. 2(a). Asimple\nexplanation of the diﬀerence would be that the second-\norder contributions in eqs. ( 11) and (12) are not enough\nfor the complete description of the spin-Peierls model\nand higher-order terms contributes signiﬁcantly for the\npresent parameter set.\nFigure2(c) shows the case of α0= 0 but with the same\nλas Fig.2(b). As is the case of Fig. 2(b), there appears5\nFIG. 2. (Color online) Dynamical spin correlation function\n(SCF) in a 16-site spin-Peierls chain. The phonon energy\nω0= 3J. (a)α0= 0.36,λJ/ω0= 0, (b) α0= 0.36,λJ/ω0=\n0.5, and (c) α0= 0,λJ/ω0= 0.5. The blue dotted lines\nin (b) and (c) show the dynamical spin correlation function\nof the eﬀective J1-J2model, where J1andJ2are evaluated\nfrom eqs. (11) and (12), respectively: J2/J1= 0.44 in (b)\nandJ2/J1= 0.17 in (c). The inset in each panel shows the\nintensity map of the correlation function. The truncation\nnumber of DMRG is set on m= 100 and maximal phonon\nnumber is two. The broadening factor is set to be γ= 0.1J.\nFIG. 3. (Color online) Dynamical spin correlation function\nin a 16-site spin-Peierls chain with α0= 0.36 andω0/J=\n1.5. The red lines represent the case of λJ/ω0= 0.5, while\nthe blue dotted lines represent the case without the couplin g.\nThe inset shows intensity map of the correlation function fo r\nλJ/ω0= 0.5.\na high-energy dispersive structure with small intensity\naroundω= 4Jinduced by the spin-phonon coupling in\nthe spin-Peierls model. The upper edge of multi-spinon\nexcitations ( ω <3J) in the spin-Peierls model is larger\nthan that of the eﬀective J1-J2model. This is again the\nsame as Fig. 2(b), indicating insuﬃcient mapping of the\nspin-Peierls model onto the J1-J2model for the present\nparameter set. We ﬁnd small spectral weights below the\ndes Cloizeaux-Pearson mode. The weights are caused by\nﬁnite-size eﬀect and decreasewith increasingsystem size.\nIn order to examine the case where phonons are in-\nside the multi-spinon continuum, we take ω0/J= 1.5\n(refs. 23-25) and show the spin correlation function for\nλJ/ω0= 0.5 (red solid lines) in Fig. 3. Making a com-\nparison with the case without the coupling (blue dot-\nted lines), we ﬁnd that phonon-induced spin excitations\nappear at ω∼2.5Jwith a dispersive structure show-\ning a minimum at k=π/2. We note that, although\nthe coupling constant λ= 0.75 is smaller than the cases\nof Fig.2(b) (λ= 1.5), the spectral intensity relative to\nmulti-spinon continuum is comparable to Fig. 2(b). This\nprobably comes from the enhancement of hybridization\nbetween spin and phonon due to overlapping of their en-\nergy scale.\nSince the phonon-induced spin excitation appears in\nbroad energy range, it is expected to be coupled with the\nmulti-spinon continuum. However, in the dimer phase\nof the spin-Peierls models, there is not only the contin-\nuum but also the state with large weight at the lower\nedge of the continuum. In order to make clear how the\ncontinuum contributes to the phonon-induced spin exci-\ntation, we examine the XYspin-Peierls model ( α0= 0\nand ∆ = 1 in eqs. ( 2) and (7)), where the XYspin chain6\nFIG. 4. (Color online) Dynamical spin correlation function in\na 16-site XYspin-Peierls chain (or dynamical charge-charge\nfunction in a 16-site Su-Schrieﬀer-Heeger chain). The inse t\nshows intensity map of the correlation function for λJ/ω0=\n0.5 andω0/J= 3.0.\nhas only continuum excitations of spinons. [37] Figure 4\nshowsthedynamicalspincorrelationfunctionforthe XY\nspin-Peierlsmodelwith λJ/ω0= 0.5andω0/J= 3.0. We\ncan see the same structure induced by the spin-phonon\ncoupling in the high energyregion as the case of the spin-\nPeierls model. Therefore, we conclude that the phonon-\ninduced spin excitation couples with the continuum of\nspinons.\nMoreover, we investigate the λandω0dependence\nof the phonon-induced spin excitation in the XYspin-\nPeierls model. We show λdependence of the dynami-\ncal spin correlation function near k=πwithλJ/ω0=\n0,1/6,1/3, and 1/2 for ﬁxed ω0/J= 3.0 in Fig. 5(a),\nandω0dependence with ω0/J= 2.0,3.0, and 4 .0 for\nﬁxedλJ/ω0= 0.5 in Fig. 5(b). We ﬁnd that the en-\nergy position of the excitation does not depend on the\nspin-phonon coupling λ, but the integral of the excita-\ntion depends on λwith a power-law behavior as seen in\nthe inset of Fig. 5(a). In Fig. 5(b), we can see that the\nexcitation is situated between ω0andω0+2Jatk∼=π.\nThe behaviors in the dynamical spin correlation func-\ntion in the XYspin-Peierls model can be explained by\nthe SSH model in eqs. ( 8) and (9). We assume that\nphonon creation and annihilation operators never have\nany ﬁnite expectation values in the ground state. This\nassumption is supported by the well-known fact that the\nground state is not in the dimer phase with the bond\nalternation for small coupling constant λ. [41] The spin\ncorrelation function is rewritten by the spinless charge-\ncharge correlation function. We obtain the imaginary\npart of the spinless charge-charge correlation function,\nχc(k,ω), within the second-order perturbation in terms2.0 6.000.32\n0.5 00.06χc(k = 16π/17, ω)\nλJ/ω0=1/6λJ/ω0=0\nλJ/ω0=1/3\nλJ/ω0=1/2 λJ/ω0integrated\n weight\nω/J3.0 4.0 5.000.16\nω0/J=3.0ω0/J=2.0\nω0/J=4.0(a)\n(b)χc(k = 16π/17, ω)\nω0 ω0+J ω0+2Jχ(2a)(π,ω)\n+χ(2b)(π,ω)a.u.\nFIG. 5. (Color online) Dynamical spin correlation function\nin a 16-site XYspin-Peierls chain at k= 16π/17∼=π. (a)\ntheλdependence. The inset shows the λdependence of the\nintegrated weight in the energyrange of3 Jand 6Jforω0/J=\n3.0. The bluecrosses representthe integrated weight obtaine d\nbysubtractingtheintegrated weight at λ= 0, andthered line\ndenotes the power-law ﬁtting, a(λJ/ω0)bwitha= 4.4×10−2\nandb= 1.8. (b)ω0dependence for λJ/ω0= 0.5. The inset\nrepresents χ(2a)\nc(π,ω)+χ(2b)\nc(π,ω) obtained by eqs. (19) and\n(20).\nofλ:\nχc(k,ω) =χ(0)\nc(k,ω)+χ(2a)\nc(k,ω)+χ(2b)\nc(k,ω)+O(λ4)\n(17)\nwith\nχ(0)\nc(k,ω) =/summationdisplay\nlδ(ω−ǫl+k+ǫl)θ(−ǫl)θ(ǫl+k)(18)\nχ(2a)\nc(k,ω) =/parenleftbiggλJ\nω0/parenrightbigg2/summationdisplay\nlq/parenleftbiggω0B(l+k,l+k+q)\nω0−ǫl+k+ǫl+k+q/parenrightbigg2\n×θ(−ǫl)θ(ǫl+k)θ(ǫl+k+q)\n×δ(ω−ω0−ǫl+k+q+ǫl) (19)\nχ(2b)\nc(k,ω) =/parenleftbiggλJ\nω0/parenrightbigg2/summationdisplay\nlq/parenleftbiggω0B(l,l+q)\nω0−ǫl+ǫl+q/parenrightbigg2\n×θ(−ǫl)θ(ǫl+k)θ(−ǫl+q)\n×δ(ω−ω0−ǫl+k+ǫl+q), (20)\nwhereǫk=−Jcos(k) andB(k,l) = sin(k)−sin(l).θ(x)7\ndenotes the step function, and we assume ω0>2J. In\nthis approximation, phonon-assisted particle-hole exci-\ntation starts from the second order of λand is given\nbyχ(2a)\nc(k,ω) andχ(2b)\nc(k,ω). Thus this excitation is\nexpected to increase as the power law λbwithb= 2.\nThis behavior explains the power-law behavior of the in-\ntegrated intensity of the excitation with ( λJ/ω0)1.8as\nshown in the inset of Fig. 5(a).\nχ(2a)\ncandχ(2b)\ncinclude the particle-hole excitation ac-\ncompanied by a phonon with the energy of ω0. The\nparticle-hole excitation can scan the full-energy range of\ncharge excitation with the width of 2 J. Therefore, the\nphonon-induced spin excitation in the XYspin-Peierls\nmodel is expected to be located between ω0andω0+2J,\nwhich is seen in Fig. 4and Fig. 5(b). At k=π,by us-\ning eqs. ( 19) and (20), we obtain a cusp-like structure\nas shown in the iset of Fig. 5(b), which is consistent\nwith the results shown in in Fig. 5. We note that this\ncusp-like structure is also similar to the phonon-assisted\nmagnon absorption observed in the 1D Mott insulator\nSr2CuO3. [42, 43] Thus we conclude that the full-energy-\nrangescanningwithparticle-holeexcitationaccompanied\nby a phonon is important to understand the phonon-\ninduced spin excitation.\n2. Phonon excitation spectrum\nWe examine how phonon excitations are inﬂuenced by\nspin-phonon coupling. Figure 6(a) shows the phonon ex-\ncitation spectrum for the Einstein phonon with ω0= 3J.\nThe introduction of spin-phonon coupling with λJ/ω0=\n0.5 into a frustrated spin model ( α0= 0.36) changes the\nphonon excitation spectrum from Fig. 6(a) to Fig. 6(b).\nTwo changes are seen in the phonon excitation spectra.\nOne is a slight shift of the phonon main peak toward\nhigher energy near q=π, accompanied by small hump\nat the high energy side as shown in the inset. The en-\nergyposition ofthe hump structure is the same as that of\nphonon-induced spin excitations seen in Fig. 2(b). The\notherchangeisthe emergenceoflow-energyphononcom-\nponents ( ω <1.5J). The strongest change in intensity\nappears near ω= 0 atq∼π. We note that, since the\nphonon excitation occurs without spin ﬂipping, i.e., in\nthe Hilbert space of zero total spin, the energy positions\nof the spin-induced phonon excitations are diﬀerent from\nthose of spin excitations shown in §IVB. Both the high-\nand low-energychanges predominantly occur near q=π,\nwhich is due to the nature of spin-phonon coupling as\nshown in Appendix. We ﬁnd that these changes are in-\nsensitive to the presence of α0and are general features\nof the spin-phonon coupling.\nV. SUMMARY\nWe have investigated spin excitation for a spin-Peierls\nchain with nearest-neighbor and next-nearest-neighborω/J0 2.0 6.0 4.0qπ(a)\n(b)0qπ\nω/Jq\n0π\n3.5 5.0PESsingle Lorentzian\nFIG. 6. (Color online) Phonon excitation spectrum in a\n16-site spin-Peierls chain ( α0= 0.36,ω0/J= 3.0) without\nspin-phonon coupling λJ/ω0= 0 (a) and with the coupling\nλJ/ω0= 0.5 (b). The inset shows the tail of main peak (red\nsolid line) together with a single Lorentzian curve obtaine d\nby ﬁtting the main peak (blue dotted line). The diﬀerence\nbetween the two lines in the inset demonstrates the presence\nof hump structure at the high-energy side of the main peak.\nHeisenberg spin exchange interactions, taking CuGeO 3\nintoconsideration. We consideragapped anddispersion-\nless (Einstein) phonon as the lattice degree of freedom.\nWe then apply a dynamical density matrix renormaliza-\ntion group method to calculate dynamical spin correla-\ntion function at zero temperature.\nWe have found a new spin excitation assisted by non-\nsoftening phonon at the energy region higher than the\nphonon energy. The new spin excitation shows a disper-\nsive feature with strong intensity near k=π. There is\nno corresponding structure in the eﬀective J1-J2model\nderived from the spin-Peierls model as expected. This\ndemonstrates the importance of treating the phonons\nquantum-mechanically.\nWe have also found the new spin excitation appears in\ntheXYspin-Peierls model. This model is equivalent to\nthe Su-Schrieﬀer-Heeger model. We have shown that the\nbehaviors of the new excitation are explained by charge-\ncharge correlation function in the SSH model.\nThe phonon excitation spectrum is also inﬂuenced by\nthe spin-phonon interaction. We have found the shift of\nmain phonon peak toward higher energy side near q=π,8\naccompanied by a new tail structure whose energy range\nis the same as the new spin excitation. In addition, new\nspin-assisted phonon structures appear at low-energy re-\ngion. The fact that the strong modiﬁcation occurs near\nq=πis understood by taking into account the momen-\ntum dependence of the spin-phonon coupling.\nThe eﬀect ofspin-phononcouplingoninelastic neutron\nscattering has been discussed so far in literature. [39]\nHowever, there is no work demonstrating the intensity\ndistributionsforthe spin-Peierlsmodelasfarasweknow.\nTherefore,webelievethatthepresentresultswillbehelp-\nful foranalyzinginelasticneutronscatteringdatain spin-\nphonon coupled systems.\nIn CuGeO 3, inelastic neutron scattering experiments\nhave clearly revealed the presence of both lowest-energy\nbranchofspin excitationwith strongintensity and multi-\nspinon continuum. [27–29] This is consistent with the re-\nsults of the J1-J2model. Experimental data has also\nshown high-energy structures above the continuum. If\nthe structures are originated from spin degrees of free-\ndom, they may be due to the coupling with phonon. It\nis desired to resolve the structure into phonon and spin\ncomponentsby either detailed analysesofmomentum de-\npendenceoftheirintensityornewpolarizedinelasticneu-\ntron scattering experiments.\nACKNOWLEDGMENT\nWe thank J. Kokalj and P. Prelovˇ sek for fruitful dis-\ncussions. We also thank R. Kajimoto, K. Ikeuchi, F.\nMizuno, M. Fujita, and M. Arai for discussions on inelas-\ntic neutron scattering data of CuGeO 3. This work was\nsupported by Nanoscience Program of Next Generation\nSupercomputing Project, the Grant-in-Aid for Scientiﬁc\nResearch (Grants No. 19052003and No. 22340097)from\nMEXT, the Grant-in-Aid for the Global COE Program\n“The Next Generation of Physics, Spun from Universal-\nity and Emergence,” and the Yukawa International Pro-\ngram for Quark-Hadron Sciences at YITP, Kyoto Uni-\nversity. A part of numerical calculations was performed\nin the supercomputing facilities in ISSP and ITC, the\nUniversity of Tokyo, YITP and ACCMS, Kyoto Univer-\nsity. The ﬁnancial support of JPSJ and MHEST under\ntheSlovenia-JapanResearchCooperativeProgramisalso\nacknowledged.Appendix A: Momentum dependence of\nspin-phonon coupling\nTo examine the nature of the spin-phonon coupling\nterm (4), we introduce Holstein-Primakoﬀbosons assum-\ning two sublattices. The corresponding boson operators\nare represented by AmandBmwith integer m. We con-\nsider the Fourier transformation of the boson operators:\n˜ak=/radicalBig\n1\nL/summationtextN\nj=1eijkajwitha2m−1=Am,a2m=Bm. By\nneglecting fourth- or more higher-order terms, the spin-\nphonon term ( 4) is rewritten by\nHsp=/summationdisplay\nqH′\nsp(q) (A1)\nwith\nH′\nsp(q)∼=iλJ\n2√\nL/summationdisplay\nk(˜bq+˜b†\n−q)\n×/bracketleftBig\nsin(q)˜a†\nk˜ak−q+sin(k)(˜a†\nk˜a†\n−k+q−˜ak˜a−k−q)/bracketrightBig\n,(A2)\nwhere˜b†\nqand˜bqare the creation and annihilation opera-\ntors of phonons in the momentum representation.\nWe can ﬁnd that the q= 0 phonon has no coupling\nwith spin in eq. ( A1). Actually, the term including the\nq= 0-phonon operators reads\nH′\nsp(q= 0) =iλJ\n2√\nL(˜b0+˜b†\n0)/summationdisplay\nksin(k)(˜a†\nk˜a†\n−k−˜ak˜a−k)\n=iλJ\n2√\nL(˜b0+˜b†\n0)/summationdisplay\nksin(−k)(˜a†\nk˜a†\n−k−˜ak˜a−k)\n=−H′\nsp(q= 0) = 0 . 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Eder: Phys. Rev. B 55(1997)\nR3358."    },    {        "title": "1511.07582v1.Relaxation_dynamics_in_an_isolated_long_range_Ising_chain.pdf",        "content": "arXiv:1511.07582v1  [quant-ph]  24 Nov 2015Relaxation dynamics in an isolated long-range Ising chain\nH. T. Ng and Jing-Ning Zhang\nCenter for Quantum Information, Institute for Interdiscip linary Information Sciences,\nTsinghua University, Beijing 100084, P. R. China\n(Dated: June 18, 2021)\nWe consider a chain of trapped ions to interact with each othe r via long-range interactions. This\nsystem can be used to simulate the long-range Ising model. We study the dynamics of quantum\ncoherence of a single spin in the chain, where the spins are in itially prepared in their upper states.\nThe relaxation dynamics exhibits due to the genuine long-ra nge interaction. The degree of quantum\ncoherence of a single spin rapidly decreases and vanishes in the steady state. However, our numerical\nresult suggests that the conventional spin chain model, whi ch truncates the interactions between\nthe distant spins, cannot show the relaxation dynamics. Thi s implies that the usual truncation\nin approximating the long-range interaction is not applica ble to describing the non-equilibrium\ndynamics. The eﬀect of the interaction range on the relaxati on dynamics is studied. The higher\nrelaxation rate will show if a system has a longer range of int eraction. However, it takes a longer\nrelaxation time in the vicinity of inﬁnite interaction rang e. We also examine the dynamics of\nquantum coherence of a block of spins. Our result may shed lig ht on the relationship between\nlong-range interaction and the coherence dynamics of a quan tum many-body system.\nPACS numbers: 03.65.Aa, 05.70.Ln, 75.10.Pq\nI. INTRODUCTION\nRecently, the long-range Ising model have been real-\nized by using a string of trapped ions [1–3] and a two-\ndimensional crystal lattice [4], respectively. The interac-\ntions between ions can be engineered such that the range\nof interaction becomes tunable. This opens the possibil-\nities to observe new phenomena [5, 6] of a many-body\nsystem with long-range interaction. More recently, the\nviolation of Lieb-Robinson bound [7] has been showed in\na chain of trapped ions [8, 9] with variable-range inter-\nactions, where the Lieb-Robinson bound [10] governs the\nspeed of information propagation in a one-dimensional\nsystem with ﬁnite-range interaction.\nNon-equilibrium dynamics of a closed many-body sys-\ntem is important in quantum statistical mechanics [11–\n13]. There are open questions remained unsolved [14–18]\nin this area. For instance, there is still no general bound\non how fast the equilibration takes place [18]. In fact, a\nsubsystem of a many-body system will lose its quantum\ncoherence [19, 20] even if the entire system is in a pure\nstate. It is natural to ask how the long-range interaction\naﬀectstherelaxationprocessofalocalsubsystem. Itmay\nbe useful to understand the role of interaction in the re-\nlaxationdynamics. Additionally, quantumcoherence[21]\nis of the essence to the applications of quantum informa-\ntion science [22] and photosynthesis for eﬃcient energy\nconversion [23]. The study of the relationship between\nquantum coherence and long-range interaction provides\ninsight into the behavior of coherence in a strongly inter-\nacting many-body system and its potential applications.\nIndeed, the long-range Ising model has been studied\nextensively [24–27]. In this paper, we study the dynam-\nics of quantum coherence of a single spin and a block\nof spins in a chain of trapped ions, where the ions in-\nteract with each other through long-range interactions.Since the interaction range between ions can be tuned\nby appropriately adjusting the laser beams, the relation-\nship of coherence dynamics and interaction range can be\nstudied experimentally. Initially, all ions are prepared in\ntheir upper spin states. The degree of quantum coher-\nence of a single spin rapidly decreases and then drops to\nnearly zero in the steady state. The relaxation dynamics\nexhibits in this long-range Ising model.\nConventionally, the long-range interaction is approxi-\nmated by keeping the interaction between a few neigh-\nboring spins only [28]. Since the interaction between the\ndistant neighbors is weak, this approximation provides a\nreasonable good estimation of the ground-state energy of\nthe exact long-range model. It should be noticed that\nthis approximation must be reexamined if the dynamics\ninvolves a lot of eigenstates. We compare the dynam-\nics of the exact model with the “approximate” model.\nOur numerical examples show that the “approximate”\nmodel cannot exhibit the relaxation dynamics. But the\nshort-time dynamics of the “approximate” model resem-\nbles the exact model if the interactions between more\ndistant neighbors are included.\nIn addition, we examine the eﬀect of interaction range\non the relaxation dynamics. If the range of interaction\nincreases, then the relaxation rate becomes higher. How-\never, when the interaction range is close to inﬁnity, the\ndynamics of quantum coherence behaves very diﬀerently.\nIt takes a longer time to lose the coherence. The system,\nwith the inﬁnite range of interaction, does not relax. We\nalso numerically study the quantum coherence of a block\nof spins in a chain. The larger size of a block of spins\nindeed gives a higher degree of quantum coherence. The\ndiﬀerent sizes of block spins have the similar rates of re-\nlaxation, and the coherence becomes steady in a long\ntime.\nThis paper is organized as follows: In Sec. II, we in-2\ntroduce the long-range Ising model. In Sec. III, we ﬁrst\nintroduce the deﬁnition on how to quantify quantum co-\nherence. We discuss the numerical results of the dynam-\nicsofquantumcoherenceofasinglespin inthechain. We\nthen compare the dynamics of the exact long-range Ising\nmodel with the “approximate” models. We also discuss\nthe relaxation dynamics of a block of spins in a chain.\nWe provide a discussion in Sec. IV and close our paper\nwith a conclusion in Sec V.\nII. SYSTEM\nWe consider a chain of trapped ions in a linear trap.\nThe two internal states of an ion form a spin. For ex-\nample, either two hyperﬁne states in171Yb+[8] or two\nelectron levels in40Ca+[9] can be used. The interactions\nbetween the spins can be produced by oﬀ-resonantlycou-\npling the two internal states to the collective motion of\nthe chain in the perpendicular direction via a laser beam\n[8, 9]. This forms a 1D spin chain with long-range inter-\naction. The Hamiltonian of a chain of trapped ions, with\nlong-ranged interaction, is written as ( /planckover2pi1= 1) [29]\nH=/summationdisplay\ni<jJijσx\niσx\nj, (1)\nwhereσx\njare Pauli spin operators, Jij=J/|i−j|αand\nαranges between 0 and 3. We denote the states |1x/angbracketrighti\nand|0x/angbracketrightias the eigenstates of spin-up and -down states\nof ioniin thex-direction.\nLet us consider the k-th eigenstate |Ek/angbracketrightof the Hamil-\ntonianHwhich can be written as the product states of\nall ions, i.e.,/producttext\ni|σi/angbracketrightiand|σi/angbracketrightiis the eigenstate of ion i\nin thex-direction and σiis either equal to zero or one.\nThe eigenenergy Ekof thek-th eigenstate |Ek/angbracketrightis given\nby\nEk=/summationdisplay\ni<jJij(2σi−1)(2σj−1), (2)\nwhereJijis positive.\nTo prepare the initial state, we consider a large mag-\nnitude of the transverse magnetic ﬁeld to be applied to\nthe ions. The Hamiltonian of the system is written as\nH′=/summationdisplay\ni<jJijσx\niσx\nj−B/summationdisplay\niσz\ni, (3)\nwhereBis the strength of the transverse ﬁeld. Initially,\nthe magnitude |B|is much largerthan Jij. The system is\nprepared in a product state of ions in the spin-up states\nas\n|Ψ(0)/angbracketright=N/productdisplay\ni=1|1z/angbracketrighti, (4)\nwhereion iisintheeigenstate |1z/angbracketrightiinthez-direction. In-\ndeed, it is the ground state ofthe Hamiltonian in Eq. (3),\nFIG. 1. (Color online) A chain of ions with long-range\ninteraction. The system is divided into two regions which ar e\n“inside” and “outside” subsystems, respectively.\nwhere|B| ≫Jij. The initial state can be expressed in\nterms of the eigenstates |Ek/angbracketrightas\n|Ψ(0)/angbracketright=N/productdisplay\nj=11√\n2(|0x/angbracketrightj+|1x/angbracketrightj), (5)\n=1\n2N/22N/summationdisplay\nk=1|Ek/angbracketright. (6)\nIt is a superposition of all eigenstates |Ek/angbracketrightwith an equal\nweight. We consider that the magnetic ﬁeld is suddenly\nramped down to zero. This initiates the system to be-\ncome out of equilibrium.\nIII. QUANTUM COHERENCE DYNAMICS\nWe consider the system to be divided into two subsys-\ntems which are “inside” and “outside” regions, respec-\ntively, as shown in Fig. 1. The “inside” system can be\ndescribed by the reduced density matrix ρin= Trout(ρ),\nwhereρinis obtained by tracing out the part of “outside”\nsystem.\nWe study the dynamics of quantum coherence of spins\nin the “inside” system. To quantify quantum coherence,\nthe coherence factor Ccan be used. It is deﬁned as [21]\nC=/summationdisplay\ni/negationslash=j|ρin\ni,j|, (7)\nwhereρinis the reduced density matrix of the system.\nThe coherence factor Cis the sum of absolute value of all\noﬀ-diagonal elements in the reduced density matrix ρin.\nThis quantity provides a proper measure of coherence of\na system [21].\nA. Relaxation of a single spin\nWe ﬁrst investigate the dynamics of coherence of a sin-\ngle spin in the chain and the other ions are treated as the\n“outside” system. The system of ion jcan be described\nby the reduced density matrix as\nρin\nj=1\n2/parenleftbigg\n1f(t)\nf(t)∗1/parenrightbigg\n. (8)\nThe coherence factor Cis given by\nC(t) =|f(t)|. (9)3\nTocalculatethe reduced densitymatrix ρin\nj, the partof\n“outside” region of the density matrix ρ=|Ψ(t)/angbracketright/angbracketleftΨ(t)|is\ntraced out. From Eq. (6), the state vector |Ψ(t)/angbracketrightis given\nby\n|Ψ(t)/angbracketright=1\n2N/22N/summationdisplay\nk=1e−iEkt|Ek/angbracketright, (10)\nwhere the eigenstate |Ek/angbracketrightcan be written as\n|Ek/angbracketright=/productdisplay\njk|σjk/angbracketrightjk\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n“inside”/productdisplay\nik/negationslash=jk|σik/angbracketrightik\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n“outside”. (11)\nFor the case of a single spin, the number of the states\n|σjk/angbracketrightjkin the “inside” region to be equal to two, i.e., |0/angbracketright\nand|1/angbracketright.\nThe density matrix is written as\nρ(t) =|Ψ(t)/angbracketright/angbracketleftΨ(t)|, (12)\n=1\n2N2N/summationdisplay\nk=12N/summationdisplay\nk′=1e−i(Ek−Ek′)t|Ek/angbracketright/angbracketleftEk′|,(13)\n=1\n2N2N/summationdisplay\nk=12N/summationdisplay\nk′=1e−i(Ek−Ek′)t|σjk/angbracketrightjkjk′/angbracketleftσjk′|/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n“inside”\n×/productdisplay\nik,ik′/negationslash=jk,jk′|σik/angbracketrightikik′/angbracketleftσik′|\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n“outside”(14)\nIf the “outside” part is traced out, the elements of den-\nsity matrix can be retained for the two eigenstates Ek\nandEk′with the same states of “outside” part, i.e.,/producttext\nik/negationslash=jk|σik/angbracketrightik. In contrast, the elements of density ma-\ntrix become zero if the two eigenstates EkandEk′with\nthe diﬀerent states of “outside” part.\nThe reduced density matrix ρin\njcan be obtained by\nsumming up all possible ˜ ρin\nl, where the reduced density\nmatrices ˜ ρin\nlhave the same state of “outside” part, and\nl= 1,...,2N−1. The density matrix ˜ ρin\nlcan be explicitly\nwritten as\n˜ρin\nl=1\n2N/parenleftbigg\n1eiωlt\ne−iωlt1/parenrightbigg\n, (15)\nwhereωl=Ek−Ek′is the eﬀective frequency, and Ek\nandEk′are the eigenenergies of the eigenstates |Ek/angbracketrightand\n|Ek′/angbracketrightwith the same states of “outside” region. Hence,\nthe reduced density matrix ρin\njis given by\nρin\nj=2N−1/summationdisplay\nl=1˜ρin\nl, (16)\n=1\n2/parenleftbigg\n1f(t)\nf∗(t) 1/parenrightbigg\n, (17)\nwhere\nf(t) =1\n2N−12N−1/summationdisplay\nl=1eiωlt. (18)-6 -4 -2 0 2 4 600.250.5\n-6 -4 -2 0 2 4 600.1250.25\n-6 -4 -2 0 2 4 600.0650.13\n-6 -4 -2 0 2 4 600.020.040 10 20 30 4000.51\n(b) (c)\n(d) (e)(a)\nFIG. 2. (Color online) In (a), coherence factor C(t) versus\ntimeJt, for the “approximate” and exact long-range models,\nN= 20 and α= 3. The coherence factor of the tenth ion\nis plotted. The diﬀerent types of interaction are denoted by\nthe diﬀerent lines: exact long-range interaction (black so lid),\nnext-next-nearest neighbor (blue dotted), next-nearest n eig-\nbor (red dashed), and nearest neighbor (green dot-dashed),\nrespectively. Histograms of the eﬀective frequencies ωlare\nshown, where N(ωl) is the number frequency of ωland is\nnormalized to be one. The diﬀerent types of interaction in\nthe diﬀerent ﬁgures: (b) nearest neighbor, (c) next-neares t\nneighbor, (d) next-next-nearest neighbor and (e) exact lon g-\nrange interaction without truncation, respectively.\nThe coherence factor Cis given by\nC(t) =1\n2N−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2N−1/summationdisplay\nl=1eiωlt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (19)\nIt is interesting to compare the relaxation dynamics of\nthe exact long-rangeIsing model with the “approximate”\nmodel, where the interactions between distant neighbors\nare truncated in the “approximate” one. Indeed, it is an\nusual approximation by including the interaction with a\nfew neighbors only in the conventional spin chain mod-\nels and the long-range interaction between the distant\nneighbors is neglected.\nHere we study the dynamics of quantum coherence of\na single spin at the centre of a chain. In Fig. 2(a), we\nplot the coherence factor C(t) versus time, for the ”ap-\nproximate” and exact long-range models. For the ex-\nact long-range model, the coherence factor Crapidly de-\ncreasesand then severalcollapsesand revivals occur with\ndecreasing magnitude. On the contrary, the coherence\nfactorsCoscillate periodically if the ions interact with\ntheir nearest neighbors only. The collapses and rivials\nare shown if the ions interact with their next-nearest and4\nnext-next-nearestneighbors,buttheirmagnitudesdonot\ndecrease in a longer time.\nIt should be noticed that if the interactions between\nmore distant neighbors are included, then the short-time\nbehaviors resemble the exact case in Fig. 2(a). The re-\nvivals of coherence appear subsequently, but they diﬀer\nin a longer time. These numerical examples suggest that\nthe ”approximate” model with truncation of long-range\ninteraction between distant neighbors cannot display the\nrelaxation dynamics.\nFrom Eq. (19), the coherence factor is a function of a\nsum of eﬀective frequencies ωl. In fact, the distribution\nof the eﬀective frequencies ωlcan indicate the important\nfeature of the coherence dynamics. In Figs. 2(b)-(e), we\nplot the histograms of the eﬀective frequencies ωl, for the\nexact model and the diﬀerent types of “approximation”\nof long-range interaction. For the case of the nearest-\nneighbor interaction, the eﬀective frequencies are equal\nto three diﬀerent values only in Fig. 2(b). As the inter-\nactions with more neighboring spins are included, a few\neﬀective frequencies distribute around the three diﬀer-\nent values. For the exact model, the three main peaks\nare shown and the eﬀective frequencies smoothly spread\naround the peaks. This means that a lot of diﬀerent ef-\nfectivefrequencies ωlcontributein thedynamicsbyusing\nthe exact model. The net eﬀect of the diﬀerent frequen-\ncies cancel out with each other, and therefore it leads to\nthe loss of quantum coherence. On the contrary, only a\nfew distinct eﬀective frequencies involvesin the time evo-\nlution of coherence factor using the “approximate” mod-\nels. Thus, the coherence factor does not relax, and the\ncollapses and revivals of coherence will repeatedly occur.\nTo facilitate our subsequent discussion, we deﬁne the\nrelaxation time tr. Within a period, the coherence factor\nC(tr) decrease to its initial value being divided by e, i.e.,\nC(0)/e, where eis equal to the Euler’s number. The\nrelaxation rate is equal to 1 /tr. Here we have assumed\nthat coherence of the system will not resurrect in that\nperiod.\nIn Fig. 3, we compare the coherence dynamics of the\nexact and “approximate” models, for the diﬀerent inter-\nactionranges. The spin undergoesthe relaxationdynam-\nics if the exact model is used. Comparing with the two\ndynamicsinFigs.3(a)and(b), thesystem, withasmaller\nα, relaxes more rapidly. When α= 0.1 is close to zero\nin Fig. 3(c), the spin takes a longer time for relaxation.\nOn the contrary, the spin does not relax in the cases of\nusing “approximation” models. Similarly, the “approxi-\nmate” model can give out a better approximation of the\nshort-time dynamics of the exact model if the interac-\ntions between more distant neighbors are included.\nWe further investigate the eﬀect of interaction range\non the coherence dynamics. In Fig. 4, we plot the co-\nherence factors versus time for the diﬀerent interaction\nranges by using the exact model only. When the range\nof interaction increases with a smaller α(α= 3,2 and\n1) in Fig. 4(a), the rate of relaxation increases. How-\never, asαapproaches zero, the coherence dynamics be-1\n0.5\n0010203040\n0102030401\n0.5\n0\n0102030401\n0.5\n0(a)\n(b)\n(c)\nFIG. 3. (Color online) Coherence factor versus time, where\nthe tenth ion is studied and N= 20. The diﬀerent interaction\nparameters are shown: (a), α= 2, (b) α= 1 and (c) α= 0.1.\nIn each ﬁgure, the diﬀerent types of interaction are shown:\nexact long-range interaction (black solid), next-next-ne arest\nneighbor (blue dotted), next-nearest neighbor (red dashed )\nand nearest neighbor (green dot-dashed), respectively.\nhaviors very diﬀerently. In Fig. 4(b), the coherence fac-\ntor decays more slowly (more revivals before vanishing)\nifαis smaller. This means that the slower relaxation\nrate when approaching to the inﬁnite interaction range.\nWhenα= 0, the coherence factor oscillates periodically.\nThis shows that the behaviors of relaxation dynamics\ngreatly depends on the interaction range.\nInFigs.4(c)-(h), weplotthehistogramsoftheeﬀective\nfrequencies ωl, for the diﬀerent interaction ranges. Dif-\nferent ranges of interaction exhibit diﬀerent patterns in\nthe histograms. When α= 3, the histogram shows three\nsharp peaks only in Fig. 4(c). In Fig. 4(d), the width\nof peaks become wider if α= 2. As αfurther decreases\nto 1 and 0.5, a single peak is shown in both ﬁgures (e)\nand (f). The peak is wider for the smaller α. However,\nwhenαis close to 0, several sharp peaks are shown in\nFigs. 4(g) and (h). The peaks becomes sharper when the\nrange of interaction is longer. This means that more ef-\nfective eigenfrequencies are degenerate as the interaction\nrange is close to inﬁnity. Thus, the relaxation does not\noccur because the system become completely degenerate5\n-20 -10 0 10 2000.020.04\n-20 -10 0 10 2000.0080.016\n-20 -10 0 10 2000.00650.013\n-20 -10 0 10 2000.0090.018\n-20 -10 0 10 2000.020.04\n-20 -10 0 10 2000.040.081\n0.5\n00 10 20 30 401\n0.5\n00 10 20 30 40(a)\n(b)\n(c) (d)\n(e) (f)\n(g) (h)\nFIG. 4. (Color online) Coherence factor versus time in (a)\nand (b), where the coherence dynamics of the tenth ion is\nstudied and N= 20. The diﬀerent lines are used to represent\nthe diﬀerent parameters α: in (a)α= 3 (red dashed), α= 2\n(blue dotted) and α= 1 (black solid); in (b) α= 0.1 (black\nsolid),α= 0.05 (blue dotted) and α= 0 (red dashed). His-\ntograms of the eﬀective frequencies ωkare shown in the lower\npart. The diﬀerent interaction parameters αare shown: (c)\nα= 3, (d) α= 2, (e) α= 1, (f) α= 0.5, (g)α= 0.1 and (h)\nα= 0.05, respectively.\nin the limit of inﬁnite interaction range.\nB. Relaxation of block of spins\nWe study the dynamics of quantum coherence of a\nblock of spins with the same initial condition in Eq. (4).\nNowweconsiderthe“inside”systemwhichcontainsmore\nthan one ion. Let us call NIbe the number of spins in\nthe “inside” system and the block of spins is located at\nthe centre of a chain.0 10 20 30 4000.51\n0 10 20 30 4000.51\n0 10 20 30 4000.51(a)\n(b)\n(c)4 6 8 1000.010.020.03\nFIG. 5. (Color online) Normalized coherence factor ˜C(t)\nversus time, for the diﬀerent parameters αandN= 20. The\ndiﬀerent parameters are used: (a) α= 3, (b) α= 2 and (c)\nα= 1. The diﬀerent lines are used to denote the diﬀerent\nsizes of blocks: NI= 10 (black solid), 8 (blue dashed), 6\n(red dotted) and 4 (green dash-dotted) respectively. In the\ninset, the normalized coherence factors ˜C(tf) are plotted ver-\nsusNI, where tf= 40J−1. The diﬀerent parameters αare\nrepresented by diﬀerent symbols: α= 3 (black circle), 2 (blue\nplus) and 1 (red upper triangle), respectively.\nFor a larger size of block, it has a higher degree of\ninitial quantum coherence C(0). In fact, C(0) can be\nevaluated by summing up all the oﬀ-diagonal elements,\nwhere the density matrix elements of the initial state are\nall equal. The initial quantum coherence C(0) is thus\ngiven by\nC(0) =2\n2NI2NI−1/summationdisplay\nk=1k, (20)\n= 2NI−1. (21)\nWe deﬁne the normalized coherence factor ˜C(t) as\n˜C(t) =C(t)\nC(0). (22)6\nIt is convenient to compare the relaxation rates for dif-\nferent block sizes.\nIn Fig. 5, we plot the normalizedcoherencefactors ver-\nsus time fordiﬀerent sizesofblock and rangeparameters.\nWe compare the time evolution of ˜Cfor the diﬀerent NI\nwith the same αin each subﬁgure. The relaxation rate is\nsimilar for the diﬀerent sizes of blocks. They all decrease\nand become steady in a long time. The system has a\nhigher relaxation rate when αis smaller. Also, the larger\nsize of block has a higher degreeof coherencethat remain\nin the steady state as shown in the inset of Fig. 5.\nIV. DISCUSSION\nLet us brieﬂy discuss the observation of relaxation dy-\nnamics with a long-range Ising model in an experiment.\nFrom the recent experiments of long-range Ising model\nusing trapped ions, the typical value ofthe spin-spin cou-\nplingJ[8] is about 100-500 Hz. The coherence lifetime\nof system [8, 9] is about 5-10 ms. Given these parame-\nters,theperiod τofcoherentdynamicsisabout0.5-5 J−1.\nThis period τis suﬃciently longto observethe relaxation\ndynamics due to the long-range interaction.\nIndeed, the long-range spin-spin couplings are gener-\nated by oﬀ-resonantly coupling to the collective motions\nof the string via a laser beam. When the transversemag-\nneticﬁeldisabsent,thespinscanbeeﬀectivelydecoupled\nfrom the phonon modes [30]. However, the phonon ex-\ncitations will arise [30] in the presence of the transverse\nmagnetic ﬁeld. The large magnetic ﬁeld is required to\nturn on shortly to prepare the initial state and it is sud-\ndenly quenchedto initiate the non-equilibrium dynamics.\nThe system then evolves in the absence of the transverse\nmagnetic ﬁeld. To avoid the intrinsic phonon eﬀects, the\nlaser should be largely detuned from the phonon modes\n[30]. Since the transverse ﬁeld is turned on initially, the\nphonon population can be kept to be low during the en-\ntire time-evolution.\nIn addition, the coherencefactorofa single spin can be\ndeterminedfromthe densitymatrixofasinglespinwhich\ncan be obtained by just measuring the spin components.Two-spin density matrix can be obtained by performing\nquantum state tomography [31, 32]. In fact, it has been\ndemonstrated in a recent experiment [9]. However, it\nmay be more challenging to construct the density matrix\nas the size of block spin goes large. The dimension of\nstate space will exponentially increase with an increasing\nnumber of spins, and thus the complete characterization\nof multi-spin density matrix becomes hard. Therefore,\nthe study of coherence may limit to a few spins only.\nV. CONCLUSION\nInsummary, wehavestudied the dynamicsofquantum\ncoherence of a single spin and block of spins in a chain of\ntrapped ions, where the ions interact with each other via\nlong-range interactions. The degree of quantum coher-\nenceofasinglespin rapidlydecreasesandnearlyvanishes\ninthesteadystate. Relaxationexhibitsinthislong-range\nIsingchain. We havecomparedthedynamicsofthe exact\nlong-range Ising model with the “approximate” models,\nwhere the interactions between the distant ions are trun-\ncatedin the“approximate”models. Ournumericalresult\nsuggests that the “approximate” model cannot show the\nrelaxation dynamics. In addition, we have examined the\neﬀect of long-range interaction on the coherence dynam-\nics. The relaxation rate becomes higher for a longer in-\nteraction range. But the spin takes a longertime to relax\nif the interaction range is in the vicinity of inﬁnite range.\nWe have also investigated the dynamics of coherence of a\nblock of spins. Our study provides a deeper insight of the\nrelationship between the long-range interaction and the\nquantum-coherence dynamics of a many-body system.\nACKNOWLEDGMENTS\nThis workwas supported in part by the National Basic\nResearch Program of China Grant No. 2011CBA00300\nand No. 2011CBA00301, and the National Natu-\nral Science Foundation of China Grant No. 11304178,\nNo. 61061130540, and No. 61361136003.\n[1] A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras\nand T. Schaetz, Nat. Phys. 4, 758 (2008).\n[2] K. Kim et al., Nature 465, 590 (2010).\n[3] R. Islam et al., Science 340, 583 (2013).\n[4] J. W. Britton et al., Nature 484, 489 (2012).\n[5] N. Laﬂorencie, I. Aﬄeck, M. Berciu, J. Stat. Mech.,\nP12001 (2005).\n[6] A. W. Sandvik, Phys. Rev. Lett. 104, 137204 (2010).\n[7] P. Hauke and L. Tagliacozzo, Phys. Rev. Lett. 111,\n207202 (2013).\n[8] P. Richerme et al., Nature 511, 198 (2014).\n[9] P. Jurcevic et al., Nature 511, 202 (2014).[10] E. Lieb and D. Robinson, Commun. Math. Phys. 28, 251\n(1972).\n[11] A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalat-\ntore, Rev. Mod. Phys. 83, 863 (2011).\n[12] J. Eisert, M. Friesdorf and C. Gogolin, Nat. Phys. 11,\n124 (2015).\n[13] T. Langen, R.Geiger, J. Schmiedmayer,Annu.Rev.Con-\ndens. Matter Phys. 6201 (2015).\n[14] C. Gogolin and J. Eisert, arXiv:1503.07538.\n[15] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).\n[16] M. Srednicki, Phys. Rev. E 50, 888 (1994).\n[17] M. Rigol, V. Dunjko and M. Olshanii, Nature 452, 854\n(2008).7\n[18] J. Goold, M. Huber, A. Riera, L. del Rio, P. Skrzypczyk,\narXiv:1505.07835.\n[19] M. C. Ba˜ nuls, J. I. Cirac, M. B. Hastings, Phys. Rev.\nLett.106, 050405 (2011).\n[20] H. T. Ng, Phys. Rev. A 92, 043634 (2015).\n[21] T. Baumgratz, M. Cramer, M. B. Plenio, Phys. Rev.\nLett.113, 140401 (2014).\n[22] M. A. Nielsen and I. L. Chuang, Quantum Computation\nand Quantum Information (Cambridge University Press,\nCambridge, 2001).\n[23] E. Romero et al., Nat. Phys. 10, 676 (2014).\n[24] T. Koﬀel, M. Lewenstein and L. Tagliacozzo, Phys. Rev.\nLett.109, 267203 (2012).\n[25] M. Foss-Feig, K. R. A. Hazzard, J. J. Bollinger and A.\nM. Rey, Phys. Rev. A 87, 042101 (2013).[26] J. Schachenmayer, B. P. Lanyon, C. F. Roos and A. J.\nDaley, Phys. Rev. X 3, 031015 (2013).\n[27] Z.-X. Gong and L.-M. Duan, New J. Phys. 15113051\n(2013).\n[28] S. Sachdev, Quantum Phase Transitions (Cambridge\nUniversity Press, Cambridge, 2001).\n[29] D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901\n(2004).\n[30] C.-C. J. Wang and J. K. Freericks, Phys. Rev. A 86,\n032329 (2012).\n[31] M. Christandl and R. Renner, Phys. Rev. Lett. 109,\n120403 (2012).\n[32] T. Sugiyama, P. S. Turner and M. Murao, Phys. Rev.\nLett.111, 160406 (2013)."    },    {        "title": "0912.3842v1.Stochastic_resonance_of_a_nanomagnet_excited_by_spin_transfer_torque.pdf",        "content": "X. Cheng et al.  1 December 18, 2009 \n Stochastic resonance of a nanomagnet excited by spi n transfer torque \n \nXIAO CHENG, CARL T. BOONE, JIAN ZHU, AND ILYA N. KR IVOROTOV j \n \nDepartment of Physics & Astronomy, University of Ca lifornia, Irvine, California 92617, USA \nj ikrivoro@uci.edu \n \n  \nSpin transfer torque from spin-polarized electrical cu rrent can excite large-amplitude \nmagnetization dynamics in metallic ferromagnets of nanoscale d imensions. Since magnetic \nanisotropy energies of nanomagnets are comparable to the therm al energy scale, temperature can \nhave a profound effect on the dynamics of a nanomagnet driven  by spin transfer torque. Here we \nreport the observation of unusual types of microwave-frequ ency nonlinear magnetization dynamics \nco-excited by alternating spin transfer torque and therm al fluctuations. In these dynamics, \ntemperature amplifies the amplitude of GHz-range prec ession of magnetization and enables \nexcitation of highly nonlinear dynamical states of magnetization  by weak alternating spin transfer \ntorque. We explain these thermally activated dynamics in ter ms of non-adiabatic stochastic \nresonance of magnetization driven by spin transfer torque. Th is type of magnetic stochastic \nresonance may find use in sensitive nanometer-scale microwave signal d etectors. X. Cheng et al.  2 December 18, 2009 \n Spin transfer torque (STT) from spin-polarized current 1,2,3,4 applied to a ferromagnet can give rise \nto several types of magnetization dynamics such as auto-oscill ations of magnetization excited by direct \ncurrent 5,6,7,8,9,10,11 , magnetization reversal driven by STT 12,13,14,15,16,17  and current-induced magnetic domain \nwall motion 18,19,20 . Practical applications of STT in non-volatile memory, microwa ve oscillators and \ndetectors are under development 21 , and non-equilibrium states of magnetization induced by STT are of \nfundamental interest in nonlinear science 22 , 23 , 24 , 25 . Alternating STT is employed in spin torque \nferromagnetic resonance (st-FMR)26,27,28  measurements of spin waves in magnetic nanostructures 26,29 . In \nthis paper, we describe unusual nonlinear GHz-range magnetizat ion dynamics that are co-excited by ac \nSTT and temperature. The pronounced thermally activated character  sets these dynamics apart from the \ntypes of excitations seen in st-FMR and conventional FMR expe riments. Temperature-dependent studies \nreveal that these dynamics arise from non-adiabatic stochastic resonance (NSR) of magnetization excited \nby ac STT. The amplitude of the observed NSR exceeds that of typi cal st-FMR-driven excitations by \nnearly two orders of magnitude for the same level of ac STT drive.  \n     We study magnetization dynamics of the permalloy (Py=Ni 81 Fe 19 ) free layer in Pt(30 nm)/ Cu(20 \nnm)/ Py(3 nm)/ Cu(6 nm)/ Co(3 nm)/ Ir 20 Mn 80 (8 nm)/ Cu(80 nm) spin valve (SV) nanopillars. All \nmagnetic layers are patterned into 120×60 nm 2 elliptical pillars with the long axis parallel to the exc hange \nbias field from Ir 20 Mn 80 . Electrical current, I, is applied between the top and the grounded bottom leads as \nshown in Fig. 1a. We report data from a single sample; nine other  samples show similar behavior. For all \nmeasurements, a magnetic field, Hr\n, is applied at 10 ° from the sample normal, with the in-plane field \ncomponent in the exchange bias direction. Fig. 1b shows resistance, R, of the SV as a function of H at \ntemperatures of 295 K and 120 K. The in-plane component of Hr\nswitches the magnetization of the \nuniaxial Py nanomagnet, Mr\n, between the high-resistance, nearly antiparallel (HR), and the low-\nresistance, nearly parallel (LR), states of the SV. The L R↔HR transition is continuous at 295 K and \nhysteretic at 120 K, indicating that the free layer is superpa ramagnetic at 295 K. Figures 1c and 1d X. Cheng et al.  3 December 18, 2009 \n illustrate the LR ↔HR transition induced by STT 1, ( )Co dc MM MIrrrr× ×⋅ ~τ , from direct current, Idc . Fig. 1e \nshows differential resistance, dV/dI,  versus H at 120 K measured for several Idc . This figure illustrates that \nthe free layer hysteresis loop width decreases with increasi ng |Idc | and becomes zero at Idc  = -2.6 mA, \nindicating that the free layer is superparamagnetic for Idc  ≤ -2.6 mA. Figure 1f shows telegraph noise of \nthe free layer between the HR and LR states at T = 120 K, H = 2.2 kOe and Idc  = -2.7 mA, which \ndemonstrates that the free layer indeed becomes superparamagnetic  at this current. This current-induced \nsuperparamagnetism results from renormalization of the amplitude  of thermal fluctuations of Mr\nby \nSTT 14,30 . For Idc  < 0, STT pushes Mr\n away from the LR state, thereby enhancing the amplitude of thermal \nfluctuations in this state, and increases the rate of thermally  activated transitions out of the LR state. In \ncontrast, the amplitude of thermal fluctuations in the HR state  is suppressed for Idc  < 0 because STT \npushes magnetization towards the HR state and thereby decreases the rate of thermally activated \ntransitions out of the HR state. As a result, the Kramers transition rates between the HR and LR states \nbecome functions of current 30 : \n( )\n\n\n\n\n\n\n\n\n−− ≈→TkH E\nIIf K\nBHR \nb\nHR LR HR \n00 1 exp     (1a), \n( )\n\n\n\n\n\n\n\n\n+− ≈→TkH E\nIIf K\nBLR \nb\nLR HR LR \n00 1 exp                                                  (1b), \nwhere ()H ELR HR \nb)( are the zero-current barriers for transition out of the HR(LR)  states, kB is the \nBoltzmann constant, ≈0f 10 9-10 10  Hz and )(\n0LR HR I = const are the critical currents 30 .   \n We measure magnetization dynamics driven by ac STT using the setup in Fig. 2a26 . In these \nmeasurements, ac STT from a square-wave modulated microwave c urrent of rms amplitude Iac  is applied \nto the SV in addition to Idc . The ac STT excites oscillations of Mr\n, which give rise to oscillations of R via \ngiant magnetoresistance,  R = R0+δRac sin( ω t+ϕ), where ϕ is the phase shift between the resistance and \ncurrent oscillations. As a result, a rectified voltage ( )ϕ δ cos \n21\nac ac RI  is generated by the SV. At Idc ≠ 0, X. Cheng et al.  4 December 18, 2009 \n an additional voltage, Idc⋅δRdc, is induced in response to Iac , where δRdc is the difference between the time-\naverage SV resistance in the presence and in the absence o f Iac 31 . The resulting rms voltage measured by \nthe lock-in amplifier is ( )dc dc ac ac dc RI RI V δπϕ δπ2cos 1+ = , where the first term is dominant for small-\namplitude oscillations of Mr\n while the second term dominates for large-amplitude oscillati ons of Mr\n, \nprovided | Idc |>>I ac 31 . In our measurements, we sweep the frequency of Iac , f, and record Vdc (f). When f \ncoincides with a spin wave eigenfrequency of the Py nanomagnet, fn, the eigenmode is resonantly excited, \nand a peak or a trough in Vdc (f) is observed at f = fn. These peaks and troughs give the st-FMR spectrum of \nspin wave eigenmodes of the nanomagnet 26,27 .  \n Figure 2c-e shows Vdc (f) at T = 120 K and H = 2.2 kOe for several values of Idc : far from (-1.5 mA, \n-3.5 mA) and at (-2.6 mA to -2.8 mA) the current-driven  LR ↔HR transition. The st-FMR spectra, Vdc (f) , \nfar from the LR ↔HR transition shown in the insets of Fig. 2c-d exhibit typical spin wa ve \nresonances 26,27,32,33 . However, for Idc  at the LR ↔HR transition, we observe qualitatively different Vdc (f). \nAt Idc  = -2.6 mA (Fig. 2c), Vdc (f) develops a negative low-frequency tail LF Vmax ≡|Vdc (0) | while at Idc  = -2.8 \nmA (Fig. 2d), Vdc (f) exhibits a positive maximum, HF Vmax , at f = 2.2 GHz.  Both LF Vmax  and HF Vmax are two \norders of magnitude greater than the amplitudes of the high-fre quency resonances, Vmax ≡ \nmax {| Vdc (f) |} f>1GHz , in the LR and HR states, indicating that large-amplitude dynami cs are excited by Iac  at \nthe LR ↔HR transition. Fig. 2e shows the response curve at Idc  = -2.7 mA in the crossover regime \nbetween the large-amplitude high- and low-frequency resonances. Fig. 2f displays Vdc (t) at Idc  = -2.8 mA \nand  f = 2.2 GHz for Iac  = 0 mA and Iac  = 0.4 mA. We also make spectral measurements of microwave \nsignal emission by the sample 5,6  at Idc  = -2.8 mA and Iac = 0 mA, and do not observe a detectable signal in \nthe 0.1-10 GHz band. These measurements show that the large-amplitude  oscillations of Mr\nare only \nexcited in the presence of Iac. \n Figure 3 further illustrates the difference of the  dynamic response at the LR ↔HR transition from \nthat in the HR and LR states. Fig. 3a-b shows the full width at hal f maximum and the amplitude, Vmax , of X. Cheng et al.  5 December 18, 2009 \n the high frequency ( f > 1 GHz) spectral peak in Vdc (f)  versus Idc . For | Idc | < 2.4 mA, the linewidth of the \npeak decreases while Vmax  increases with increasing | Idc |. This behavior is due to renormalization of the \neffective damping by dc STT 26,34 . However, for -3.1 mA < Idc  < -2.6 mA, both the amplitude and the line \nwidth increase dramatically, signaling a transition to a ne w dynamic regime. Fig. 3c-d shows the \ndependence of Vmax  on Iac for several values of Idc . For Idc  far from the LR ↔HR transition, Vmax (Iac ) is \nquadratic. In this small-amplitude regime, δRdc≈031  and ac ac RI V δπ1\nmax ≈  ~2\nac I. In contrast, at the \nLR↔HR transition, Vmax (Iac ) crosses over from quadratic to linear behavior and eventually s aturates at a \nvalue close to \nπ22 RIdc ∆, where ∆R = 35 m Ω is the resistance difference between the HR and LR states  at \nIdc  = -2.8 mA. This type of response indicates that Iac  induces a transition from the HR state with \nresistance R0+∆R to a state with time-average resistance of ≈ R0+∆R/2. The large-amplitude high- and \nlow-frequency responses of the types shown in Fig. 2c-d are observed in a strip in the ( Idc ,H ) plane, which \ncoincides with the region of the LR ↔HR transitions as illustrated in Fig. 3e. \n The origin of the large-amplitude dynamics at the LR ↔HR transition is revealed by the \ntemperature dependence of Vdc (f). Fig. 3f shows the maximum amplitudes of the high, HF Vmax ~≡ \nmax{ Vdc (f,H)} f, H, and low, LF Vmin ~≡ min{ Vdc (0,H)} H, frequency resonances versus T. Below a threshold \ntemperature, Tth (I dc ), HF Vmax ~ and LF Vmin ~are small and Vdc (f) shows small-amplitude eigenmode resonances. \nAbove Tth (I dc ), HF Vmax ~ and LF Vmin ~rapidly rise to their maximum values at T = ()dc LF HF \nSR I T)( and the response \ncurves become similar to those in Fig. 2c-d. For T > TSR (I dc ), HF Vmax ~ and LF Vmin ~slowly decrease. This type \nof temperature dependence of the amplitude of motion is a salient feat ure of stochastic resonance (SR)36 .  \nSR is an effect of noise-induced amplification of the response o f a nonlinear system to a weak \nperiodic drive 35,36,37 . To describe SR, we consider the magnetic energy 38  of the Py nanomagnet (Fig. 4): \n  ( )dHHMMNM Errrrtr\n+⋅−⋅⋅ =π2 ,     (2)  X. Cheng et al.  6 December 18, 2009 \n where Nt\n={N x, N y, N z} is the diagonal demagnetization tensor of the free layer 38 , Hr\n is the external field, \ndHr\n~Co Mr is the stray field from the Co layer and Co Mr is the magnetization of Co. Fig. 4b-d shows that \nE(θ,φ) is an asymmetric double-well potential with the wells corresponding to the HR and LR states.  \nAccording to Eq. (1), the Kramers transition rates between t he HR and LR states depend on STT: \nKHR →LR  decreases and KLR→HR increases with increasing | Idc |. Therefore, equal dwell times in the HR and \nLR states ( KHR →LR = K LR→HR ≡ KE) can be achieved via tuning of Idc  even though the double-well potential \nis asymmetric. Analysis of Eq. (1) and previous experiments 30  show that KHR →LR =K LR→HR is observed on a \nline H=H E(I dc ) in the (I dc ,H) plane, and KE(I dc , H E(I dc )) on this line exponentially increases with | Idc |. \nAlternating current, Iac , applied in addition to Idc periodically modulates KHR →LR  and KLR→HR, and, due to \nthe exponential sensitivity of the transition rates to current, small Iac  can induce periodic transitions \nbetween the HR and LR states at the frequency of Iac . In this case, the LR ↔HR transitions are random at \nIac = 0 and nearly periodic at Iac ≠0. The periodic LR ↔HR transitions induced by Iac  cease if T becomes \ntoo low ( T < T th (I dc )) so that KHR →LR  or KLR→HR becomes small compared to f. This temperature-induced \namplification of the amplitude of motion of Mr\nat T > T th (I dc ) under the action of weak ac drive is SR 39 .  \n Low-frequency (adiabatic) SR driven by ac STT can explain the  large low-frequency tail of Vdc (f)  \nin Fig. 2c. Indeed, such a tail is observed when the system is near the LR end of the LR ↔HR transition \n(Fig. 2b) where KHR →LR  >> KLR→HR at Iac=0. According to Eq. (1), KLR→HR is much more sensitive to small \nvariations of current than KHR →LR  because LR \nbHR \nb E E << . Therefore, in the presence of Iac , KHR →LR  is nearly \ntime-independent while KLR→HR oscillates with the frequency of the ac drive. This implies t hat at large \nenough Iac ,  KHR →LR  >> KLR→HR(t) for a fraction of the Iac period, while for another fraction of the period \nKHR →LR  << KLR→HR(t) . If T is high enough so that KHR →LR  >> f and max{ KLR→HR(t) }t >> f, then the \nLR→HR →LR transition takes place in almost every cycle of Iac. This results in large-amplitude \nresistance oscillations at low f and gives rise to the large low-frequency tail in Vdc (f) . The negative sign of \nVdc (0) shows that the oscillations of Mr\nare in phase with the ac STT oscillations (more negative curr ent X. Cheng et al.  7 December 18, 2009 \n favors the HR state), as expected for adiabatic SR.  Figure 3 f confirms the SR nature of the effect as the \nresonance turns on only at T > Tth , and quickly reaches the maximum amplitude at the SR tempera ture, \nTSR . The turn-on of the SR is sharp in T due to the exponential dependence of the transitions rates on 1/ T. \nThe slow decay of Vdc (0) for T > T SR  is due to partial thermal randomization of the LR ↔HR transitions 36 . \nThe adiabatic SR is observed only for Idc  at the LR end of the LR ↔HR transition because the \nsystem returns to the LR state with current-sensitive KLR→HR in almost every period of Iac. In contrast, for \nIdc  = -2.8 mA at the HR end of the LR ↔HR transition, a low-frequency Iac  does not induce the HR →LR \ntransition because KHR →LR  is weakly sensitive to Iac, and the system remains in the HR state. For this Idc , \nonly a signature of small-amplitude intra-well resonance in the  HR state is expected in Vdc (f) . However, \nFig. 2d shows that this is not the case. Although the low frequency tail in Vdc (f)  disappears at Idc  = -2.8 \nmA, a peak due to unexpected high-frequency ( f = 2.2 GHz) large-amplitude dynamics is observed. \nThis surprising high-frequency dynamics can be explained if Iac  excites large-amplitude \noscillations of magnetization with a ~180 ° phase shift with respect to the ac STT drive. Fig. 4e illust rates \nhow such a phase shift can result in a non-zero time-average compone nt of STT perpendicular to the \nsample plane and thereby stabilize large-amplitude precession o f Mr\n on an out-of-plane trajectory 40 . Since \nthe angle between Mr\nand Co Mris not zero for H > 0, there is a non-zero component of STT perpend icular \nto the sample plane, ( )zCo z MM Mrrr\n× ×~τ . For Idc  < 0, τz > 0 near the LR state and τz < 0 near the HR state. \nTherefore, if Mr\n oscillates on a large-amplitude trajectory passing near both t he HR and LR states, the \ntime-average τz due to Idc  is close to zero. In contrast, Iac at the frequency of the oscillations of Mr\n on the \nlarge-amplitude trajectory generates positive time-averag e τz if the phase shift between the oscillations of \nMr\n and ac STT is ~ 180 °. Therefore, for motion of Mr\n phase locked to the ac STT with a ~ 180 ° phase \nshift, the STT from Iac  always pushes Mr\n in the direction perpendicular to the sample plane (increases  \nMz), towards higher energy precessional trajectories. Figure 4c- d shows that for large enough Mz, Mr\ncan \nprecess on large-amplitude trajectories encircling both the L R and the HR energy minima. The frequency X. Cheng et al.  8 December 18, 2009 \n of precession on such high-energy trajectories can be lower th an the frequency of the trajectories \nencircling only the HR or the LR energy minima 5. Such a large-amplitude dynamic state (D) stabilized by \nIac is consistent with the data in Fig. 2d showing a resonance with the amplitude c orresponding to peak-to-\npeak resistance oscillations similar to ∆R and a frequency below the eigenmode frequencies in the LR and \nHR states. The width of the resonance peak in the D state is determined by the bandwidth of phase \nlocking of the oscillations of Mr\n to Iac  rather than by damping. This explains the large line width of  the \nresonance in Fig. 2d, and the dependence of the line width on Idc  in Fig. 3a. This also explains the increase \nof the line width with Iac shown in the inset of Fig. 3c because larger Iac  increases the phase locking \nbandwidth 25 . The saturation of Vmax  with Iac in Fig. 3c is due to the saturation of the amplitude of \nresistance oscillations in the D state at a value close to ∆R (Vmax  ≈ \nπ22 RIdc ∆). \nSurprisingly, the amplitude of the D-state resonance in Vdc (f) has almost the same temperature \ndependence as the adiabatic SR low frequency tail, Vdc (0), (Fig. 3f). Therefore, the D state is thermally \nactivated and the observed D-state resonance belongs to the class of SR phenomena. Since the frequency \nof the D state is greater than the Kramers transition rate s between the HR and LR states, the D-state \nresonance is a high-frequency or non-adiabatic stochastic resona nce (NSR) 41,42,43,44 . We now discuss the \norigin of the NSR effect in our SV system and explain why NSR is  only seen near the HR end of the \nLR↔HR transition. To understand NSR, we consider thermally activa ted transitions among the HR, LR \nand D states. The time-average magnetic energies as well as the  widths of the energy distributions in these \nstates depend on STT as illustrated in Fig. 4b. Consideration of the available traj ectories of Mr\n in the HR, \nLR and D states gives the relation between the time-average e nergies in these states: 〈ED〉 > 〈EHR 〉 > 〈ELR 〉. \nAt the HR end of the LR ↔HR transition, where the NSR is observed, K LR →HR >>K HR →LR , and thus if \nMr\nfalls from the D state to the low-energy LR state, it is rapidly returned to the higher-energy HR state \nas illustrated in Fig. 4f. Since 〈EHR 〉, is close to 〈ED〉, small STT from Iac  pushing Mr\n towards the D state \nis sufficient to supply energy 〈ED〉-〈EHR 〉 and thereby induce the HR →D transition in the half the period of X. Cheng et al.  9 December 18, 2009 \n Iac  for which τz(t) >0. STT from  Iac  constantly supplies energy to the D state but not to the HR and L R \nstates, for which ac STT adds energy for half a period of Iac  and removes energy for the other half. Due to \nthe energy supplied by Iac  to the D state, this state becomes the most stable of the three non-equilibrium \nstates (HR, LR and D) at sufficiently large Iac  and T resulting in K HR →D>> K D→HR  and K HR →D>> K D→LR . \nAs a result, the system spends most of its time in the D sta te and a large-amplitude high-frequency peak \nappears in Vdc (f)  (Fig. 2d). Energy considerations also explain why NSR is not se en at the LR end of the \nLR ↔HR transition. In the LR state, the energy gap to the D stat e is large 〈ED〉-〈ELR 〉 > 〈ED〉-〈EHR 〉 and the \nenergy supplied by ac STT is insufficient to induce the LR →D transition. A detailed quantitative \nunderstanding of NSR requires a Fokker-Planck description of the transiti on rates 23,45 . \nThe large rectified voltage due to NSR can be employed for mi crowave signal detection 46 . Indeed, \nby using high-magnetoresistance MgO magnetic tunnel junctions (M TJ) 47,48 , a rectified voltage of greater \nthan 0.1 V in response to microwave currents of ~ 100 µA may be expected at NSR. The expected \nsensitivity of such an MTJ-based detector, ~20 mV/ µW, is greater than that of Schottky diode detectors.  \nIn conclusion, we observe adiabatic and non-adiabatic SR of magnetiz ation co-excited by ac STT \nand temperature in nanoscale spin valves. Our work demonstrates that combined dc  and ac STT applied to \na nanomagnet stabilize unusual dynamic states of magnetization f ar from equilibrium. The rates of \nthermally activated transitions from these states are sensi tive to the amplitude and phase of the applied \nSTT, and thus nanomagnets driven by ac and dc STT 24,25  provide a convenient playground for studies of \nthermodynamic processes far from equilibrium. The amplitude of mag netization oscillations in the non-\nadiabatic SR regime is nearly two orders of magnitude gre ater than that in the FMR regime for the same \nlevel of the ac STT drive. The non-adiabatic SR dynamics giv e rise to large rectified voltages generated \nby spin valves in response to applied microwave currents, and th us non-adiabatic SR can be utilized in \nsensitive microwave signal detectors of nanoscale dimensions.  \n \n X. Cheng et al.  10 December 18, 2009 \n Acknowledgements \nThis work was supported by the NSF (grants DMR-0748810 and ECCS-0701458) and by the \nNanoelectronics Research Initiative through the Western Instit ute of Nanoelectronics. We thank D. Ralph \nand R. Buhrman, in whose labs the samples were made, for helpful dis cussions and assistance with the \nsample preparation. We acknowledge the Cornell Nanofabrication Fa cility/NNIN and the Cornell Center \nfor Nanoscale Systems, both supported by the NSF, which facilities were used f or the sample fabrication. \nAuthor contributions \nX.C. collected and analyzed data and wrote the paper; I.K. made the s amples and wrote the paper. \nAll authors contributed to the data collection and the preparation of the  manuscript. X. Cheng et al.  11 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1| Spin valve magneto-resistance and current-induced switching.  a , Schematic view of the \nspin valve with approximate directions of magnetization of the pin ned, Co Mr\n, and the free, Mr\n, layers in \nthe high resistance (HR) state of the spin valve. External magnetic field, Hr\n, is applied at 10 ° to the \nsample plane normal. b, Resistance R as a function of H at two temperatures: 295 K and 120 K. R as a \nfunction of direct current, Idc , at 295 K ( c) and 120 K ( d). e, Differential resistance, dV/dI,  as a function of \nH at 120 K measured for several values of Idc . f, Telegraph noise of R between the high (HR) and low (LR) \nresistance states at T = 120 K, H = 2.2 kOe and Idc  = -2.7 mA.  \n \n IrMn Co Cu Py \nMmH 10 o\nMco M\n-2 0 2 424.2 24.4 24.6 \n-2 0 2 420.2 20.4 20.6 \n0 1 2 320.2 20.4 20.6 20.8 \n0 5 10 20.27 20.28 20.29 20.30 20.31 20.32 (a) \n0.0 0.5 1.0 1.5 2.0 2.5 3.0 24.28 24.30 24.32 24.34 \n120 K 295 K \nField (kOe) R (���ΩΩΩ)\n20.20 20.22 20.24 20.26 20.28 \n (b) \n \n \noffset 0.1 ΩΩ ΩΩ2.0 kOe \n1.8 kOe \n1.7 kOe \n1.5 kOe \n1.0 kOe (c) 295 K R (ΩΩΩΩ)\nDirect current (mA)  \n 2.2 kOe \n2.0 kOe \n1.5 kOe \n1.0 kOe \n0.75 kOe offset 0.1 ΩΩ ΩΩ(d) 120 K R(ΩΩΩΩ)\nDirect current (mA)  \n \noffset 0.1 ΩΩ ΩΩ(e) 120 K \n-2.6 mA \n-2.4 mA \n-2.0 mA \n-1.0 mA dV/dI (ΩΩΩΩ)\nField (kOe) 0 mA  \n (f) 120 K   H = 2.2 kOe \n                 I dc = - 2.7 mA \n              R (ΩΩΩΩ)\nTime (ms) X. Cheng et al.  12 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2| Measurements of magnetization dynamics driven by ac spin transfer torque.  a , Spin \ntorque FMR (st-FMR) measurement setup. b, R versus Idc  at T = 120 K, H = 2.2 kOe. Symbols mark the \ndirect current values for which st-FMR data are shown in c-e. Measured response curves, Vdc (f) , at the \nLR↔HR transition:  c, Idc  = -2.6 mA, d, Idc  = -2.8 mA,  e, Idc  = -2.7 mA. Insets: st-FMR spectra far from \nthe LR ↔HR transition ( c,  Idc  = -1.5 mA) and ( d , Idc  = -3.5 mA). f, Vdc (t)  at Idc  = -2.8 mA, for Iac = 0 (open \ncircles) and   Iac = 0.4 mA at f = 2.2 GHz (solid squares). The data in c-e are measured at T = 120 K, H = \n2.2 kOe, Iac = 0.4 mA. \n \n \n \n \n Microwave \nsignal \ngenerator \nLock-in Amplifier DC \ncurrent \nsource rf signal \nIdc \nVdc Reference H         10 o\n-3 -2 -1 020.24 20.26 20.28 20.30 20.32 \n0 1 2 3 4 5-15 -10 -5 0\n0 1 2 3 4 50510 15 20 \n0 1 2 3 4 5-10 -5 05\n0.0 0.5 1.0 0510 15 20 25  - 2.8 mA \n -  2.7 mA \nLR (a) \n R (ΩΩΩΩ)\nDirect current (mA)  - 2.6 mA (b) H= 2.2 kOe \nHR \n-0.10 -0.05 0.00 \n \n   \n5.5 4.5 3.5  I dc = - 1.5 mA  \n Idc  = - 2.6 mA (c) Vdc  (µµµµV)\nFrequency (GHz) 2 3 4-0.3 -0.2 -0.1 0.0 \n \n   Idc = - 2.8 mA \nIdc = - 3.5 mA  \n (d)  Vdc  (µµµµV)\nFrequency (GHz)  \n (e) Idc  = - 2.7 mA Vdc (µµµµV)\nFrequency (GHz)  \nIdc = - 2.8 mA \n Vdc  (µµµµV)\nTime (s)  Iac = 0.4 mA \n I ac  = 0 mA (f) X. Cheng et al.  13 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3| Dependence of ac-STT-driven magnetization dynamic s on current and temperature. a,  \nFull width at half maximum and b, the amplitude, Vmax , of the high-frequency peak in Vdc (f)  response \ncurves as functions of Idc . Grey bands in (a) and (b) mark the crossover regime between t he large-\namplitude high- and low-frequency peaks. In this crossover regime, br oadband response curves Vdc (f)  \nsuch as that shown in Fig 2(e) are observed. Inset in b: blow-up of the low-current region of Vmax (Idc ). c, \nThe dependence of  Vmax  on ac drive current, Iac , for three values of Idc : far from (-1.5 mA, -3.5 mA) and at \n(-2.8 mA) the LR ↔HR transition. Inset: Vdc (f)  response curves in the regime of large Iac  for Idc =-2.8 mA . \nd, Quadratic dependence of Vmax  on Iac  for Idc  far from the LR ↔HR transition. Lines are quadratic fits to \nthe data. e, Phase diagram of the system in the ( H, Idc ) plane. Grey bands mark regions of the LR ↔HR \nresistance transitions at fixed field and temperature. Solid  squares and open circles mark the direct current \nvalues at which maximum rectified signal due to adiabatic and non-adiabatic SR are observed at fixed \nfield and temperature. f, Temperature dependence of the amplitudes of the high frequency HF Vmax ~and low \nfrequency LF Vmin ~st-FMR signals at two values of Idc  : -2.2 mA (open symbols) and -2.8 mA (solid symbols). -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 0.0 0.5 1.0 \n-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 0510 15 20 \n0.0 0.2 0.4 0.6 0.8 0510 15 20 \n-3 -2 -1 01.6 1.8 2.0 2.2 2.4 \n0 100 200 300 -20 -10 010 20 0.0 0.2 0.4 0.6 0.8 0.0 0.3 0.6 0.9 Broadband Response \n(f) (e) (d) (c) (b)   FWHM (GHz) \nDirect current (mA) (a) \nBroadband Response \n-2.0 -1.5 -1.0 0.00 0.15 0.30 \nA\n  Vmax  (µµµµV)\nDirect current (mA) \nIdc =\n0 1 2 3 40510 15 20 \n  Vdc  (µµµµV)\nFrequency (GHz)  0.8  mA \n 0.6  mA \n 0.4  mA Iac  = \n  \n - 2.8 mA \n - 1.5 mA \n - 3.5 mA Vmax  (µµµµV)\nAlternating current (mA) Idc =\n295 K 200 K \n  \n Non-adiabatic SR \n Adiabatic SR \n     HR     LR transition Field (kOe) \nDirect current (mA) 120 K \n - 2.8 mA  VLF \nmin  - 2.2 mA VLF \nmin  - 2.8 mA  VHF \nmax  - 2.2 mA VHF \nmax Voltage ( µµµµV)\nTemperature (K)  - 1.5 mA \n - 3.5 mA \n  Vmax  (µµµµV)\nAlternating current (mA) \n~~~~X. Cheng et al.  14 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4| Stochastic resonance energy diagrams.  a, Spherical coordinate system in which the energy \nof the Py nanomagnet,  E( θ,ϕ), is described by Eq. (2). b, E( θ,ϕ) as a function of ϕ for θ = 3 π/8, the \napproximate equilibrium polar angle of the Py magnetization in t he HR and LR states. This energy is a \ndouble-well potential with the minima corresponding to the HR and LR states. Schematic energy \ndistributions of magnetization in the HR, LR and the dynamic st ate D at non-zero Idc  and Iac  are shown. c,  \n3D sketch of the energy surface E( θ,ϕ); thick solid lines schematically show magnetization traject ories \nwith time-average energies, 〈ELR 〉, 〈EHR 〉, 〈ED〉 in the LR, HR and D states. d, Contour plot of E( θ,ϕ); \ndashed lines schematically show 〈ELR 〉, 〈EHR 〉, 〈ED〉. e, Alternating STT, τac , always pushes magnetization \nout of the sample plane towards high energy trajectories of the D state if the phase of the magnetization \noscillations is ~ 180 ° with respect to the ac STT drive. This τac  stabilizes the large-amplitude dynamic \nstate D. f,  Sketch of the Kramers transition rates among the HR, LR a nd D states at the non-adiabatic \nstochastic resonance. Solid arrows show the dominant path bringing magnetization into the D state. \nDashed arrows show transitions out of the D state. \n(a) (b) \n(e) (d) (c) \n(f) E\nLR HR D\nKLR     HR KD     LR KD    HR KHR    D HR \nMco H10 o\nττ ττac ττ ττac D\nLR MLR HR Dx\nyz\nφφ φφθθ θθ\nMco M\neasy axis \neasy plane H\n10 o\nφφ φφππ ππ/2 0E\nππ ππθ = θ = θ = θ = 3/8 ππ ππ\nLR HR \nLR HR D\n0 ππ ππππ ππ/4 \nφφ φφ ππ ππ/2 \nθθ θθππ ππ/2 D\nHR LR 33 33ππ ππ/8 X. Cheng et al.  15 December 18, 2009 \n                                                            \n1 Slonczewski, J. C. Current-driven excitation of ma gnetic multilayers. J. Magn. Magn. Mater.  159 , L1-L7 (1996). \n2 Berger, L. Emission of spin waves by a magnetic mu ltilayer traversed by a current. Phys. Rev. B  54 , 9353-9358 \n(1996). \n3 Stiles, M.D. & Zangwill, A., Anatomy of spin-trans fer torque, Phys. Rev. B 66 , 014407 (2002). \n4 Tserkovnyak, Y., Skadsem, H.J., Brataas, A. & Baue r, G.E.W. 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O. Box 3049, 67663 Kaiserslautern, Germany\n(Dated: August 5, 2022)\nWe analyze theoretically the demagnetization dynamics in a ferromagnetic model system due\nto the interplay of spin-orbit coupling and electron-electron Coulomb scattering. We compute the\nk-resolved electronic reduced spin-density matrix including precessional dynamics around internal\nspin-orbit and exchange \felds as well as the electron-electron Coulomb scattering for densities and\nspin coherences. Based on a comparison with numerical solutions of the full Boltzmann scattering\nintegrals, we establish that the k-resolved reduced spin-density matrix dynamics are well described\nusing a simpler generalized relaxation-time ansatz for the reduced spin-density matrix. This ansatz\nallows one to relate the complicated scattering dynamics underlying the demagnetization dynamics\nto a physically meaningful momentum relaxation time \u001c. Our approach reproduces the behaviors\nof the demagnetization time \u001cm/1=\u001cand\u001cm/\u001cfor the limits of short and long \u001c, respectively,\nand is also valid for the intermediate regime. The ansatz thus provides a tool to include the\ncorrect demagnetization behavior in approaches that treat other contributions to the magnetization\ndynamics such as transport or magnon/phonon dynamics.\nI. INTRODUCTION\nThe relaxation of electrically or optically induced elec-\ntronic spin polarizations in semiconductors and simple\nmetals has been studied for more than 50 years and\nhas important connections to the spin-dependent dynam-\nics of electrons in ferromagnets. Spin relaxation dy-\nnamics in semiconductors have often been interpreted in\nterms of three di\u000berent \\classical\" mechanisms: Elliott-\nYafet(EY), Dyakonov-Perel(DP), and Bir-Aronov-Pikus,\nwhich were invented, respectively, for semiconductors\nwith degenerate bands, for electronic bands with small\nspin splitting and small-band gap systems with electron-\nhole exchange interactions, see Refs. [1, 2] for a gen-\neral overview. The most widely applicable EY and\nDP mechanisms were based originally on a combina-\ntion of spin-orbit coupling with electron-impurity and\nelectron-phonon scattering. In semiconductor spintron-\nics, it was realized about 20 years ago that Coulomb scat-\ntering, which arises from the spin-independent interac-\ntion that does not directly couple electrons to the lattice\nvia phonons or impurities, can also contribute to spin\nrelaxation. More precisely, it leads to spin dephasing in\nthe presence of a k-dependent spin-orbit induced splitting\nbetween\"and#states, which can be described in terms\nof ak-dependent internal e\u000bective magnetic \feld [3, 4].\nIn ferromagnetic metals, a pronounced quenching of\nthe magnetization, which is mainly related to d-band\nelectrons, can be observed after excitation with an ul-\ntrashort optical pulse. While this is a more sizable e\u000bect\nthan the relaxation of an induced spin polarization of a\nsmall density of excited electrons in semiconductor s- or\np-like bands, the concept of Elliott-Yafet spin dynamics\n\u0003hcsch@physik.uni-kl.devia electron-phonon scattering [5] was introduced early\non as a mechanism to explain the reduction of spin angu-\nlar momentum observed in the demagnetization process\nof ferromagnets [6].\nThe present paper is concerned with the character-\nistics of magnetization dynamics that are caused by a\n\\spin-relaxation like\" approach to magnetization dynam-\nics. Compared to semiconductors, ferromagnetic metals\npossess a more complicated ground state with correlated\nd-electron bands at the Fermi level, more complicated\nelementary excitations (magnons) and arguably a dif-\nferent electron-phonon coupling (spin-lattice coupling).\nThe mechanism of incoherent electronic dynamics [7, 8]\ntogether with spin-orbit coupling considered in this pa-\nper thus competes with or complements other mecha-\nnisms, such as coherent electronic dynamics [9] coupling\nFermi-level electrons to more tightly bound orbitals, di-\nrect angular momentum transfer to phonons [10, 11] and\nmagnon interactions [12, 13], to name only a few. The\ndominant scattering mechanisms contributing to the in-\ncoherent electron dynamics arise from the interaction\nwith phonons and other electrons. Electron-phonon scat-\ntering is often regarded as important because it can lead\nto electronic spin \rips via coupling to the lattice, [6]\nwhich \fts into a picture of a three-temperature model\nas the spin-lattice coupling. Theoretical calculations in-\ndicate that the spin-dependent electron-phonon inter-\naction, in which the phonons directly change electron\nspin, gives only a small contribution to electronic dy-\nnamics [14, 15]. Instead, the main impact of electron-\nphonon scattering is related to providing an essentially\nspin-independent momentum scattering process that af-\nfects the spin in combination with electronic precessional\nspin dynamics [16]. If the electron-phonon scattering\ncontributes to magnetization dynamics mainly because\nit acts as a momentum scattering channel for electrons,\nthen the electron-electron scattering provides an addi-arXiv:2208.02356v1  [cond-mat.mtrl-sci]  3 Aug 20222\ntional momentum scattering channel that should be even\nmore important for highly excited electrons because it\ncan act on an even shorter timescale of 10 femtoseconds.\nThe demagnetization dynamics corresponding to the lat-\nter mechanism have so far been investigated at the level\nof Fermi's Golden Rule rates for transitions between spin-\nmixed states due to the Coulomb interaction [7, 8]. This\napproach can explain a sizable contribution to demag-\nnetization, in particular, if a dynamical Stoner exchange\nsplitting is included [8, 17].\nIn this paper we investigate the electron-electron scat-\ntering contribution to the spin-dependent dynamics in\na ferromagnetic model system using a similar approach\nas we have employed for electron-phonon scattering [16].\nThat is, we go beyond Fermi's Golden Rule rates for\nCoulomb scattering between electronic distributions in\nk-space and include the precessional dynamics of co-\nherences, i.e., the o\u000b-diagonal components of the spin-\ndensity matrix, around anisotropic e\u000bective spin-orbit\n\felds.\nUsing a screening parameter to control the strength\nof the electron-electron Coulomb scattering, we \fnd that\nthe in\ruence of this scattering mechanism, including its\ne\u000bects on the precessional dynamics, can be captured\nwell using an extended relaxation time ansatz with a sin-\ngle e\u000bective momentum relaxation time \u001cfor any given\ninteraction strength. The ansatz and the e\u000bective re-\nlaxation time provide an arguably more general descrip-\ntion of relaxation processes than what can be obtained\nmicroscopically from our simple model band structure.\nIn terms of this relaxation time we can consistently de-\nscribe a whole range of di\u000berent demagnetization behav-\niors from a proportionality to \u001c\u00001to the proportionality\nto\u001c, including the important intermediate regime, which\nhas, to the best of our knowledge, not been mapped out\nin a ferromagnetic material yet. For semiconductors and\nnon-magnetic metals, similar scalings of the spin relax-\nation rates/times have been found in their dependence of\nquasiparticle broadening [18, 19] and doping concentra-\ntions [20].\nII. THEORETICAL APPROACH\nOur theoretical approach to determine the demagneti-\nzation dynamics and the quantities involved in the elec-\ntronic dynamics under the in\ruence of internal spin-orbit\n\felds and electron-electron scattering proceeds by \frst\ndetermining the single-particle energies and states of the\nBloch electrons in our model band structure and then\nsetting up and numerically solving the dynamical equa-\ntions for the reduced electronic spin-density matrix in-\ncluding electron-electron Coulomb scattering. We then\nintroduce here a relaxation time ansatz that can approxi-\nmate the spin-conserving electron-electron Coulomb scat-tering well. The ansatz involves only a single relaxation\ntime and introduces a time-dependent e\u000bective quasi-\nequilibrium spin-density matrix, to which the system\nevolves during demagnetization and remagnetization.\nA. Hamiltonian and Dynamical Equation\nThe derivation of equations of motion (EOMs) for the\nelectron-electron interaction is closely related to what we\nhave presented in Refs. 21 and 16, but the general ap-\nproach has been well established for semiconductor spin-\ntronics earlier [22]. Here, we only give a short overview of\nthe model system, which uses a two-dimensional k-space\nto keep the single-particle band structure simple. The\nsingle-particle contribution to the system Hamiltonian is\ngiven by\n^H(k) =^Hkin(k) +^HSO(k) +^HStoner: (1)\nwith the e\u000bective-mass contribution ^Hkin(k) =~2k2\n2m\u0003.\nThe spin-orbit contribution is assumed to be of the\nRashba form and can be written in terms of the vector\nof Pauli matrices ^~ \u001b\n^HSO(k) =\u000b\u0010\n^~ \u001b\u0002k\u0011\n\u0001ez=\u000b(^\u001bxky\u0000^\u001bykx):(2)\nIts strength is controlled by the Bychkov-Rashba param-\neter\u000b. The mean-\feld Hubbard contribution leads to\na Stoner contribution ^HStoner =Umwhich depends on\nthe magnetization m, see Eq. (7), and an e\u000bective on-site\ninteraction energy U.\nFigure 1 illustrates the important features of the\nmodel. The model band structure exhibits a k-dependent\nband splitting and k-dependent Bloch spinors, which\nwe denote by*and+to indicate that they are not\npure spin states. The splitting \u0001 Ek\u0011\"k*\u0000\"k+of\nthe bands shown in Fig. 1(a) ranges from \u0001 Ek=0=\n400 meV to \u0001 Ek=10 nm\u00001= 725 meV for di\u000berent k-\nvalues. Fig. 1(b) and (c) depict the k-local spin expecta-\ntion valueshk\u0016j~ \u001bjk\u0016i=2 for the Bloch spinors jk\u0016i: (b)\nthe longitudinal component vs. kand (c) the components\nperpendicular to zvs. polar angle for a \fxed k. Around\nk= 0 states are essentially \"and#spin states, but the\nmixing increases with increasing k, or equivalently, en-\nergy. For highly excited electrons at k-states as shown in\nFig. 1(c), the Bloch states are considerably spin-mixed.\nThe electronic quantum state is described by the re-\nduced spin-density matrix \u001a\u0016\u00160\nk=h^cy\nk\u0016^ck\u00160i, where ^c(y)\nk\u0016\nis the annihilation (creation) operator of an electron\nwith momentum kin band\u0016. In the equation of mo-\ntion (EOM) for the spin-density matrix, we include the\nelectron-electron Coulomb interaction at the level of 2nd\nBorn scattering integrals, which can be derived using\nGreen function or reduced density-matrix techniques [23{\n25],3\n−10 0 10\nk(nm−1)024energy (eV)(a)\n−10 0 10\nk(nm−1)−0.50.00.5sz\nk/¯h(b)\n0 1 2\nϕ/π−0.50.00.5s(x/y)\nk/¯h(c)\nFIG. 1. Self-consistently calculated k-dependent band structure (a) with the corresponding spin structure in zdirection (b).\nThe'-dependent spin structure in x(red) andy(blue) direction are shown for a \fxed k= 10 nm\u00001(c). The solid lines\ncorrespond to the energies and k-local quantization axes in the lower band, the dashed lines to those in the upper band. The\nparameters used throughout this paper are: Stoner interaction parameter U= 400 meV, Rashba parameter \u000b= 30 meVnm\u00001,\nelectron density ne= 0:7 nm\u00001. At each k-point, the quantization axis of the upper and lower bands point in opposite\ndirections.\n@\n@t\u001a\u0016\u00160\nk=i\n~(\"k\u0016\u0000\"k\u00160)\u001a\u0016\u00160\nk\n+\u0019\n~X\nlqX\n\u00161\u00162\u00163\u00164\u00165\u00166\u00167\u0010\nV\u0016\u00161\u00162\u00163\nklq\u0011\u0003\u0010\nV\u00164\u00165\u00166\u00167\nklq\u0000V\u00164\u00165\u00166\u00167\nl+qlk\u0000l\u0011\n\u000e\u0010\n\u0001E\u00164\u00165\u00166\u00167\nklq\u0011\nh\n\u001a\u00163\u00167\nk+q\u001a\u00162\u00166\nl\u0010\n\u000e\u00161\u00165\u0000\u001a\u00165\u00161\nl+q\u0011\u0010\n\u000e\u00160\u00164\u0000\u001a\u00164\u00160\nk\u0011\n\u0000\u001a\u00164\u00160\nk\u001a\u00165\u00161\nl+q(\u000e\u00162\u00166\u0000\u001a\u00162\u00166\nl)\u0010\n\u000e\u00163\u00167\u0000\u001a\u00163\u00167\nk+q\u0011i\n+\u0019\n~X\nlqX\n\u00161\u00162\u00163\u00164\u00165\u00166\u00167V\u00160\u00161\u00162\u00163\nklq\u0010\nV\u00164\u00165\u00166\u00167\nklq\u0000V\u00164\u00165\u00166\u00167\nl+qlk\u0000l\u0011\u0003\n\u000e\u0010\n\u0001E\u00164\u00165\u00166\u00167\nklq\u0011\nh\n\u001a\u00167\u00163\nk+q\u001a\u00166\u00162\nl\u0010\n\u000e\u00165\u00161\u0000\u001a\u00161\u00165\nl+q\u0011\n(\u000e\u00164\u0016\u0000\u001a\u0016\u00164\nk)\u0000\u001a\u0016\u00164\nk\u001a\u00161\u00165\nl+q(\u000e\u00166\u00162\u0000\u001a\u00166\u00162\nl)\u0010\n\u000e\u00167\u00163\u0000\u001a\u00167\u00163\nk+q\u0011i\n:(3)\nThe \frst row describes a coherent precession of\nthe o\u000b-diagonal contributions of the spin-density ma-\ntrix, i.e., the coherences \u001a\u0016\u00160\nk;\u00166=\u00160, due to\nthe splitting between the bands \u0016and\u00160atk.\nThe remaining terms are electron-electron scatter-\ning contributions with the Coulomb-matrix elements\nV\u00161\u00162\u00163\u00164\nklq=Vqhk\u00161jk+q\u00164ihl+q\u00162jl\u00163iwhereVqde-\nnotes a screened Coulomb potential depending on the\nmomentum qtransferred from the electron with initial\nmomentum kto the electron with \fnal momentum l, i.e.,\nk!k+qandl+q!l. To obtain the Boltzmann-like\nscattering integrals in Eq. 3 one has to employ a Markov\napproximation not only for real occupation-number dis-\ntributions but also for complex coherences with a preces-\nsional contribution stemming from the \frst term. This\nprecessional frequency is removed by transforming to a\nrotating frame, in which the Markov approximation can\nbe made, and then transforming back. [22, 26, 27]\nSince the k-space for the single-particle states de\fned\nabove is 2-dimensional, we use the screened Coulomb po-\ntential in two dimensions in the form Vq=e2\n2V\"0\"b(q+\u0014)[28] with the elementary charge e, the normalization vol-\numeV, the dielectric constant \"0, the screening constant\n\"band the screening parameter \u0014. This is not an essential\nrestriction of our dynamical approach to 2-dimensional\nphysics, it serves here only to simplify the numerical cal-\nculations, in particular the sums over momenta land\nkon the RHS of Eq. (3), which have to be calculated\nin every time step. Below we analyze the dependence\nof the dynamics in dependence of the inverse screen-\ning length, which is determined by the band structure,\ncarrier density, and possibly by the dielectric environ-\nment. In our simpli\fed band structure we essentially\nregard this as a model parameter and choose values on\nthe order of 20 nm\u00001and smaller. This value is consis-\ntent with a calculation of the screening parameter \u0014for\nelectrons in parabolic bands [23, 28] in the 2d-limit via\n\u0014=m\u0003e2=(2\u0019~2\"b\"0)f(k= 0) with relative background\nscreening constant \"b= 1.\nOne goal of this paper is to compare and contrast the\nelectronic dynamics described by the full density-matrix\nwith those obtained using occupation numbers . In prin-4\nciple, an approach that uses only occupation numbers\nn\u0016\nk:=\u001a\u0016\u0016\nk, i.e., the diagonal elements of the density\nmatrix, is an approximation to the full density matrix.\nIn this case, the Boltzmann scattering integrals are es-\nsentially rates as one would obtain from Fermi's Golden\nRule, which connect non-pure spin states and thus lead to\nspin-\rip transitions [6, 29]. Because the Coulomb inter-\naction is spin-independent, the Coulomb scattering alone\ncannot cause a transition that changes the magnetiza-\ntion. For electron scattering dynamics between non-pure\nspin states, the restriction to occupation dynamics, which\nneglects the in\ruence of the o\u000b-diagonal parts of the spin-\ndensity matrix, there is no conservation of ensemble spin\nand one obtains demagnetization due to Coulomb scat-\ntering [7, 8]. In order to elucidate this, we also evaluate\nEq. (3) at the level of Fermi's Golden Rule for occupa-\ntions. This approach is often called \\Boltzmann scat-\ntering\", but this may cause confusion in our case be-\ncause we also have Boltzmann-like scattering integrals for\nallelements of the spin-density matrix in the complete\nEOM (3). To di\u000berentiate between the full spin-density\nmatrix calculation and the calculation that uses only the\noccupations, we refer to them as \\generalized Boltzmann\nscattering\" (or simply \\full\") and \\occupation-number\napproximation\", respectively.\nThe numerical solution of the EOM (3) requires a con-\nsiderable accuracy to keep the numerical errors from ac-\ncumulating over the demagnetization and remagnetiza-\ntion dynamics, which would spoil the important conser-\nvation laws. We thus use a Runge-Kutta-type integration\nmethod developed by Dormand and Prince [30] with a dy-\nnamical time-step control to keep a high accuracy while\nalso optimizing the CPU time.\nB. Relaxation-time approximation\nIn addition to the dynamics of the spin-density ma-\ntrix with generalized Boltzmann scattering, which is non-\nlocal ink-space, we will use a relaxation-time ansatz that\nis designed speci\fcally for spin-polarized systems with\nspin-orbit coupling. In Ref. [31] we applied the ansatz in\nthe context of optically driven dynamics mainly to sim-\nplify the numerical e\u000bort. Here, we stress that it allows\none to replace the complexity of the scattering integrals\nby introducing a single physically meaningful relaxation\ntime that characterizes the complex, k-dependent scat-\ntering dynamics. It thus provides a simple and intuitive,\nbut also accurate description of this scattering process\nthat should also have applications to calculations involv-\ning transport and/or in combination with other scatter-\ning mechanisms, such as electron-magnon scattering.\nThe ansatz is based on suitably de\fned quasi-\nequilibrium density matrices of the general form\n~\u001aeq=f(T;\u0016;\u0010z) (4)\nwherefis a Fermi-Dirac distribution depending on tem-\nperatureT, chemical potential \u0016and spin accumula-tion\u0010. The parameters T,\u0016and\u0010are determined such\nthat the distribution reproduces a prescribed charge den-\nsity, energy density and spin polarization. In this work,\nwe only consider the spin polarization in z-direction due\nto the symmetries of our model system. The grand\ncanonical Hamiltonian for non-interacting electrons cor-\nresponding to Eq. (4) is\n^K=^H+\u0016^N\u0000\u0010z^\u001bz; (5)\nwhere ^His the many-particle Hamiltonian correspond-\ning to (1) discussed above and ^Nis the particle number\noperator.\nThe expectation values of the particle density, the spin\npolarization and the energy density used in the quasi-\nequilibrium distribution (4) will be obtained from those\nof the non-equilibrium density matrix \u001aas it arises during\nthe dynamics and are calculated as follows. The electron\ndensity is given by\nne=1\nVX\n\u0016X\nk\u001a\u0016\u0016\nk(6)\nthe spin polarization/magnetization mby\nm=1\nVneX\n\u0016\u00160X\nkhk\u0016j^szjk\u00160i\u001a\u0016\u00160\nk(7)\nand the energy density \"by\n\"=1\nVX\n\u0016X\nk\"k\u0016\u001a\u0016\u00160\nk: (8)\nHere,Vis the normalization volume. Note, in particu-\nlar, that the spin-polarization dynamics also include the\nmicroscopic coherences \u001a\u0016\u00160\nk; \u00166=\u00160;and that, since we\nwill only be discussing the relative change of the spin\npolarization further below, magnetization and spin po-\nlarization are interchangeable.\nOur relaxation-time approximation consists of replac-\ning the scattering integrals by the following contribution\nto the equation of motion for the spin-density matrix\n@\n@t\u001a\u0016\u00160\nk\f\f\f\nrel=\u0000\u001a\u0016\u00160\nk\u0000~\u001a\u0016\u00160\nk\n\u001c: (9)\nHere, ~\u001a\u0016\u00160\nkis the quasi-equilibrium reduced spin-density\nmatrix introduced in Eq. (4), which is diagonal in the\neigenbasis of the grand-canonical single-particle hamilto-\nnian ^K, see Eq. (5). The dynamics of the spin-density\nmatrix\u001a\u0016\u00160\nk, however, is calculated in the eigenbasis of\nthe regular single-particle Hamiltonian, so that one must\ntransform the density matrix accordingly, i.e., from the\neigenbasis of ^Kto that of ^H. Due to the transformation\nbetween the ^Kand ^Hbases, ~\u001a\u0016\u00160\nk, which is diagonal in\nthe grand-canonical basis, has o\u000b-diagonal elements in\nthe basis of ^Hand therefore also describes the in\ruence\nof scattering processes on the o\u000b-diagonal elements of5\nthe spin-density matrix, which are needed for the correct\ndetermination of the ensemble spin expectation value,\ni.e., the magnetization. With this approach, our Eq. (9)\nemploys only a single relaxation time \u001cand mimics in-\ncoherent electron-electron scattering as it conserves the\nrespective conservation laws. Note that the relaxation\ntime is by construction independent of kand energy and\nacts in a di\u000berent way compared to relaxation times\nthat are usually introduced as energy-dependent out-\nscattering rates and then averaged over suitably chosen\nquasi-equilibrium distribution functions [32, 33]. Such an\nansatz can also model spin-conserving electron-phonon\nscattering if one uses a di\u000berent quasi-equilibrium distri-\nbution with a \fxed temperature of the phonon bath Tpn.\nIn this case, one only needs to determine \u0016and\u0010in order\nto conserve density and spin polarization.\nC. Initial conditions and model excitation\nThe magnetization dynamics discussed below start\nfrom a magnetic equilibrium state and since the Stoner\ncontribution of the Hamiltonian (1) depends on the spin-\npolarization mof the system, this equilibrium state is\ndetermined self-consistently as follows: We start from an\narbitrary value of minz-direction to set the preferen-\ntial direction and iteratively (i) calculate the new band\nstructure according to m, (ii) populate this band struc-\nture with electronic equilibrium (Fermi) distributions\n\u001a\u0016\u00160\nk=1\ne\f(\"k\u0016\u0000\u0016)+ 1\u000e\u0016\u00160: (10)\nby adjusting the chemical potential \u0016Csuch that our de-\nsired electronic density nefor a given equilibrium tem-\nperatureTeqis reproduced and (iii) calculate the new\nspin polarization mnewin this band structure and repeat\nsteps (i){(iii) until the spin polarization di\u000berence \u0001 m\nbetween two consecutive iterations is small enough (we\nchose \u0001m< 10\u00009).\nIn order to achieve a comparison of the di\u000berent mag-\nnetization dynamics we employ a simple model excitation\nby an ultrashort optical pulse. As before [16, 34] we as-\nsume that the electronic energy is instantaneously raised\nto an excitation temperature Tex\u001dTeq= 100 K and the\nexcited electrons are distributed according to Eq. (10)\nwith the excitation temperature in the self-consistently\ndetermined band structure.\nThe instantaneous heating leads to di\u000berent chemical\npotentials for the *and+bands as well as to a small\nchange of the spin polarization mdue to the k-dependent\nspin-mixing of the states. This approach does not change\nthe electronic density in each band and is numerically\nsimple and controllable by a single parameter Tex; it is\ndesigned to capture qualitatively the e\u000bect of an ultra-\nshort pulse that deposits energy in the electronic system,\nsee, e.g., Ref. [14] for a more detailed description of this\nprocess. With this model of the excitation process we ne-\nglect optically driven interband coherences that may beexcited by the excitation pulse, as the purpose of this pa-\nper is to analyze the possible dynamics in the incoherent\nregime.\nWe also choose a high excitation temperature Texof\n4000 K in order to clearly exhibit the dynamical features.\nThe temperature Texis purely a measure of the excita-\ntion of the electronic system and is well above the Curie\ntemperature, which is TC\u00191030 K for the model pa-\nrameters listed in Fig. 1.\nIII. RESULTS\nWe start with the generalized Boltzmann scatter-\ning. The main quantity of interest to us is the time-\ndependence of the magnetization m, i.e., the spin po-\nlarization of the electrons in the split bands. As it is\nour goal to compare the dynamics for di\u000berent ratios\nof typical scattering times to typical precession times,\nwe choose here to vary only the screening parameter \u0014,\nwhich changes the matrix element of the Coulomb inter-\naction and thus the k-dependent scattering rates. Using\nthis one adjustable parameter to control the Coulomb\nscattering rates, we intend to illustrate the range of pos-\nsible behaviors of the magnetization dynamics and how\nwell these can be captured with our extended relaxation\ntime ansatz. We therefore make no e\u000bort here to connect\n\u0014to a variable that can be tuned in experiment.\nFigure 2 shows the demagnetization dynamics ob-\ntained for the excitation conditions discussed in Sec. II C.\nShown are the relative magnetization changes result-\ning from the full calculation (solid lines) and the cal-\nculation using the occupation-number approximation\n(dashed lines) for small, intermediate and large screening\nparameters \u0014= 2 nm\u00001,\u0014= 10 nm\u00001and\u0014= 20 nm\u00001.\nWe do not include electron-phonon coupling, or any other\ncoupling to an energy bath that would absorb the en-\nergy transferred to the electronic system by the opti-\ncal excitation over time. The system therefore stays in\nthe demagnetized state even for times t&0:1 ps. For\nthe occupation-number approximation a smaller screen-\ning parameter \u0014(faster scattering) always leads to faster\nspin/magnetization dynamics. In the full calculation,\nthis behavior is also visible but less pronounced and the\nmagnetization dynamics only coincide for larger \u0014(slower\nscattering) with those of the occupation-number approx-\nimation. For values of \u0014.5 nm\u00001the two calculations\ndeviate at shorter times and the demagnetization dynam-\nics of the full calculation are considerably slower. This is\nshown more prominently in Fig. 3, which plots the same\ncurves vs. a logarithmic time scale to display the behavior\nat very short time scales. Around and below the preces-\nsion timeTp=h=\u0001Ek\u001910 fs one observes that Tpef-\nfectively sets a lower limit for any magnetization change\nby this mechanism. The reason for this discrepancy is\nthat the occupation-number dynamics neither includes\nprecessional spin dynamics nor its dephasing due to the\nCoulomb interaction. The Golden-Rule like scattering in6\n0 25 50 75 100\nt(fs)−50−250magnetization change (%)κ= 20 nm−1\nκ= 10 nm−1\nκ= 2 nm−1\nFIG. 2. Relative magnetization change vs. time for di\u000berent\nscreening parameters \u0014for the full calculation (solid lines)\nand the occupation-number approximation (dashed lines).\nthe occupation-number dynamics simply gets faster for\nlarger interaction matrix elements.\nIn an earlier paper [16] we have studied the in\ruence\non spin dephasing of electron-longitudinal-phonon scat-\ntering, for which the typical momentum scattering rates\nare longer than the precession times of electronic spins\naround typical exchange \felds (determined by the ex-\nchange splitting between the bands). For this mismatch\nof scattering and precession times, we found that the\noccupation-number approximation agreed well with the\nfull calculation, and that it is thus justi\fed to use an\nElliott-Yafet like description, i.e., an incoherent scat-\ntering process that leads to a spin change, or, as it\nis often called, a spin \rip, and exhibits a linear time-\ndependence of demagnetization times on typical electron-\nphonon scattering times. For the electron-electron scat-\ntering considered in this paper, the occupation-number\napproximation reproduces the result of the full calcula-\ntion only for stronger screening e\u000bects. The discrepancy\nbetween the occupation-number approximation and the\nfull dynamics should be even more pronounced in systems\nwith a smaller splitting, where the precession frequencies\nare smaller.\nIn Fig. 4 we turn to a comparison of the full calculation\nwith the relaxation-time ansatz. We start with the de-\nmagnetization characteristics as obtained for two di\u000ber-\nent screening parameters and suitably chosen relaxation\ntimes\u001cfor each screening parameter. We obtain a good\nagreement of the demagnetization dynamics between the\n\u0014= 20 nm\u00001and\u001c= 13 fs curves as well as between the\n\u0014= 2 nm\u00001and\u001c= 1:7 fs curves. This is already a sat-\nisfying result, as a look back to Fig. 2 shows that such a\ngood agreement for di\u000berent \u0014values cannot be achieved\nusing the occupation-number approximation. However,\nthe magnetization change is a k-integrated quantity, and\nwe would also like to compare the k-resolved dynamics.\nFigure 5 compares the results of the full calculation\nand the relaxation-time approximation for the dynamics\n100101102\nt(fs)−50−250magnetization change (%)κ= 20 nm−1\nκ= 10 nm−1\nκ= 2 nm−1FIG. 3. Same data as in Fig. 2 displayed with a logarithmic\ntime axis to emphasize the short-time dynamics. Regardless\nof\u0014, there is only a marginal magnetization change for the\nfull calculation (solid lines) on short times <4 fs since the\nbuild-up of the coherences is limited by the precession time\nof\u001910 fs. The demagnetization process in the occupation-\nnumber approximation starts at arbitrarily early times for\nstronger scattering, i.e., smaller \u0014.\nof the coherence \u001a*+\nkat~k= 3:3 nm\u00001for di\u000berent values\nof\u0014. This particular ~kis located near the Fermi edge of\nthe lower band so that its dynamics play an important\nrole during the whole demagnetization process. The dif-\nferentk-dynamics for the two cases that arise in the full\nmicroscopic calculation are reproduced well by the cal-\nculation with the extended relaxation time ansatz. For\nthe full calculation and the relaxation-time ansatz pre-\ncessional dynamics of the coherence are clearly visible at\nearly times t.30 fs for\u0014= 20 nm\u00001and the \ft with\n\u001c= 13 fs. There are two contributions we want to dis-\ncuss here. First, the precessional dynamics is driven by\ntransitions k!k+qwhich conserve the vector spin and\nthus lead to a mismatch of the spin with the local quanti-\nzation axis at k+q. Second, the scattering also dephases\nthe precession of the spin coherences toward a \fnite value\nat aroundt= 30 fs, at which time the demagnetization\nis not even half completed (see Fig. 2). After that time,\nthere is only a slow relaxation of the spin coherence. This\nresult is qualitatively similar to that obtained for spin-\nconserving electron-phonon scattering [16].\nFor\u0014= 2 nm\u00001in the full calculation, the respec-\ntive\u001c= 1:7 fs that \fts best in the relaxation time ap-\nproach becomes shorter by a factor of 8, indicating a\nmuch faster scattering. In this case one cannot discern a\nprecessional motion in the coherence, but rather strongly\ndamped dynamics with one intermittent maximum. The\nwhole spin-density dynamics here occurs on essentially\nthe same time scale as the corresponding demagnetiza-\ntion curve in Fig. 4. In both cases the relaxation-time\nansatz comes close to the full calculation even though it\nreplaces the complicated electron-electron scattering dy-\nnamics ink-space by a k-local expression with a constant,7\n0 25 50 75 100\nt(fs)−50−250magnetization change (%)κ= 20 nm−1\nτ= 13 fs\nκ= 2 nm−1\nτ= 1.7 fs\nFIG. 4. Relative magnetization change vs. time for the\nfull calculation with two di\u000berent screening parameters (solid\nlines) and two relaxation time calculations (dashed lines).\nThe remaining parameters are as in Fig. 2.\ni.e.,k-independent, relaxation time. That the quantita-\ntive agreement of the k-resolved and k-integrated quan-\ntities is so good is likely also due to our model excitation\nwhich creates excited electron distributions that are close\nto hot Fermi-Dirac distributions. For excitations that ex-\nhibit stronger non-equilibrium characteristics, the quan-\ntitative agreement may not be quite as good, but the\nrelaxation time approximation should still capture the\nmost important features of the spin-dependent electronic\nscattering dynamics.\nUp to now we have demonstrated that the relaxation-\ntime ansatz reproduces the magnetization dynamics\ndue to precessional dynamics and scattering quite\nwell. Because the complicated scattering dynamics are\nparametrized by the relaxation time \u001cwe can use it to\nstudy the in\ruence of scattering on the electronic spin-\ndependent dynamics via this single parameter. For a\nrange of relaxation times \u001c, we calculate the demag-\nnetization curves as in Fig. 4 and subsequently extract\nthedemagnetization time \u001cmby \ftting the demagne-\ntization dynamics by an exponential function m(t) =\nb\u0000aexp(t=\u001cm). Fig. 6 plots the demagnetization times\nobtained from this \ftting procedure vs. the relaxation\ntime\u001cas black diamonds. For long relaxation times \u001c,\nthe demagnetization time increases with \u001c, but for short\nrelaxation times, they increase with decreasing \u001c. The\nscaling for small and large \u001cis in agreement with that\nfound in Ref. 20 for electronic dynamics in quantum\nwells. Importantly, there exists a minimum of \u001cm(\u001c)\nfor intermediate \u001c. Calculations using the occupation-\nnumber approximation (not shown) miss the behavior\nat small\u001cand yield only a steady decrease of \u001cmwith\ndecreasing relaxation times \u001c. This behavior is already\nevident in Fig. 2 where the occupation-number approxi-\nmation leads to ever faster demagnetization.\nThe quantitative dependence of \u001cmon\u001cis \ft to a func-\n−2−10Re(ρ⇑⇓\n˜k)κ= 20 nm−1\nτ= 13 fs\nκ= 2 nm−1\nτ= 1.7 fs\n0 25 50 75 100\nt(fs)012Im(ρ⇑⇓\n˜k)FIG. 5. Real (top) and imaginary part (bottom) of the coher-\nence at ak-point ~k= 3:3 nm\u00001near the Fermi edge vs time\nfor the full calculation with two di\u000berent screening parame-\nters (solid lines) and two relaxation time calculations (dashed\nlines). The remaining parameters are as in Fig. 2.\ntion of the form\n\u001cm=A\n\u001c+B\u001c ; (11)\nwhich is shown as red solid line in Fig. 6. We see a re-\nmarkably good agreement of the \ft and the extracted\ndemagnetization times from the relaxation time calcu-\nlations. The parameter A= 10:1 fs2can be put into\ncorrespondence with the precession frequency of the spin-\nsplitting ~\n = \u0001EkviaA'\n\u00002. In the short \u001climit\nwe thus obtain a connection \u001cm\u0019\n\u00002=\u001c. The form of\nEq. (11) has already been suggested by Refs. 18 and 19.\nIt can be obtained, for instance, from a thermal Green\nfunction approach to spin relaxation in semiconductors\nand metals. The relation of the spin relaxation time to\na characteristic time \u001c0=\r\u00001in Refs. 18 and 19 is very\nsimilar to ours, but there the momentum scattering rates\nare related to lifetimes at the Fermi energy. Our results\nsuggest that the scaling of the demagnetization time with\n\u001cis rather robust and also valid for excited systems with\nelectronic populations far away from the Fermi energy if\n\u001cis interpreted as in Eqs. (4) and (9).\nWhile the behavior for short \u001cis reminiscent of typical\nspin dephasing mechanisms of spintronics, the same mi-\ncroscopic interplay of precessional spin dynamics around\ninternal \felds with a spin-independent scattering mech-\nanism is behind the demagnetization dynamics for the\nwhole range of\u001cshown in Fig. 6. In particular for larger \u001c\nwe obtain the inverse relation \u001cm/\u001c. Such a relation\nis usually associated with spin relaxation as it occurs via\nspin-\rip transitions due to an explicitly spin-dependent\ninteraction. Fig. 6 shows clearly that both behaviors\noccur as limiting cases for small and large \u001c, respec-\ntively, for electrons in a ferromagnetic band structure\nwith spin-orbit coupling. For intermediate \u001cof about\n1 to 10 fs, a minimum of demagnetization times occurs\nwhich is also very well described by the \ft curve. The8\n10−1100101102\nτ(fs)101102103τm(fs)\nﬁt:τm≈10.1fs2\nτ+ 3.56τ\ndata\nFIG. 6. Demagnetization time vs e\u000bective scattering time \u001c\nwith a \ft. The remaining parameters are as in Fig. 2.\nrange around the minimum is likely close to the realis-\ntic range for metallic systems. We believe that the result\ncontained in Fig. 6 gives an accurate and intuitive picture\nof electron-electron scattering dynamics in highly excited\nferromagnets by identifying \u001cas a physically meaningful\nparameter. One can thus use it as a \ft parameter also for\nsystems that are not described by the model band struc-\nture used in this paper. This makes it possible to extract\n\u001cfrom measured data \u001cmdata via a \ft, or obtain \u001cm(\u001c)\nfrom numerical calculations solving the full Boltzmann\nscattering problem. Fig. 6 is particularly important for\n\fts to experimental \u001cmdata. If one does not include the\nnonlinear regime at small \u001cand assumes a linear rela-\ntion between \u001cmand\u001cone would greatly overestimate\nthe actual momentum relaxation time and miss the con-\ntribution from the precessional dynamics completely.\nIV. CONCLUSION\nIn this paper we discussed the spin-dependent inco-\nherent carrier dynamics due to electron-electron scatter-ing in a ferromagnetic model system with spin-orbit cou-\npling, which provides an extension of our earlier study\nof electron-phonon scattering in this system. We de-\nscribed a dynamical calculation using the spin-density\nmatrix, spin-orbit coupling and electron-electron scat-\ntering, which is non-local in k-space, and compared the\nnumerical results with a calculation using only occu-\npation numbers and with a generalized relaxation-time\nansatz. We found that the calculation using only occu-\npation numbers failed to capture the demagnetization be-\nhavior for weak screening, i.e., strong scattering because\nthe precessional dynamics around spin-orbit \felds is ne-\nglected. The comparison with the generalized relaxation-\ntime ansatz showed a very good agreement both for weak\nand strong Coulomb scattering, i.e., in situations where\nprecessional dynamics of the o\u000b-diagonal part of the re-\nduced spin-density matrix are clearly visible and also in\ncases where they are suppressed by scattering/dephasing.\nThis suggests that the relaxation-time ansatz can cap-\nture essential properties of the incoherent spin-dependent\ndynamics using a k-local expression with a single mo-\nmentum relaxation time \u001c. Such a simpler form should\nbe useful in numerical calculations for more compli-\ncated problems, in which scattering/dephasing due to\nthe Coulomb interaction plays a role, such as transport\nand/or electronic dynamics due to coupling to magnons.\nIn terms of the momentum relaxation time we were able\nto \ft the calculated demagnetization times using a sum\nof terms proportional to \u001cand\u001c\u00001. The\u001cmvs.\u001crela-\ntionship is much simpler than the microscopic electron\ndynamics and such a \ft should be possible regardless of\nthe details of the underlying band structure.\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation) -\nTRR 173/2 - 268565370 Spin+X (Project B03).\n[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Spintronics: Fun-\ndamentals and applications, Reviews of Modern Physics\n76, 323 (2004).\n[2] J. Fabian, A. Matos-Abiague, C. Ertler, and P. Stano,\nSemiconductor Spintronics, acta physica slovaca 57, 565\n(2007).\n[3] M. Wu and C. Ning, Dyakonov-Perel E\u000bect on Spin De-\nphasing in n-Type GaAs, physica status solidi (b) 222,\n523 (2000).\n[4] M. M. Glazov and E. L. Ivchenko, Dyakonov{Perel Spin\nRelaxation Controlled by Electron{Electron Scattering,\nJournal of Superconductivity 16, 735 (2003).\n[5] Y. 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Koch, Quantum Theory of the Opti-\ncal and Electronic Properties of Semiconductors , 5th ed.\n(World Scienti\fc, Singapore, 2009).\n[29] D. Steiauf and M. F ahnle, Elliott-Yafet mechanism and\nthe discussion of femtosecond magnetization dynamics,\nPhysical Review B 79, 140401 (2009).\n[30] J. Dormand and P. Prince, A family of embedded Runge-\nKutta formulae, Journal of Computational and Applied\nMathematics 6, 19 (1980).\n[31] C. Scholl, S. Vollmar, and H. C. Schneider, O\u000b-resonant\nall-optical switching dynamics in a ferromagnetic model\nsystem, Physical Review B 99, 224421 (2019).\n[32] P. H. Song and K. W. Kim, Spin relaxation of conduction\nelectrons in bulk III-V semiconductors, Physical Review\nB66, 035207 (2002).\n[33] Z. G. Yu, S. Krishnamurthy, M. van Schilfgaarde, and\nN. Newman, Spin relaxation of electrons and holes\nin zinc-blende semiconductors, Physical Review B 71,\n245312 (2005).\n[34] S. Vollmar, D. J. Hilton, and H. C. Schneider, General-\nized Elliott-Yafet spin-relaxation time for arbitrary spin\nmixing, Physical Review B 96, 075203 (2017)."    },    {        "title": "1312.7303v1.Spin_Transfer_Torque_and_Electric_Current_in_Helical_Edge_States_in_Quantum_Spin_Hall_Devices.pdf",        "content": "Spin Transfer Torque and Electric Current in Helical Edge States in Quantum Spin\nHall Devices\nQinglei Meng, Smitha Vishveshwara, and Taylor L. Hughes\nDepartment of Physics, University of Illinois, 1110 West Green St, Urbana IL 61801\nWe study the dynamics of a quantum spin Hall edge coupled to a magnet with its own dynamics.\nUsing spin transfer torque principles, we analyze the interplay between spin currents in the edge\nstate and dynamics of the axis of the magnet, and draw parallels with circuit analogies. As a\nhighlighting feature, we show that while coupling to a magnet typically renders the edge state\ninsulating by opening a gap, in the presence of a small potential bias, spin-transfer torque can\nrestore perfect conductance by transferring angular momentum to the magnet. In the presence of\ninteractions within the edge state, we employ a Luttinger liquid treatment to show that the edge,\nwhen subject to a small voltage bias, tends to form a unique dynamic rotating spin wave state that\nnaturally couples into the dynamics of the magnet. We brieﬂy discuss realistic physical parameters\nand constraints for observing this interplay between quantum spin Hall and spin-transfer torque\nphysics.\nThe study of symmetry protected topological insula-\ntors has led to a series of remarkable theoretical and\nexperimental discoveries[1–3]. The initial prediction of\nthe time-reversal invariant quantum spin Hall (QSH)\ninsulator[4, 5] was soon realized in HgTe/CdTe quan-\ntum wells[6, 7] and, more recently, in InAs/GaSb quan-\ntum wells[8, 9]. One of the most exciting features of the\nquantum spin Hall insulator is the presence of robust,\ngapless edge states with counter propagating modes with\nopposite spin-polarization. The edge states form a so-\ncalledhelicalliquidwhichisanewclassof1Dliquidsthat\nis perturbatively stable as long as time-reversal symme-\ntry is preserved[10, 11]. As predicted, experiments show\nthat each edge, when biased, exhibits a quantized two-\nterminal, longitudinal conductivity of e2=h;even in the\npresence of disorder, as long as time-reversal symmetry\nis not broken[3, 6, 7, 12, 13].\nSome of the most interesting observable predictions\nconcerning helical modes involve proximity coupling of\nthe edge states to magnetic and/or superconducting\nlayers that act to de-stabilize the edge and open a\ngap[14–16]. For example, in the non-interacting limit,\nthe helical liquid is simply a 1D Dirac fermion and it\nis well-known that the domain-walls of mass-inducing\nperturbations in this system lead to topological bound\nstates[17, 18]. These bound states can be the source\nof fractional charges, for the case of a magnetic do-\nmain wall[14, 17, 18], or Majorana zero modes in a\nsuperconductor-magnet interface[16]. Our focus is on\nthe coupling of magnetic perturbations to the helical liq-\nuid. Several works have discussed the coupling of dy-\nnamical magnets to the QSH edge states leading to vari-\nous eﬀects like adiabatic charge pumping[14, 19], a spin-\nbattery eﬀect[20], and many related eﬀects in 3D time-\nreversal invariant topological insulators[21–26].\nIn this article we consider the transport properties of\na helical liquid in proximity to a dynamical ferromag-\nneticislandandshowthatthiscoupledmagnet-QSHedge\nsystem can exhibit a rich range of behavior due to spin-transfer torque physics. In the generic case, the magnetic\nisland opens a gap and acts as a barrier for the helical\nliquid via the Zeeman coupling (if its magnetization has\na component perpendicular to the spin-polarization of\nthe helical liquid). Essentially the island provides a local\nmechanism for up-spin right-moving modes to backscat-\nter to down-spin left-moving modes. This is a process\nthat is usually strongly suppressed as it would require\nthe electrons on one edge to scatter across the gapped\ninterior of the sample to the opposite edge. Thus, in light\nof the gap formation, we would naively expect that if the\nedge is coupled to the magnetic island and then voltage\nbiased, then as long as the voltage does not exceed the\nmagnet-induced gap then there will be no longitudinal\nconductance on this edge. During this process we would\nexpect charge to build up near the island until the po-\ntentialcounter-actstheappliedvoltagetoreachasteady-\nstate, i.e., we should have capacitor-like physics. In this\narticle we show that, unexpectedly, once the magnet is\ndynamically inﬂuenced by a spin-transfer torque applied\nby the scattering edge states, then the edge can begin\nconducting current even at zero-temperature when the\napplied voltage is smaller than the magnet-induced gap.\nThe resulting electrical behavior is then inductor-like.\nThe basic idea is as follows. A standard unbiased QSH\nbar carries gapless chiral edge currents of opposite spin\n(say polarized along ^z) traveling in opposite directions,\nthuscarryingzerochargecurrentbuttwoquantizedunits\nof spin current, if we ignore spin relaxation for now. The\npresence of a magnet, as in Fig. 1, couples the left and\nright movers and gaps these modes if the magnetization\nis not entirely along ^z:The gap renders the QSH edge a\ncharge insulator, in that there is no initial charge current\nfor a bias voltage with associated energy less than that of\nthe magnet-induced gap. However, because of the spin-\nmomentum locking, taking into account the spin degree\noffreedomyieldsmorecomplexbehavior. Inthepresence\nof such a bias the edge initially carries excess spin current\nononesideofthemagnet. Thisimbalanceofspin-currentarXiv:1312.7303v1  [cond-mat.mes-hall]  27 Dec 20132\n7QSHMagnet\nx < 0 x > 0IS(x<) IS(x>)\n7RR\n L(a)\n(b)\nFigure 1. Quantum spin Hall insulator system coupled to a\nmagnetic island on a single edge in the presence of a ﬁnite\nbias voltage. (a) We schematically illustrate the basic setup\nand zoom in the region of the island. (b) Circuit analogy for\nthe Hall bar and magnet. Resistors represent the quantized\nh=e2resistance of each edge and the inductor represents the\neﬀect of the magnetic island.\non the sides of the magnet provides a spin torque that\nresults in the transfer of angular momentum and subse-\nquent bias-controlled dynamics of the magnet. Again,\nbecause of the spin-momentum locking, the induced dy-\nnamics of the magnet in turn aﬀects the QSH dynamics.\nFor instance, in spite of the charge gap exceeding the\napplied voltage, the magnet induces a charge current in\nthe edge as it rotates[14] due to the spin-transfer torque.\nHence, we show that the magnet can act as an inductive\ncircuit element instead of a capacitive element.\nIn what follows, we model the QSH-magnet coupled\nsystem and explore its dynamics employing spin-transfer\ntorque methods. We analyze the approach to steady\nstate, the nature thereof and characteristic relaxation\ntimes, and draw parallels with electrical circuit analo-\ngies. Applicable to experimental realizations, we esti-\nmate the eﬀect from typical parameters of the QSH in\nHgTe/CdTe quantum wells and with the magnetic sys-\ntemofK2CuF 4[27]. Wethenstudytheinterplaybetween\nmagnetization dynamics and bias voltage in the presence\nofinteractionsintheQSHedgestates. Previously, within\na Luttinger liquid framework, we have shown an insta-\nbility towards an unusual spin-density wave ordering[28];\nhere we ﬁnd that the bias voltage endows this textured\nphase with unique dynamics.\nBeginning with the free helical liquid, let us con-\nsider the QSH edge description which has the associated\nHamiltonian[10, 11]\nH0=\u0002\ndx~vh\n y\nL\"(x)(i@x) L\"(x)\u0000 y\nR#(x)(i@x) R#(x)i\n:\n(1)\nAs shown in Fig. 2, these correspond to linearly dis-\npersing edge states moving along the x-direction with\nspeedv;where the operator  R\"(L#)annihilates an elec-\ntron moving to the right(left) with up(down) spin. Theproximity coupling between the magnet with magnetiza-\ntion\u0000 !M= (Mx;My;Mz)and the QSH edge is given by\nthe usual Zeeman coupling\nHM=\u0000\u00160\u0016B\u0000 !M\u0001\u0000 !\u001b (2)\nwhere\u00160is the the vacuum permeability, \u0016Bis the Bohr\nmagneton and the Pauli matrices\u0000 !\u001b= (\u001bx;\u001by;\u001bz)act\non the space  = [ R\"(x); L#(x)]T. In the region\nnear the magnet, the QSH edge spectrum is eﬀectivelyq\n(vp\u0000\u00160\u0016BMz)2+ (\u00160\u0016B)2(M2x+M2y)which has an\nexcitation gap induced by the magnet with magnitude\n\u0001 = 2\u00160\u0016B(M2\nx+M2\ny)1=2:\nLet us now consider the eﬀect of a voltage bias V,\nwhich, for instance, we apply at the lead on the left in\nFig. 1. Initially the spin currents in the left and right\nregions of the magnet are diﬀerent, giving rise to a spin\ncurrent imbalance \u0001\u0000 !IS= [\u0000 !IS(x<)\u0000\u0000 !IS(x>)], where\u0000 !IS(x) =~\n2 y1\n2(v\u001bz\u0000 !\u001b+\u0000 !\u001bv\u001bz) , andx<(x>)is on the\nleft (right) of the magnet as indicated in Fig. 1a. Be-\ncause of the spin-momentum locking of the helical liquid,\nthe spin current imbalance generically depends on the\nrotation frequency of the magnet. For simplicity, let us\nconsider the case when the magnet rotates in-plane at a\nfrequency(\nM\n2\u0019). WecantransformthefulledgeHamilto-\nnianH=H0+HMtotherotatingframeviathetransfor-\nmationH0=UHUy\u0000iU@Uy, whereU=ei\nMt\n2\u001bz[20]. In\nthe new basis, the Hamiltonian takes the resultant form\nHrot=\u0014~(\u0000iv@x\u0000\nM\n2)\u0000\u00160\u0016BMs\n\u0000\u00160\u0016BMs ~(iv@x+\nM\n2)\u0015\n;(3)\nwhere there is a rotation-induced voltage shift of ~\nM=e\nwhich is opposite for each spin component (see Fig. 2b).\nAfter imposing the appropriate boundary conditions and\nmatching ﬁelds at the interfaces between the unper-\nturbed helical liquid and the magnet, we ﬁnd the initial\nspin current imbalance\n\u0001\u0000 !IS(t= 0) =eV\u0000~\nM\n2\u0019^z: (4)\nHere have assumed that the length of magnet, LM, is\nlong enough , LM\u001d~v\n\u00160\u0016BMs, that a low-energy electron\nincident on the magnet barrier is completely reﬂected;\naccounting for tunneling requires a simple modiﬁcation.\nThe spin-current imbalance applies a torque on the\nmagnet and we can appeal to spin transfer torque (STT)\nphysics to analyze the coupled dynamics between QSH\nedge currents and the magnet. Applying the well-\nestablished STT formalism[29, 30], the dynamics is de-\nscribed by the Landau-Lifshitz equation\n\r\u00001@t\u0000 !M=\u0000D\u0000 !M\u0002Mz^z+1\nVM^M\u0002(\u0001\u0000 !IS\u0002^M)(5)\nwhere\ris the gyromagnetic ratio of the magnet, VMis\nthe volume of the magnet, and ^Mis the unit vector di-\nrected along the magnetization\u0000 !M. The ﬁrst term on the3\nright-hand side accounts for the easy-plane anisotropy\nenergy1\n2DM2\nzVMof the magnet [31]. The source of the\nanisotropy can be either intrinsic, as for an easy-plane\nmagnet, or induced by the coupling to the edge states\nitself, though the latter eﬀect is weak compared to usual\nmagnetic energy scales. Thus we would generally desire\nthe intrinsic anisotropy to be large to observe interest-\ning dynamics since the edge coupling is usually small.\nThe second term on the right-hand side accounts for the\ntorque due to the spin current imbalance \u0001\u0000 !ISderived\nabove. Since the magnitude of the magnetization is ef-\nfectively ﬁxed, the spin torque along the direction of the\nmagnetization has no eﬀect; only the transverse part of\nthis imbalanced spin current exerts the torque on the\nmagnetization. We will see that the eﬀect of this term is\nto drive the edge from an insulating state to a conducting\nstate.\nIn general, the dynamics derived from substituting the\nspin imbalance expression of Eq. (4) into the dynami-\ncal equation of motion Eq. (5) has no simple solution.\nHowever, in most of the physical cases of interest we can\nmake the approximation that the magnet always stays\nin-plane, i.e. Mz\u001cMS, whereMSis the magnitude\nof the spontaneous magnetization. This condition holds\nfor small enough bias voltages, i.e., bias voltages that are\nsmall compared to the magnet-induced gap, as will be\njustiﬁed in the proposed experimental setup to follow.\nWith this approximation, we obtain the simple solution:\n\u0001\u0000 !IS=eV\n2\u0019e\u0000\r2D~\n2\u0019VMt^z\nIC=e2V\nh(1\u0000e\u0000\r2D~\n2\u0019VMt)\nMx+iMy=MSei\u0001t\n0\nMdt0\nMz=2\u0019\ne\rDIC (6)\nwhereIC=e yv\u001bz is the charge current on the edge.\nThus, we can immediately see that the dynamics involves\na characteristic relaxation time \u001c=2\u0019VM\n\r2D~. The smaller\nthe magnet and larger the anisotropy, the faster the re-\nlaxation.\nWe can simply illustrate the consequences of the dy-\nnamics. The STT on the magnet due to the spin cur-\nrent imbalance ( \u0001\u0000 !IS) decays to zero, while the in-plane\nmagnetizationbeginstorotate; therotationfrequencyin-\ncreases to the constant valueeV\nh. Interestingly, in spite\nof the magnet-induced gap, the charge current ramps up\nto its quantized saturation value ofe2\nhV, rendering the\nmagnetic barrier transparent to charge. The spin trans-\nfer torque provides a magnetization along the zdirection,\nwhich reaches a new equilibrium valueeV\n\rD~. In fact, this\nzdirection magnetization acts as an eﬀective magnetic\nﬁeld causing the in-plane magnetization to precess. The\ncharge current that ﬂows here is essentially due to the\nsame charge-pumping mechanism reported in Ref. 14 fora rotating magnet. However, for our case the magnetiza-\ntiondynamicsandtherotationfrequencyareintrinsically\ncontrolled by the applied bias voltage.\nAn even simpler picture for understanding the dynam-\nics involves representing the geometry in Fig. 1a as an\neﬀective electrical circuit analog shown in Fig. 1b. The\nupper and lower QSH edges in Fig. 1 each provide a re-\nsistance of R=h\ne2. What our dynamical solution has\nshown is that the coupling of the edge states to the mag-\nnet can eﬀectively be represented by an inductor with\ninductance L=\u001cR. Hence, for this set up, charge is\nonly transported through the lower edge initially, which\nyields an eﬀective conductance of e2=h. As with a real\ninductor, which stores energy in an induced ﬁeld, here\nthe energy is stored in the form of the anisotropy en-\nergy of the easy-plane magnet. Over time, the inductive\ncomponentbecomestransparent, allowingcurrenttopass\nthrough. In the ﬁnal steady state, the upper and lower\nedges both conduct perfectly and the conductance of the\nsystemrisesandsaturatestoitsquantizedvalueof 2e2=h:\nLet us brieﬂy consider a physical magnetic system, for\nwhich we focus on K2CuF 4, known for its large easy-\nplane anisotropy[27, 32]. This material possesses a spon-\ntaneous magnetization of \u00160Ms= 0:124T;a gyromag-\nnetic ratio \r=\u00002\u00021011s\u00001T\u00001, and an out-of-plane\nanisotropy ﬁeld BA= 0:280T, i.e.D=BA\nMs= 2:26\u00160.\nFor a typical magnet of volume VM= 104nm\u0002102nm\u0002\n102nm these parameters provide a relaxation time esti-\nmate of\u001c= 10\u00001s:In order to be consistent with our\napproximation that Mz\u001cMS,we require an applied\nvoltage to be smaller than 1mV . This constraint con-\nﬁnes the rotational frequency of the magnet ( \nM=eV\nh)\nandtheassociatedradiationtolieinthemicrowaverange\nwhich indicates that microwave cavity resonator experi-\nments may be useful for the observation of this eﬀect.\nWhile we have so far presented a clean, optimistic\ndescription of the eﬀect, it must be mentioned that in\naddition to the primary contribution to the dynamics\nstemming from spin transfer torque, one also expects two\nsources of dissipation: (i) Gilbert damping of the magne-\ntization dynamics and (ii) spin-relaxation of the helical\nliquid due to spin-orbit scattering. Gilbert damping con-\ntributesanadditionalterm\u000b\nMs\r\u0000 !M\u0002d\u0000 !M\ndttotheright-hand\nside of Eq. (5) , where \u000bis the damping constant. As\nshown in Appendix A we ﬁnd that the damping provides\nan additional channel for relaxation, modifying the relax-\nation rate in Eq. (6) to \u001c\u00001=\r2D(~\n2\u0019VM+\u000bMS\nj\rj). More\nimportantly, it also changes the precession frequency to\n\nM=eV=(~+\u000b2\u0019VMMS=j\rj). The eﬀects of spin-orbit\nscattering will cause the spin carried by the helical liq-\nuid to relax as the charge current is carried from the\nleadstothemagneticisland. Thiswillreducetheamount\nof spin-current imbalance by a geometry and impurity-\ndependent factor \u0010and subsequently the precession fre-\nquency will be reduced by the same factor. Both of these4\nE\nkeVE\nk2k F(a) (b)\nFigure 2. (a) Spectrum of the free helical liquid at ﬁnite\nchemical potential. With repulsive interactions present, sys-\ntem forms a gapped spin-density wave order parameter with\nwave vector 2kF, which nests the Fermi-points. (b) Spectrum\nof a free current-carrying helical liquid in the presence of a ﬁ-\nnite bias voltage. Alternatively, when coupled to a magnet,in\nthe rotating frame of the magnet, the helical edges show a\nrelative chemical potential shift. The magnetic order param-\neter that can eﬀectively gap the associated Fermi-points has\nto thus connect states at diﬀerent energies, exhibiting ﬁnite\nfrequency dynamics.\neﬀects alter \nM;and since the charge current is sim-\nplye\nM;these two sources of dissipation will reduce the\nsaturation conductance of the magnet-coupled edge from\nits quantized value. Notably, experiments indicate that\nspin-orbit scattering eﬀects do not dominate the spin\nphysics in HgTe/CdTe quantum wells[33], however the\nGilbert damping of the magnet will surely reduce the ef-\nfective edge conductance, though hopefully not below an\nobservable value.\nSo far we have neglected interactions within the QSH\nedges; we now examine the stability of the magnetization\ndynamics presented above in the presence of interactions.\nFirst we will consider the possibility of new interaction-\ndriven phenomena, initially analyzing the QSH system\nin and of itself without the coupling to the magnet. As\ndone previously[10, 11, 28, 34, 35], the interacting he-\nlical liquid can be explored within a Luttinger liquid\nframeworkthroughthebosonizationofthefermionﬁelds.\nWe can use the boson ﬁelds \u001eand\u0012and the correspon-\ndence R\"(x)\u0018e\u0000i(\u001e(x)\u0000\u0012(x));  L#(x)\u0018ei(\u001e(x)+\u0012(x))to\nbosonize the helical liquid. The Hamiltonian in Eq. (1),\nalong with interactions, can be bosonized to yield the\nLuttinger liquid Hamiltonian[36]\nH=1\n2\u0019\u0002\ndxh\nuK(r\u0012)2+u\nK(r\u001e)2+ 2\u0016r\u001ei\n;(7)\nwhereu=v((1 +g4\n2\u0019v)2\u0000(g2\n2\u0019v)2)1=2is the renormalized\nvelocity,K= (1+g4\n2\u0019v\u0000g2\n2\u0019v\n1+g4\n2\u0019v+g2\n2\u0019v)1=2is the Luttinger parame-\nter, and the g2;g4represent the standard interaction cou-\npling constants[36]. Values of K < (>)1represent repul-\nsive (attractive) interactions, and here we only consider\nrepulsive interactions. We have also included a chemical\npotential (\u0016) term to account for the edge Fermi-level\nnot lying exactly at the Dirac point, a condition that\nleads to interesting physics in the presence of interac-\ntions (see Fig. 2a for an illustration in the free case). Forrepulsive interactions, the system is unstable to sponta-\nneously breaking time-reversal symmetry and generating\nin-plane ferromagnetic order which opens a gap at the\nFermi-energy when \u0016= 0:If the chemical potential is\nnot exactly tuned to be at the Dirac point, the system\ninstead exhibits a spatial oscillation of the in-plane mag-\nnetic order, forming spin density wave (SDW) [28, 36].\nFor the remainder of the calculations it is convenient\nto transform into the Lagrangian formulation, yielding\nthe Lagrangian associated with Eq. 7\nL\r=1\n2\u0019uK((@t\u001e)2\u0000u2(r\u001e)2\u00002\u0016uKr\u001e)+@t(\r\u001e):(8)\nHere the last term corresponds to a total derivative,\nwhich does not aﬀect the classical equations of motion,\nbut allows us to add a non-vanishing charge current. The\nparameter\rrepresentsanadditionalfreedomthatshould\nbe ﬁxed by a physical quantity, which we choose to be\nthe particle current operator jgiven by\nj=\u001c@t\u001e\n\u0019\u001d\n;\nthus ﬁxing the choice: \r=\u0000j=Ku.\nThe eﬀect of repulsive interactions on the helical liq-\nuid can be seen by evaluating the appropriate suscep-\ntibilities. The primary quantity of interest is the sus-\nceptibility of the operator O+(x;t)\u0011 y\nR\"(x;t) L#(x;t)\nwhich is related to the in-plane magnetization of the\nedge state via m+(x;t)\u0011mx(x;t) +imy(x;t) =\n2\u0016BhO+(x;t)i. To evaluate the spin susceptibility as-\nsociated with the in-plane magnetization, \u001fm(x;t) =\n\u0000i~\u0012(t)h[O+(x;t);Oy\n+(0;0)]i, it is easiest to ﬁrst shift\nthe\u001eﬁeld in the Lagrangian in Eq. (8) as ~\u001e(x;t) =\n\u001e(x;t)+\u0016Kx=u +\u0019uK\rt, andthenemploystandardLut-\ntinger liquid techniques. As a function of temperature\nTwe ﬁnd that the Fourier-transformed susceptibility in\nmomentum and frequency space diverges as\n\u001fm(!=\u00002\u0019j;k =\u00002K\nu\u0016)\u0018T2K\u00002\nnear (!c;kc) = (\u00002\u0019j;\u00002K\u0016=u ):\nThe divergence of the spin susceptibility is indicative\nof an intrinsic instability towards a magnetically ordered\nphase in the presence of repulsive interactions. As we\ndiscussed in previous work [28], the ﬁnite momentum at\nwhich the spin susceptibility diverges indicates that for\n\u00166= 0SDW order is preferred in which the in-plane mag-\nnetization spatially rotates over a length scale \u0018\u0019u\nK\u0016. A\nnew eﬀect is that, in the presence of an injected cur-\nrent, the susceptibility diverges at ﬁnite-frequency, i.e.,\nthe SDW order rotates at the frequency 2\u0019jas a func-\ntion of time. Thus, the edge can be carrying current and\nin a gapped, intrinsically-magnetized state if the SDW\norder rotates as a function of time.\nWe can heuristically illustrate why the time oscilla-\ntion of the SDW occurs by resorting to the free-fermion5\ndescription ( K= 1) where the current jinduced by\na bias voltage Vcan be determined by the ﬁlling of\nthe single-particle energy spectrum as shown in Fig.\n2b. In the presence of the repulsive interaction term\nHint= y\nR\" R\" y\nL# L#=O+(x;t)Oy\n+(x;t)the system\nwill try to develop in-plane magnetic order hO+(x;t)i, in\norder to induce a mass term  y\nR\" L#hOy\n+(x;t)i, that will\nopen up a gap and lower the energy of the system. No-\ntice that the most eﬃcient way to lower the energy is to\nopen up the gap at the Fermi points. When the current\nvanishes this implies that SDW order will form with a\nwave-vector that nests the two degenerate Fermi-points\n(see Fig. 2a). However, when there is ﬁnite current in\nthis system, then, eﬀectively, the two Fermi points are\nnot at the same energy. In order to couple these two\nFermi points that lie at diﬀerent energies, the SDW has\nto have a time dependent part hO+(x;t)i\u0018eieVt= ~which\nis exactly why we observe a divergent spin susceptibility\nat ﬁnite frequency.\nFinally, we revisit the coupling to the external magnet\nin the presence of interactions. Since the external mag-\nnetic island has been assumed to be uniform, we expect\nthat to achieve the strongest coupling between QSH edge\n(with SDW order) and the external magnet, the length\nof the magnet should be smaller than the SDW wave-\nlength\u0018\u0019u\nK\u0016:The presence of a magnet, as in the non-\ninteractingcase, opensupagapinthehelicalliquid. This\nis easy to see in the Luttinger liquid formalism, where the\ncoupling between the edge state and external magnet in\nEq.(2) has the Sine-Gordon form cos(2\u001e\u0000\u0012H), where\u0012H\nis the angle of the in-plane magnetization. This coupling\nis relevant in the renormalization group sense, and hence\nlocks the phase 2\u001e=\u0012Hat low temperature. In previous\nwork, we have analyzed the static eﬀect of external mag-\nnets on the helical liquid at ﬁnite \u0016[28]. For the dynamic\nsituation we are considering in this work, one can derive\nthe particle current as j=D\n@t\u001e\n\u0019E\n=@t\u0012H\n2\u0019which is simply\nthe adiabatic charge pumping on the QSH edge as de-\nrived in Ref. [14, 19, 37], but now including interactions.\nIf the magnet is not initially rotating then, just as in\nthe non-interacting case, we expect an initial spin cur-\nrent imbalance across the magnet when a voltage is ap-\nplied. We can calculate this spin current imbalance,\nwhich, due to spin-momentum locking is proportional\nto the charge density diﬀerence across the the magnet,\n\u0001Iz\nS=~\n2v(\u001a(x<)\u0000\u001a(x>)) =K\n2\u0019v\nueV. Thus, even with in-\nteractionsthereisaninitialspin-currentimbalancewhich\nwill apply a STT to the magnet. While a full analysis\nof the spin-transfer torque in the presence of interactions\nis beyond the scope of this work, we expect that just as\nwith the non-interacting case, the excess spin current,\nnow accompanied by an in-plane magnetization rotation\noftheedge, transfersangularmomentumtothemagnetic\nregion. Once again, as with the non-interacting case, in\nsteady state, a charge current will ﬂow as the magnetevolves to a steady-state of rotation at a rate propor-\ntional to the applied voltage.\nApplications - The unique combination of QSH physics\nand spin transfer torque gives rise to new ways of prob-\ning and manipulating the QSH edge, particularly by ex-\nploiting well-characterized magnetic materials and their\ninformation storage and access properties. i) Microwave\nresonator - We saw above that an excess QSH spin cur-\nrent produced by a voltage bias Vinduces the magnet to\nprecessatafrequency eV=h. Thisprecessionwouldresult\nin microwave radiation of about 24GHz for typical bias\nvoltages of order 0:1meV. In principle, one can envision\nputting an array of QSH-coupled magnets in a microwave\nresonator to generate a voltage-tunable microwave laser.\nii) Spin polarization detector- Thus far, we have assumed\nthat the QSH spin axis coincides with that of the easy\nplane of the spin magnet. In principle, the two need not\nbe aligned, eﬀectively giving the excess QSH spin cur-\nrent components in the xy-plane and in turn aﬀecting\nthe dynamics of the magnet. Analyzing this dynamics\nwould provide information on spin polarization in the\nQSH system. iii) AC QSH circuit - Information on the\nQSH edges can also be obtained by charge current mea-\nsurements from the perspective of the circuit analogy of\nFig. 1. The circuit description can be taken further by\nincluding a conventional capacitance element to produce\noscillatory charge and spin currents. In conclusion, here\nwe have presented an initial glimpse of the rich physics\nthat can emerge through the interplay of QSH edge and\nspin-transfer torque physics.\nWe are grateful to E. Johnston-Halperin for insightful\ncomments. For support, we acknowledge the U.S. De-\npartment of Energy, Division of Materials Sciences un-\nder Award No. DE-FG02-07ER46453 (T. L. H. and S.\nV.) and the National Science Foundation under Grant\nNo. DMR-0906521 (Q. M.).\nAppendix A: Gilbert damping\nNow we include the Gilbert damping term in the spin\ntransfer torque analysis of the Landau-Lifshitz-Gilbert\nequation:\n\r\u00001@t\u0000 !M=\u0000D\u0000 !M\u0002Mz^z+1\nVM^M\u0002(\u0001\u0000 !IS\u0002^M)\n+\u000b\nMs\r\u0000 !M\u0002d\u0000 !M\ndt(A1)\nwhere the last term is due to Gilbert damping. Now\nwe can write Eq. A1 in terms of components, and fur-\nthermore continue our approximation from the body of\nthe text where we assume Mz\u001cMSfor total in-plane\nmagnetization MS:Additionally, making an ansatz that\nMx=MScos\u0012(t); My=MSsin\u0012(t);6\n\r\u00001@tMz=eV\u0000~_\u0012(t)\n2\u0019VM+\u000b\n\rMS_\u0012(t)\n\r\u00001@tMx=\u0000DMSsin\u0012(t)Mz\u0000\u000b\n\r_\u0012(t) cos\u0012(t)Mz\n\r\u00001@tMy=DMScos\u0012(t)Mz\u0000\u000b\n\r_\u0012(t) sin\u0012(t)Mz:\nIn general the dynamics can be complicated, even af-\nter assuming Mz\u001cMS:Let us consider our physi-\ncal system of interest K2CuF 4where we estimate that\nDMS= 0:28T:Thus for small voltages V < 1mV , then\nDMS>\u0000\u000b\n\r\nM, as long as \u000b<10\u00001:With this approx-\nimation we have\n\r\u00001@tMz=eV\u0000~_\u0012(t)\n2\u0019VM+\u000b\n\rMS_\u0012(t)\n\r\u00001@tMx=\u0000DMSsin\u0012(t)Mz\n\r\u00001@tMy=DMScos\u0012(t)Mz:\nFrom here we see that Gilbert damping will just provide\nanother channel for the damping of the imbalanced spin\ncurrent, which will decrease relaxation time to\n\u001c=\u0014\u0012~\n2\u0019VM+\u000bMS\nj\rj\u0013\n\r2D\u0015\u00001\n(A2)\nand decrease the rotation frequency to\n\nM=eV\n~+\u000b2\u0019VMMS=\r: (A3)\n[1] M. 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Such a non-Kramers pseudo-spin 1/2 is the\nconsequence of crystalline symmetries as opposed to the Kramers doublet arising from time-reversal invariance,\nand is necessarily a composite of spin and orbital degrees of freedom. We investigate possible spin-orbital\nliquids with fermionic spinons for such non-Kramers pseudo-spin 1/2 systems on the Kagome lattice. Using the\nprojective symmetry group analysis, we ﬁnd tennew phases that are not allowed in the corresponding Kramers\nsystems. These new phases are allowed due to unusual action of the time reversal operation on non-Kramers\npseudo-spins. We compute the spin-spin dynamic structure factor that shows characteristic features of these\nnon-Kramers spin-orbital liquids arising from their unusual coupling to neutrons, which is therefore relevant for\nneutron scattering experiments. We also point out possible anomalous broadening of Raman scattering intensity\nthat may serve as a signature experimental feature for gapless non-Kramers spin-orbital liquids.\nPACS numbers:\nI. INTRODUCTION\nThe low energy magnetic degrees of freedom of a Mott\ninsulator, in the presence of strong spin-orbit coupling, are\ndescribed by states with entangled spin and orbital wave\nfunctions.1,2In certain crystalline materials, for ions with\neven numbers of electrons, a low energy spin-orbit entan-\ngled “pseudo-spin”-1/2 may emerge, which is not protected\nby time-reversal symmetry (Kramers degeneracy)12but rather\nby the crystal symmetries.3,4Various phases of such non-\nKramers pseudo-spin systems on geometrically frustrated lat-\ntices, particularly various quantum paramagnetic phases, are\nof much recent theoretical and experimental interest in the\ncontext of a number of rare earth materials including frus-\ntrated pyrochlores5–9and heavy fermion systems.10,11\nIn this paper, we explore novel spin-orbital liquids that may\nemerge in these systems due to the unusual transformation of\nthe non-Kramers pseudo-spins under the time reversal trans-\nformation. Contrary to Kramers spin-1/2, where the spins\ntransform as S!\u0000Sunder time reversal,12here only one\ncomponent of the pseudo-spin operators changes sign under\ntime reversal:f\u001b1;\u001b2;\u001b3g!f\u001b1;\u001b2;\u0000\u001b3g.3,4This is be-\ncause, due to the nature of the wave-function content, the\n\u001b3component of the pseudo-spin carries a dipolar magnetic\nmoment while the other two components carry quadrupolar\nmoments of the underlying electrons. Hence the time rever-\nsal operator for the non-Kramers pseudo-spins is given by\nT=\u001b1K(whereKis the complex conjugation operator),\nwhich allows for new spin-orbital liquid phases. Since the\nmagnetic degrees of freedom are composed out of wave func-\ntions with entangled spin and orbital components, we prefer\nto refer the above quantum paramagnetic states as spin-orbital\nliquids, rather than spin liquids.\nSince the degeneracy of the non-Kramers doublet is pro-\ntected by crystal symmetries, the transformation properties of\nthe pseudo-spin under various lattice symmetries intimatelydepend on the content of the wave-functions that make up\nthe doublet. To this end, we focus our attention on the ex-\nample of Praseodymium ions (Pr3+) in a localD3denviron-\nment, which is a well known non-Kramers ion that occurs in\na number of materials with interesting properties.5–7Such an\nenvironment typically occurs in Praseodymium pyrochlores\ngiven by the generic formulae Pr 2TM2O7, where TM(= Zr,\nSn, Hf, or Ir) is a transition metal. In these compounds, the\nPr3+ions host a pair of 4 felectrons which form a J= 4\nground state manifold with S= 1andL= 5, as expected due\nto Hund’s rules. In terms of this local environment we have\na nine fold degeneracy of the electronic states.4This degener-\nacy is broken by the crystalline electric ﬁeld. The oxygen and\nTM ions form a D3dlocal symmetry environment around the\nPr3+ions, splitting the nine fold degeneracy. A standard anal-\nysis of the symmetries of this system (see appendix A) shows\nthat theJ= 4 manifold splits into three doublets and three\nsinglets ( \u0000j=4= 3Eg+ 2A1g+A2g) out of which one of the\ndoublets is found to have the lowest energy, usually well sepa-\nrated from the other crystal ﬁeld states.4This doublet (details\nin Appendix A), formed out of a linear combination of the\nJz=\u00064withJz=\u00061andJz=\u00062states, is given by\nj\u0006i=\u000bjm=\u00064i\u0006\fjm=\u00061i\u0000\rjm=\u00072i:(1)\nThe non-Kramers nature of this doublet is evident from the\nnature of the “spin” raising and lowering operators within\nthe doublet manifold; the projection of the angular momen-\ntum raising and lowering operators to the space of doublets\nis zero (PJ\u0006Pj\u001bi= 0 wherePprojects into the doublet\nmanifold). However, the projection of the Jzoperator to this\nmanifold is non-zero, and describes the z component of the\npseudo-spin ( \u001b3). In addition, there is a non-trivial projection\nof the quadrupole operators fJ\u0006;Jzgin this manifold. These\nhave off-diagonal matrix elements, and are identiﬁed with the\npseudo-spin raising and lowering operators ( \u001b\u0006=\u001b1\u0006i\u001b2).\nIn a pyrochlore lattice the local D3daxes point to the centre\nof the tetrahedra.4On looking at the pyrochlore lattice alongarXiv:1304.2766v1  [cond-mat.str-el]  9 Apr 20132\nthe [111] direction, it is found to be made out of alternate\nlayers of Kagome and triangular lattices. For each Kagome\nlayer (shown in Fig. 1) the local D3daxes make an angle of\ncos\u00001(p\n2=3)with the plane of the Kagome layer. We imag-\nine replacing the Pr3+ions from the triangular lattice layer\nwith non-magnetic ions so as to obtain decoupled Kagome\nlayers with Pr3+ions on the sites. The resulting structure is\nobtained in the same spirit as the now well-known Kagome\ncompound Herbertsmithite was envisioned. As long as the lo-\ncal crystal ﬁeld has D3dsymmetry, the doublet remains well\ndeﬁned. A suitable candidate non-magnetic ion may be iso-\nvalent but non-magnetic La3+. Notice that the most extended\norbitals in both cases are the ﬁfth shell orbitals and the crystal\nﬁeld at each Pr3+site is mainly determined by the surrounding\noxygens and the transition metal element. Hence, we expect\nthat the splitting of the non-Kramers doublet due to the above\nsubstitution would be very small and the doublet will remain\nwell deﬁned. In this work we shall consider such a Kagome\nlattice layer and analyze possible Z2spin-orbital liquids, with\ngapped or gapless fermionic spinons .\nThe rest of the paper is organized as follows. In Sec. II, we\nbegin with a discussion of the symmetries of the non-Kramers\nsystem on a Kagome lattice and write down the most general\npseudo-spin model with pseudo-spin exchange interactions up\nto second nearest neighbours. In Sec. III formulate the pro-\njective symmetry group (PSG) analysis for singlet and triplet\ndecouplings. Using this we demonstrate that the non-Kramers\ntransformation of our pseudo-spin degrees of freedom under\ntime reversal leads to a set of ten spin-orbital liquids which\ncannot be realized in the Kramers case. In Sec. IV we de-\nrive the dynamic spin-spin structure factor for a representa-\ntive spin liquid for the case of both Kramers and non-Kramers\ndoublets, demonstrating that experimentally measurable prop-\nerties of these two types of spin-orbital liquids differ qualita-\ntively. Finally, in Sec. V, we discuss our results, and propose\nan experimental test which can detect a non-Kramers spin-\norbital liquid. The details of various calculations are discussed\nin different appendices.\nII. SYMMETRIES AND THE PSEUDO-SPIN\nHAMILTONIAN\nSince the local D3daxes of the three sites in the Kagome\nunit cell differ from each other a general pseudo-spin Hamil-\ntonian is not symmetric under continuous global pseudo-spin\nrotations. However, it is symmetric under various symmetry\ntransformations of the Kagome lattice as well as time rever-\nsal symmetry. Such symmetry transformations play a major\nrole in the remainder of our analysis. We start by describing\nthe effect of various lattice symmetry transformations on the\nnon-Kramers doublet.\nWe consider the symmetry operations that generate the\nspace group of the above Kagome lattice. These are (as shown\nin Fig. 2(a))\n\u000fT1,T2: generate the two lattice translations.\n\u000f\u001b=C0\n2I: (not to be confused with the pseudo-spin\nFIG. 1: A Kagome layer, in the pyrochlore lattice environment. We\nconsider sites labelled z and z’ replaced by non-magnetic ions, de-\ncoupling the Kagome layers. The local axis at the u,v and w sites\npoint towards the center of the tetrahedron on which these lie.\nu\nvw\nFIG. 2: (color online) (a) The symmetries of the Kagome lattice.\nAlso shown are the labels for the sublattices and the orientation of the\nlocal z-axis. (b) Nearest and next nearest neighbour bonds. Colors\nrefer to the phases \u001er;r0and\u001e0\nr;r0, with these being 0 on blue bonds,\n1 on green bonds and 2 on red bonds.\noperators which come with a superscript) where Iis the\nthree dimensional inversion operator about a plaquette\ncenter andC0\n2refers to a two-fold rotation about a line\njoining two opposite sites on the plaquette.\n\u000fS6=C2\n3I: whereC3is the threefold rotation op-\nerator about the center of a hexagonal plaquette of the\nKagome lattice.\n\u000fT=\u001b1K: Time reversal.\nHere, we consider a three dimensional inversion operator\nsince the local D3daxes point out of the Kagome plane. The\nabove symmetries act non-trivially on the pseudo-spin degrees\nof freedom, as well as the lattice degrees of freedom. The ac-\ntion of the symmetry transformations on the pseudo-spin op-\nerators is given by,\nS6:f\u001b3;\u001b+;\u001b\u0000g!f\u001b3;\u0016!\u001b+;!\u001b\u0000g;\nT:f\u001b3;\u001b+;\u001b\u0000g!f\u0000\u001b3;\u001b\u0000;\u001b+g;\nC0\n2:f\u001b3;\u001b+;\u001b\u0000g!f\u0000\u001b3;\u001b\u0000;\u001b+g;\nT1:f\u001b3;\u001b+;\u001b\u0000g!f\u001b3;\u001b+;\u001b\u0000g;\nT2:f\u001b3;\u001b+;\u001b\u0000g!f\u001b3;\u001b+;\u001b\u0000g; (2)\n(!= \u0016!\u00001=ei2\u0019\n3). Operationally their action on the doublet\n(j+i j\u0000i )can be written in form of 2\u00022matrices. The\ntranslationsT1;T2act trivially on the pseudo-spin degrees of3\nfreedom, and the remaining operators act as\nT=\u001b1K; \u001b =\u001b1; S 6=\u0014\n\u0016!0\n0!\u0015\n; (3)\nwhereKrefers to complex conjugation. The above expres-\nsions can be derived by examining the effect of these operators\non the wave-function describing the doublet (Eq. 1).\nWe can now write down the most generic pseudo-spin\nHamiltonian allowed by the above lattice symmetries that is\nbilinear in pseudo-spin operators. The form of the time-\nreversal symmetry restricts our attention to those products\nwhich are formed by a pair of \u001b3operators or those which\nmix the pseudo-spin raising and lowering operators. Any\nterm which mixes \u001b3and\u001b\u0006changes sign under the sym-\nmetry, and can thus be excluded. Under the C3transforma-\ntion about a site, the terms C3:\u001b3\nr\u001b3\nr0!\u001b3\nC3(r)\u001b3\nC3(r0)and\nC3:\u001b+\nr\u001b\u0000\nr0!\u001b+\nC3(r)\u001b\u0000\nC3(r0). However, the term \u001b+\nr\u001b+\nr0(and\nits Hermitian conjugate) gain additional phase factors when\ntransformed; under the C3symmetry transformation, this term\nbecomesC3:\u001b+\nr\u001b+\nr0!\u0016!\u001b+\nC3(r)\u001b+\nC3(r0). In addition, un-\nder the\u001bsymmetry, this term transforms as \u001b:\u001b+\nr\u001b+\nr0!\n\u001b\u0000\n\u001b(r)\u001b\u0000\n\u001b(r0). Thus the Hamiltonian with spin-spin exchange\ninteractions up to next-nearest neighbour is given by\nHeff=JnnX\nhr;r0i[\u001b3\nr\u001b3\nr0+ 2(\u000e\u001b+\nr\u001b\u0000\nr0+h:c:)\n+2q(e2\u0019i\u001er;r 0\n3\u001b+\nr\u001b+\nr0+h:c:)]\n+JnnnX\nhhr;r0ii[\u001b3\nr\u001b3\nr0+ 2(\u000e0\u001b+\nr\u001b\u0000\nr0+h:c:)\n+2q0(e2\u0019i\u001e0\nr;r 0\n3\u001b+\nr\u001b+\nr0+h:c:)]; (4)\nwhere\u001eand\u001e0take values 0, 1 and 2 depending on the bonds\non which they are deﬁned (Fig. 2(b)).\nIII. SPINON REPRESENTATION OF THE PSEUDO-SPINS\nAND PSG ANALYSIS\nHaving written down the pseudo-spin Hamiltonian, we now\ndiscuss the possible spin-orbital liquid phases. We do this in\ntwo stages in the following sub-sections.\nA. Slave fermion representation and spinon decoupling\nIn order to understand these phases, we will use the\nfermionic slave-particle decomposition of the pseudo-spin op-\nerators. At this point, we note that the pseudo-spins satisfy\nS= 1=2representations of a “SU(2)” algebra among their\ngenerators (not to be confused with the regular spin rotation\nsymmetry). We represent the pseudo-spin degrees of freedom\nin terms of a fermion bilinear. This is very similar to usual\nslave fermion construction for spin liquids13,14. We take\n\u001b\u0016\nj=1\n2fy\nj\u000b[\u001a\u0016]\u000b\ffj\f; (5)where\u000b;\f=\";#is deﬁned along the local zaxis andfy(f)\nis anS= 1=2fermionic creation (annihilation) operator. Fol-\nlowing standard nomenclature, we refer to the f(fy)as the\nspinon annihilation (creation) operator, and note that these\nsatisfy standard fermionic anti-commutation relations. The\nabove spinon representation, along with the single occupancy\nconstraint\nfy\ni\"fi\"+fy\ni#fi#= 1; (6)\nform a faithful representation of the pseudo-spin-1/2 Hilbert\nspace. The above representation of the pseudo-spins, when\nused in Eq. 4, leads to a quartic spinon Hamiltonian. Fol-\nlowing standard procedure,13,14this is then decomposed us-\ning auxiliary ﬁelds into a quadratic spinon Hamiltonian (af-\nter writing down the corresponding Eucledian action). The\nmean ﬁeld description of the phases is then characterized by\nthe possible saddle point values of the auxiliary ﬁelds. There\nare eight such auxiliary ﬁelds per bond, corresponding to\n\u001fij=hfy\ni\u000bfj\u000bi\u0003;\u0011ij=hfi\u000b\u0002\ni\u001c2\u0003\n\u000b\ffj\fi\u0003; (7a)\nEa\nij=hfy\ni\u000b[\u001ca]\u000b\ffj\fi\u0003;Da\nij=hfi\u000b\u0002\ni\u001c2\u001ca\u0003\n\u000b\ffi\fi\u0003;\n(7b)\nwhere\u001ca(a= 1;2;3) are the Pauli matrices. While Eq. 7a\nrepresents the usual singlet spinon hopping (particle-hole) and\npairing (particle-particle) channels, Eq. 7b represents the cor-\nresponding triplet decoupling channels. Since the Hamilto-\nnian (Eq. 4) does not have pseudo-spin rotation symmetry,\nboth the singlet and the triplet decouplings are necessary.16,17\nFrom this decoupling, we obtain a mean-ﬁeld Hamiltonian\nwhich is quadratic in the spinon operators. We write this com-\npactly in the following form17(subject to the constraint Eq.6)\nH0=X\nijJij~fy\niUij~fj; (8)\n~fy\ni=h\nfy\ni\"fi#fy\ni#\u0000fi\"i\n; (9)\nUij=\u0018\u000b\f\nij\u0006\u000b\u0000\f; (10)\n\u0006\u000b=\u001a\u000b\nI; \u0000\f=I\n\u001c\f; (11)\nwhere\u001a\u000bare the Identity (for \u000b= 0) and Pauli matrices\n(\u000b= 1;2;3) acting on pseudo-spin degrees of freedom, and\n\u001c\u000brepresents the same in the gauge space. We immediately\nnote that\n\u0002\n\u0006\u000b;\u0000\f\u0003\n= 08\u000b;\f: (12)\nThe requirement that our H0be Hermitian restricts the coefﬁ-\ncients\u0018ijto satisfy\n\u001800\nij;\u0018ab\nij2=;\u0018a0\nij;\u00180b\nij2<: (13)\nfora;b2 f 1;2;3g. The relations between \u0018ijs and\nf\u001fij;\u0011ij;Eij;Dijgare given in Appendix C.17As a straight\nforward extension of the SU(2)gauge theory formulation for4\nspin liquids,13,18we ﬁnd that H0is invariant under the gauge\ntransformation\n~fj!Wj~fj; (14)\nUij!WiUijWy\nj; (15)\nwhere theWimatrices are SU(2) matrices of the form Wi=\nei~\u0000\u0001~ ai(~\u0000\u0011(\u00001;\u00002;\u00003)). Noting that the physical pseudo-\nspin operators are given by\n~ \u001bi=1\n4~fy\ni~\u0006~fi; (16)\nEq. 12 shows that the spin operators, as expected, are gauge\ninvariant. It is useful to deﬁne the “ \u0006-components” of the Uij\nmatrices as follows:\nUij=V\u000b\nij\u0006\u000b; (17)\nwhere\nV\u000b\nij=\u0018\u000b\f\nij\u0000\f=\u0014J\u000b\nij0\n0J\u000b\nij\u0015\n; (18)\nand\nJ\u000b\nij=\u0014\u0018\u000b0\nij+\u0018\u000b3\nij\u0018\u000b1\nij\u0000i\u0018\u000b2\nij\n\u0018\u000b1\nij+i\u0018\u000b2\nij\u0018\u000b0\nij\u0000\u0018\u000b3\nij\u0015\n: (19)\nUnder global spin rotations the fermions transform as\n~fi!V~fi; (20)\nwhere V is an SU(2) matrix of the form V=ei~\u0006\u0001~b(~\u0006\u0011\nf\u00061;\u00062;\u00063g). So whileV0\nij(the singlet hopping and pairing)\nis invariant under spin rotation, fV1\nij;V2\nij;V3\nijgtransforms as\na vector as expected since they represent triplet hopping and\npairing amplitudes.\nB. PSG Classiﬁcation\nWe now classify the non-Kramers spin-orbital liquids based\non projective representation similar to that of the conven-\ntional quantum spin liquids.13Each spin-orbital liquid ground\nstate of the quadratic Hamiltonian (Eq 11) is character-\nized by the mean ﬁeld parameters (eight on each bond,\n\u001f;\u0011;E1;E2;E3;D2;D2;D3, or equivalently Uij). However,\ndue to the gauge redundancy of the spinon parametrization\n(as shown in Eq. 15), a general mean-ﬁeld ansatz need not\nbe invariant under the symmetry transformations on their own\nbut may be transformed to a gauge equivalent form without\nbreaking the symmetry. Therefore, we must consider its trans-\nformation properties under a projective representation of the\nsymmetry group.13For this, we need to know the various pro-\njective representations of the lattice symmetries of the Hamil-\ntonian (Eq. 4) in order to classify different spin-orbital liquid\nstates.\nOperationally, we need to ﬁnd different possible sets of\ngauge transformations fGGgwhich act in combination withthe symmetry transformations fSGgsuch that the mean-ﬁeld\nansatzUijis invariant under such a combined transformation.\nIn the case of spin rotation invariant spin-liquids (where only\nthe singlet channels \u001fand\u0011are present), the above statement\nis equivalent to demanding the following invariance:\nUij= [GSS]Uij[GSS]y=GS(i)US(i)S(j)Gy\nS(j);(21)\nwhereS2SGis a symmetry transformation and GS2GG\nis the corresponding gauge transformation. The different pos-\nsiblefGSj8S2SGggive the possible algebraic PSGs that\ncan characterize the different spin-orbital liquid phases. To\nobtain the different PSGs, we start with various lattice sym-\nmetries of the Hamiltonian. The action of various lattice\ntransformations15is given by\nT1:(x;y;s )!(x+ 1;y;s);\nT2:(x;y;s )!(x;y+ 1;s);\n\u001b:(x;y;u )!(y;x;u );\n(x;y;v )!(y;x;w );\n(x;y;w )!(y;x;v );\nS6:(x;y;u )!(\u0000y\u00001;x+y+ 1;v);\n(x;y;v )!(\u0000y;x+y;w);\n(x;y;w )!(\u0000y\u00001;x+y;u); (22)\nwhere (x;y)denotes the lattice coordinates and s2fu;v;wg\ndenotes the sub-lattice index (see ﬁgure 2).\nIn terms of the symmetries of the Kagome lattice, these op-\nerators obey the following conditions\nT2=\u001b2= (S6)6=e;\ng\u00001T\u00001gT=e8g2SG;\nT\u00001\n2T\u00001\n1T2T1=e;\n\u001b\u00001T\u00001\n1\u001bT2=e;\n\u001b\u00001T\u00001\n2\u001bT1=e;\nS\u00001\n6T\u00001\n2S6T1=e;\nS\u00001\n6T\u00001\n2T1S6T2=e;\n\u001b\u00001S6\u001bS6=e: (23)\nIn addition, these commutation relations are valid in terms of\nthe operations on the pseudo-spin degrees of freedom, as can\nbe veriﬁed from Eq. 3.\nIn addition to the conditions in Eq. 23, the Hamiltonian is\ntrivially invariant under the identity transformation. The in-\nvariant gauge group (IGG) of an ansatz is deﬁned as the set of\nall pure gauge transformations GIsuch thatGI:Uij!Uij.\nThe nature of such pure gauge transformations immediately\ndictates the nature of the low energy ﬂuctuations about the\nmean ﬁeld state. If these ﬂuctuations do not destabilize the\nmean-ﬁeld state, we get stable spin liquid phases whose low\nenergy properties are controlled by the IGG. Accordingly,\nspin liquids obtained within projective classiﬁcation are pri-\nmarily labelled by their IGGs and we have Z2;U(1)and\nSU(2)spin liquids corresponding to IGGs of Z2;U(1)and5\nSU(2)respectively. In this work we concentrate on the set of\nZ2“spin liquids” (spin-orbital liquids with a Z2IGG).\nWe now focus on the PSG classiﬁcation. As shown in Eq.\n2, in the present case, the pseudo-spins transform non-trivially\nunder different lattice symmetry transformations. Due to the\npresence of the triplet decoupling channels the non-Kramers\ndoublet transforms non-trivially under lattice symmetries (Eq.\n3). Thus, the invariance condition on the Uijs is not given by\nEq. 21, but by a more general condition\nUij= [GSS]Uij[GSS]y=GS(i)\u001eS\u0002\nUS(i)S(j)\u0003\nGy\nS(j):\n(24)\nHere\n\u001eS\u0002\nUS(i)S(j)\u0003\n=DSUS(i)S(j)Dy\nS; (25)\nandDSgenerates the pseudo-spin rotation associated with the\nsymmetry transformation ( S) on the doublet. The matrices\nDShave the form\nDS6=\u00001\n2\u00060\u0000ip\n3\n2\u00063; (26)\nD\u001b=DT=i\u0003\u00061;DT1=DT2= \u00060: (27)\nUnder these constraints, we must determine the relations\nbetween the gauge transformation matrices GS(i)for our set\nof ansatz. The additional spin transformation (Eq. 25) does\nnot affect the structure of the gauge transformations, as the\ngauge and spin portions of our ansatz are naturally separate\n(Eq. 12). In particular, we can choose to deﬁne our gauge\ntransformations such that\nGS:Uij=GS:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\u0018\u000b\f\nij\u0006\u000bGy\nS(i)\u0000\fGS(j);\n(28)\nS:Uij=S:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\u0018\u000b\f\nS(i)S(j)DS\u0006\u000bDy\nS\u0000\f;(29)\nwhere we have used the notation GS:Uij\u0011Gy\nS(i)UijGS(j)\nand so forth. As a result, we can build on the general con-\nstruction of Lu et al.15to derive the form of the gauge trans-\nformation matrices. The details are given in Appendix B.\nA major difference arises when examining the set of alge-\nbraic PSGs for Z2spin liquids found on the Kagome lattice\ndue to the difference between the structure of the time re-\nversal symmetry operation on the Kramers and non-Kramers\npseudo-spin- 1=2s. In the present case, we ﬁnd there are 30\ninvariant PSGs leading to thirty possible spin-orbital liquids.\nThis is in contrast with the Kramers case analysed by Lu et\nal.,15where tenof the algebraic PSGs cannot be realized as\ninvariant PSGs, as all bonds in these ansatz are predicted to\nvanish identically due to the form of the time reversal oper-\nator, and hence there are only twenty possible spin liquids.\nHowever, with the inclusion of spin triplet terms and the non-\nKramers form of our time reversal operator, these ansatz are\nnow realizable as invariant PSGs as well. The time reversal\noperator, as deﬁned in Appendix B, acts as\nT:\u0018\u000b\f\nij\u0006\u000b\u0000\f!~\u0018\u000b\f\nij\u0006\u000b\u0000\f; (30)where ~\u0018\u000b\f=\u0018\u000b\fif\u000b2f1;2gand~\u0018\u000b\f=\u0000\u0018\u000b\fif\u000b2f0;3g.\nThe projective implementation of the time-reversal symmetry\ncondition (Eq. 23) takes the form (see Appendix B)\n[GT(i)]2=\u0011TI8i; (31)\nwhereGT(i)is the gauge transformation associated with time\nreversal operation and \u0011T=\u00061for aZ2IGG.\nTherefore, the terms allowed by the time reversal symmetry\nto be non zero are, for \u0011T= 1,\n\u001810;\u001811;\u001812;\u001813;\u001820;\u001821;\u001822;\u001823; (32)\nand for\u0011T=\u00001, with the choice GT(i) =i\u00001(see appendix\nB),\n\u001802;\u001803;\u001810;\u001811;\u001820;\u001821;\u001832;\u001833: (33)\nThis contrasts with the case of Kramers doublets, in which no\nterms are allowed for \u0011T=\u00001, and for\u0011T=\u00001the allowed\nterms are\n\u001802;\u001803;\u001812;\u001813;\u001822;\u001823;\u001832;\u001833: (34)\nFurther restrictions on the allowed terms on each link arise\nfrom the form of the gauge transformations deﬁned for the\nsymmetry transformations. All nearest neighbour bonds can\nthen be generated from Uijdeﬁned on a single bond, by per-\nforming appropriate symmetry operations.\nUsing the methods outlined in earlier works (Ref. 13, 15)\nwe ﬁnd the minimum set of parameters required to stabilize Z2\nspin-orbital liquids. We take into consideration up to second\nneighbour hopping and pairing amplitudes (both singlet and\ntriplet channels). The results are listed in Table I.\nThe spin-orbital liquids listed from 21\u000030are not allowed\nin the case of Kramers doublets and, as pointed out before,\ntheir existence is solely due to the unusual action of the time-\nreversal symmetry operator on the non-Kramers spins. Hence\nthese tenspin-orbital liquids are qualitatively new phases that\nmay appear in these systems. Of these ten phases, only two\n(labelled as 21and22in Table I) require next nearest neigh-\nbour amplitudes to obtain a Z2spin-orbital liquid. For the\nother eight , nearest neighbour amplitudes are already sufﬁ-\ncient to stabilize a Z2spin-orbital liquid.\nIt is interesting to note (see below) that bond-pseudo-spin-\nnematic order (Eq. 35 and Eq. 36) can signal spontaneous\ntime-reversal symmetry breaking. Generally, since the triplet\ndecouplings are present, the bond nematic order parameter for\nthe pseudo-spins21,22\nQ\u000b\f\nij=h\u0010\nS\u000b\niS\f\nj+S\f\niS\u000b\nj\u0011\n=2\u0000\u000e\u000b\f(~Si\u0001~Sj)=3i;(35)\nas well as vector chirality order\n~Jij=h~Si\u0002~Sji; (36)\nare non zero. Since the underlying Hamiltonian Eq. 4) gener-\nally does not have pseudo-spin rotation symmetry, the above\nnon-zero expectation values do not spontaneously break any6\nTABLE I: Symmetry allowed terms: We list the terms allowed to be non-zero by symmetry, for the 30 PSGs determined by Yuan-Ming Lu et\nal15. The PSGs listed together are those with \u001112=\u00061and all other factors equal. Included are terms allowed on nearest and next-nearest\nneighbour bonds, as well as chemical potential terms \u0000which can be non zero on all sites for certain spin-orbital liquids. Also included is the\ndistance of bond up to which we must include in order to gap out the gauge ﬂuctuations to Z2via the Anderson-Higgs mechanism13. Only\nPSGs 9 and 10 can not host Z2spin-orbital liquids with up to second nearest neighbour bonds.\nNo. \u0003s n.n. n.n.n. Z2\n1-2 \u00002;\u00003\u001810;\u001821;\u001802;\u001803;\u001832;\u001833\u001810;\u001821;\u001802;\u001803;\u001832;\u001833n.n.\n3-4 0\u001810;\u001821;\u001802;\u001803;\u001832;\u001833\u001810;\u001821n.n.\n5-6 \u00003\u001810;\u001821;\u001802;\u001803;\u001832;\u001833\u001810;\u001821;\u001803;\u001833n.n.\n7-8 0 \u001811;\u001820\u001811;\u001820;\u001802;\u001803;\u001832;\u001833n.n.n.\n9-10 0 \u001811;\u001820\u001811;\u001820-\n11-12 0 \u001811;\u001820\u001810;\u001811;\u001802;\u001832n.n.n.\n13-14 \u00003\u001810;\u001811;\u001803;\u001833\u001810;\u001821;\u001802;\u001803;\u001832;\u001833n.n.\n15-16 \u00003\u001810;\u001811;\u001803;\u001833\u001810;\u001821;\u001803;\u001833n.n.\n17-18 0\u001810;\u001811;\u001803;\u001833\u001810;\u001811;\u001802;\u001832n.n.\n19-20 0\u001810;\u001811;\u001803;\u001833\u001810;\u001821n.n.\n21-22 0\u001810;\u001821;\u001822;\u001823\u001810;\u001821;\u001822;\u001823n.n.n.\n23-24 0\u001810;\u001821;\u001822;\u001823\u001810;\u001811;\u001812;\u001823n.n.\n25-26 0\u001811;\u001812;\u001813;\u001820\u001813;\u001820;\u001821;\u001822n.n.\n27-28 0\u001811;\u001812;\u001813;\u001820\u001810;\u001811;\u001813;\u001822n.n.\n29-30 0\u001811;\u001812;\u001813;\u001820\u001811;\u001812;\u001813;\u001820n.n.\npseudo-spin rotation symmetry. However, because of the un-\nusual transformation property of the non-Kramers pseudo-\nspins under time reversal, the operators corresponding to\nQ13\nij;Q23\nij;J1\nij;J2\nijare odd under time reversal, a symmetry of\nthe pseudo-spin Hamiltonian. Hence if any of the above op-\nerators gain a non-zero expectation value in the ground state,\nthen the corresponding spin-orbital liquid breaks time rever-\nsal symmetry. While this can occur in principle, we check\nexplicitly (see Appendix C) that in all the spin-orbital liquids\ndiscussed above, the expectation values of these operators are\nidentically zero. This provides a non-trivial consistency check\non our PSG calculations.\nWe now brieﬂy dicuss the effect of the ﬂuctuations about\nthe mean-ﬁeld states. In the absence of pairing channels (both\nsinglet and triplet) the gauge group is U(1). In this case, the\nﬂuctuations of the gauge ﬁeld about the mean ﬁeld (Eq. 15)\nare related to the scalar pseudo-spin chirality ~S1\u0001~S2\u0002~S3,\nwhere the three sites form a triangle.19Such ﬂuctuations are\ngapless in a U(1)spin liquid. It is interesting to note that the\nscalar spin-chirality is odd under time-reversal symmetry and\nit has been proposed that such ﬂuctuations can be detected\nin neutron scattering experiments in presence of spin rotation\nsymmetry breaking.20In the present case, however, due to the\npresence of spinon pairing, the gauge group is broken down\ntoZ2and the above gauge ﬂuctuations are rendered gapped\nthrough Anderson-Higg’s mechanism.13\nIn addition to the above gauge ﬂuctuations, because of thetriplet decouplings which break pseudo-spin rotational sym-\nmetry, there are bond quadrupolar ﬂuctuations of the pseudo-\nspinsQ\u000b\f\nij(Eq. 35), as well as vector chirality ﬂuctuations ~Jij\n(Eq. 36)21,22on the bonds. These nematic and vector chirality\nﬂuctuations are gapped because the underlying pseudo-spin\nHamiltonian (Eq. 4) breaks pseudo-spin-rotation symmetry.\nHowever, we note that because of the unusual transformation\nof the non-Kramers pseudo-spins under time reversal (only\nthez\u0000component of pseudo-spins being odd under time re-\nversal),Q13\nij;Q23\nij;J1\nijandJ2\nijare odd under time reversal.\nHence, while their mean ﬁeld expectation values are zero (see\nabove), the ﬂuctuations of these quantities can in principle lin-\nearly couple to the neutrons in addition to the z\u0000component\nof the pseudo-spins.\nHaving identiﬁed the possible Z2spin-orbital liquids, we\ncan now study typical dynamic structure factors for these\nspin-orbital liquids. In the next section we examine the typi-\ncal spinon band structure for different spin-orbital liquids ob-\ntained above and ﬁnd their dynamic spin structure factor.7\nFIG. 3: The spin structure factor for an ansatz in spin liquid 17, with\nthe spin variables transforming as a Kramers doublet.\nFIG. 4: The spin structure factor for an ansatz in spin liquid 17, with\nthe spin variables transforming as a non-Kramers doublet.\nIV . DYNAMIC SPIN STRUCTURE FACTOR\nWe compute the dynamic spin structure factor\nS(q;!) =Zdt\n2\u0019ei!tX\nijeiq\u0001(ri\u0000rj)X\na=1;2;3h\u001ba\ni(t)\u001ba\nj(0)i;\n(37)\nfor an example ansatz of our spin liquid candidates, in order to\ndemonstrate the qualitative differences between the Kramers\nand non-Kramers spin-orbital liquids. In the above equation,\nthe pseudo-spin variables are deﬁned in a global basis (with\nthe z-axis perpendicular to the Kagome plane). In computing\nthe structure factor for the non-Kramers example, we include\nonly the\u001b3components of the pseudo-spin operator in the lo-\ncal basis, since only the z-components carry magnetic dipole\nmoment (see discussion before). Hence, only this component\ncouples linearly to neutrons in a neutron scattering experi-\nment.\nEq. 37 fails to be periodic in the ﬁrst Brillouin zone of\nthe Kagome lattice16, as the term ri\u0000rjin eq. 37 is a half-\ninteger multiple of the primitive lattice vectors when the sub-\nlattices of sites i and j are not equal. As such, we examine the\nstructure factor in the extended brillouin zone, which consists\nof those momenta of length up to double that of those in theﬁrst brillouin zone. We plot the structure factor along the cut\n\u0000!M0!K0!\u0000, whereM0=2MandK0=2K. We\nexamine the structure factors for two ansatz of spin liquid #\n17 which has both Kramers and non-Kramers analogues.\nAs expected, we ﬁnd that the structure factor has greater\nweight in the case of a Kramers spin liquid. This is par-\ntially due to the fact that the moment of the scattering par-\nticle couples with all components of the spin, rather than sim-\nply thez-component. In addition, we note that the presence\nof terms allowed in the non-Kramers spin-orbital liquid in-\nduce the formation of a gap, which is absent for the Kramers\ncase with up to second nearest neighbour singlet and triplet\nterms in this particular spin-orbital liquid. Qualitative and\nquantitative differences such as these, which can be observed\nin these structure factors between Kramers and non-Kramers\nspin-orbital liquids, provides one possible distinguishing ex-\nperimental signature of these states. We shall not pursue this\nin detail in the present work.\nV . DISCUSSION AND POSSIBLE EXPERIMENTAL\nSIGNATURE OF NON-KRAMERS SPIN-ORBITAL LIQUIDS\nIn this work, we have outlined the possible Z2spin-orbital\nliquids, with gapped or gapless fermionic spinons, that can be\nobtained in a system of non-Kramers pseudo-spin-1/2s on a\nKagome lattice of Pr+3ions. We ﬁnd a total of thirty , 10 more\nthan in the case of corresponding Kramers system, allowed\nwithin PSG analysis in presence of time reversal symmetry.\nThe larger number of spin-orbital liquids is a result of the dif-\nference in the action of the time-reversal operator, when real-\nized projectively. We note that the spin-spin dynamic struc-\nture factor can bear important signatures of a non-Kramers\nspin-orbital liquid when compared to their Kramers counter-\nparts. Our analysis of the number of invariant PSGs leading to\npossibly different spin-orbital liquids that may be realizable in\nother lattice geometries will form interesting future directions.\nWe now brieﬂy discuss an experiment that can play an im-\nportant role in determining non-Kramers spin-orbital liquids.\nSince the non-Kramers doublets are protected by crystalline\nsymmetries, lattice strains can linearly couple to the pseudo-\nspins. As we discussed, the transverse ( xandy) components\nof the pseudo-spins f\u001b1;\u001b2gcarry quadrupolar moments and\nhence are even under the time reversal transformation. Fur-\nther, they transform under an Egirreducible representation\nof the local D3dcrystal ﬁeld. Hence any lattice strain which\nhas this symmetry can linearly couple to the above two trans-\nverse components. It turns out that in the crystal type that\nwe are concerned, there is indeed such a mode related to the\ndistortion of the oxygen octahedra. Symmetry considerations\nshow that the linear coupling is of the form Eg1\u001b1+Eg2\u001b2\n(fEg1;Eg2gbeing the two components of the distortion in\nthe local basis). The above mode is Raman active. For\na spin-liquid, we expect that as the temperature is lowered,\nthe spinons become more prominent as deconﬁned quasipar-\nticles. So the Raman active phonon can efﬁciently decay into\nthe spinons due to the above coupling channel. If the spin\nliquid is gapless, then this will lead to anomalous broaden-8\ning of the above Raman mode as the temperature is lowered,\nwhich, if observed, can be an experimental signature of the\nnon-Kramers spin-orbital liquid. The above coupling is for-\nbidden in Kramers doublets by time-reversal symmetry and\nhence no such anomalous broadening is expected.\nAcknowledgments\nWe thank T. Dodds, SungBin Lee, A. Paramekanti and J.\nRau for insightful discussions. This research was supported\nby the NSERC, CIFAR, and Centre for Quantum Materials at\nthe University of Toronto.\nAppendix A: Crystal Field Effects\nIn this appendix, we explore the breaking of the J= 4spin\ndegeneracy by the crystalline electric ﬁeld. The oxygen and\nTM ions form a D3dlocal symmetry environment around the\nPr3+ions, splitting the ground state degeneracy of the elec-\ntrons. This symmetry group contains 6 classes of elements: E,\n2C3,3C0\n2,i,2S6, and 3\u001bd, where theC3are rotations by 2\u0019=3\nabout the local z axis, the C0\n2are rotations by \u0019about axis per-\npendicular to the local z axis, iis inversion, S6is a rotation\nby4\u0019=3combined with inversion and \u001bdis a reﬂection about\nthe plane connecting one corner and the opposing plane, run-\nning through the Prmolecule about which this is measured\n(or, equivalently, a rotation about the x axis combined with\ninversion). For our J=4 manifold, these have characters given\nby\n\u001f(4)(E) = 2\u00034 + 1 = 9 = \u001f(4)(i) (A1)\n\u001f(4)(C3) =\u001f(4)(2\u0019\n3) =sin(3\u0019)\nsin(\u0019=3)= 0 =\u001f(4)(S6)(A2)\n\u001f(4)(\u001bd) =\u001f(4)(\u0019) =sin(9\u0019=2)\nsin(\u0019=2)= 1 =\u001f(4)(C0\n2)(A3)\nwhere the latter equalities are given by the fact that our J=4\nmanifold is inversion symmetric. Thus, decomposing this in\nterms ofD3dirreps, our l=4 manifold splits into a sum of\ndoublet and singlet manifolds as\n\u0000l=4= 3Eg+ 2A1g+A2g: (A4)\nTo examine this further, we need to consider the matrix ele-\nments of the crystal ﬁeld potential between the states of differ-\nent angular momenta. We know that this potential must be in-\nvariant under all group operations of D3d, so we can examine\nthe transformation properties of individual matrix elements,\nhmjVjm0i. Under the C3operation, these states of ﬁxed m\ntransform as\nC3jmi=e2\u0019im\n3jmi=!mjmi (!=e2\u0019i\n3) (A5)\nand thus the matrix elements transform as\nC3:hmjVjm0i!hmj(C3)\u00001VC3jm0i=!m0\u0000mhmjVjm0i:\n(A6)By requiring that this matrix be invariant under this transfor-\nmation, we can see that this potential only contains matrix\nelements for mixing of states which have the z-component of\nangular momentum which differ by 3. Thus, our eigenstates\nare mixtures of the jm= 4i,jm= 1i, andjm=\u00002istates,\nof thejm= 3i,jm= 0i, andjm=\u00003istates, and of the\njm=\u00004i,jm=\u00001i, andjm= 2istates.\nIn addition to this, we have the transformation properties\nTjmi= (\u00001)mj\u0000mi (A7)\nand\n\u001bjmi= (\u00001)mj\u0000mi (A8)\n(where the operators for time reversal and reﬂection are\nbolded for future clarity). Inversion acts trivially on these\nstates, as we have total angular momentum even. Thus our\ntime-reversal and lattice reﬂection (about one axis) symme-\ntries give us doublet states of eigenstates \u000bjm= 4i+\fjm=\n1i\u0000\rjm=\u00002iand\u000bjm=\u00004i\u0000\fjm=\u00001i\u0000\rjm= 2i\n(with\u000b,\f,\r2< in order to respect the time reversal symme-\ntry) for the three eigenstates of V in these sectors. The eigen-\nstates of thejm= 3i,jm= 0i, andjm=\u00003iportion of V\nmust therefore split into three singlet states, by our represen-\ntation theory argument A4. Due to the expected strong Ising\nterm in our potential, we expect the eigenstate with maximal J\nto be the ground state, meaning that to analyze the properties\nof this ground state we are interested in a single doublet state,\none with large \u000b(close to one). We will restrict ourselves\nto this manifold from this point forward, and deﬁne the two\nstates in this doublet as\nj+i=\u000bjm= 4i+\fjm= 1i\u0000\rjm=\u00002i (A9)\nj\u0000i=\u000bjm=\u00004i\u0000\fjm=\u00001i\u0000\rjm= 2i:(A10)\nWe shall also refer to states of angular momentum jm=ni\nasjnifor simplicity of notation.\nAppendix B: Gauge transformations\nWe begin by describing the action of time reversal on our\nansatz. The operation is antiunitary, and must be combined\nwith a spin transformation \u001b1in the case of non-Kramers dou-\nblets. As a result, the operation acts as T:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\n\u0018\u000b\f\u0003\nij\u00061\u0006\u000b\u0003\u00061\u0000\f\u0003. However, we can simplify this consid-\nerably by performing a gauge transformation in addition to\nthe above transformation, which yields the same transforma-\ntion on any physical variables. The gauge transformation we\nperform isi\u00002, which changes the form of the time reversal\noperation to T:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\u0018\u000b\f\u0003\nij\u00061\u0006\u000b\u0003\u00061\u00002\u0000\f\u0003\u00002=\n~\u0018\u000b\f\nij\u0006\u000b\u0000\f, where ~\u0018\u000b\f=\u0018\u000b\fif\u000b2f1;2gand~\u0018\u000b\f=\u0000\u0018\u000b\fif\n\u000b2f0;3g.\nOn the Kagome lattice, the allowed form of the gauge trans-\nformations has been determined by Yuan-Ming Lu et al.15For\ncompleteness, we will reproduce that calculation, valid also9\nfor our spin triplet ansatz, here. The relations between the\ngauge transformation matrices,\n[GT(i)]2=\u0011TI; (B1)\nG\u001b(\u001b(i))G\u001b(i) =\u0011\u001bI; (B2)\nGy\nT1(i)Gy\nT(i)GT1(i)GT(T\u00001\n1(i)) =\u0011T1TI; (B3)\nGy\nT2(i)Gy\nT(i)GT2(i)GT(T\u00001\n2(i)) =\u0011T2TI; (B4)\nGy\n\u001b(i)Gy\nT(i)G\u001b(i)GT(\u001b\u00001(i)) =\u0011\u001bTI; (B5)\nGy\nS6(i)Gy\nT(i)GS6(i)GT(S\u00001\n6(i)) =\u0011S6TI; (B6)\nGy\nT2(T\u00001\n1(i))Gy\nT1(i)GT2(i)GT1(T\u00001\n2(i)) =\u001112I; (B7)\nGS6(S\u00001\n6(i))GS6(S\u00002\n6(i))GS6(S3\n6(i))\n\u0002GS6(S2\n6(i))GS6(S6(i))GS6(i) =\u0011S6I; (B8)\nGy\n\u001b(T\u00001\n2(i))Gy\nT2(i)G\u001b(i)GT1(\u001b(i)) =\u0011\u001bT1I; (B9)\nGy\n\u001b(T\u00001\n1(i))Gy\nT1(i)G\u001b(i)GT2(\u001b(i)) =\u0011\u001bT2I; (B10)\nGy\n\u001b(S6(i))GS6(S6(i))G\u001b(i)GS6(\u001b(i)) =\u0011\u001bS6I; (B11)\nGy\nS6(T\u00001\n2(i))Gy\nT2(i)GS6(i)GT1(S\u00001\n6(i)) =\u0011S6T1I;(B12)\nGy\nS6(T\u00001\n2T1(i))Gy\nT2(T1(i))GT1(T1(i))\nGS6(i)GT2(S\u00001\n6(i)) =\u0011S6T2I; (B13)\nare valid for our case as well, due to the decoupling of spin\nand gauge portions of our ansatz. In the above, the relations\nare valid for all lattice sites i= (x;y;s ), I is the 4x4 identity\nmatrix, and the GSmatrices are gauge transformation matri-\nces generated by exponentiation of the \u0000matrices. The \u0011’s\nare\u00061, the choice of which characterize different spin liquid\nstates. In deriving this form of the commutation relations, we\nhave included a gauge transformation i\u00002in our deﬁnition of\nthe time reversal operator, as this simpliﬁes the effect of the\noperator on the mean ﬁeld ansatz.\nWe turn next to the calculation of the gauge transforma-\ntions. We look ﬁrst at the gauge transformations associated\nwith the translations. We can perform a site dependent gauge\ntransformation W(i), under which the gauge transformations\nassociated with the translational symmetries transform as\nGT1(i)!W(i)GT1(i)Wy(i\u0000^x) (B14)\nGT2(i)!W(i)GT2(i)Wy(i\u0000^y): (B15)\nAs such, we can choose a gauge transformation W(i) to sim-\nplify the form of GT1andGT2. Using such a transformation,\nalong with condition B7, we can restrict the form of these\ngauge transformations to be\nGT1(i) =\u0011iy\n12I GT2(i) =I: (B16)\nTo preserve this choice, we can now only perform gauge\ntransformations which are equivalent on all lattice positions\n(W(x;y;s ) =W(s)) or transformations which change the\nshown matrices by an IGG transformation.\nNext, we look at adding the reﬂection symmetry \u001b. Given\nour formulae for GT1andGT2, along with the relations be-tween the gauge transformations, we have that\nGy\n\u001b(T\u00001\n2(i))G\u001b(i)\u0011x\n12=\u0011\u001bT1I (B17)\nGy\n\u001b(T\u00001\n1(i))G\u001b(i)\u0011y\n12=\u0011\u001bT2I: (B18)\nDeﬁningG\u001b(0,0,s) =g\u001b(s), we have, by repeated application\nof the above,\nG\u001b(0;y;s) =\u0011y\n\u001bT1g\u001b(s) (B19)\nG\u001b(x;y;s ) =\u0011y\n\u001bT1\u0011xy\n12\u0011x\n\u001bT2g\u001b(s): (B20)\nNext, using\nG\u001b(\u001b(i))G\u001b(i) =\u0011\u001bI (B21)\nwe ﬁnd that\n\u0011\u001bI=G\u001b(y;x;\u001b (s))G\u001b(x;y;s ) (B22)\n= (\u0011\u001bT1\u0011\u001bT2)x+yg\u001b(\u001b(s))g\u001b(s): (B23)\nSince this is true for all x and y, \u0011\u001bT1\u0011\u001bT2= 1 and thus\n\u0011\u001bT1=\u0011\u001bT2andg\u001b(\u001b(s))g\u001b(s) =\u0011\u001bI(where\u001b(u) =\nu;\u001b(v) =wand\u001b(w) =v). Our ﬁnal form for the gauge\ntransformation is\nG\u001b(x;y;s ) =\u0011x+y\n\u001bT1\u0011xy\n12g\u001b(s): (B24)\nNext we look at adding the S6symmetry to our calcula-\ntion. We can do an IGG transformation, taking GT1(T1(i))\nto\u0011S6T2GT1(T1(i)), with the net effect being that \u0011S6T2be-\ncomes one (previous calculations are unaffected). We now\nhave that\nGy\nS6(T\u00001\n2T1(i))GS6(i)\u0011y\n12=I (B25)\nGy\nS6(T\u00001\n2(i))GS6(i)\u0011\u0000x\u00001\n12 =\u0011S6T1I (s=u;v)(B26)\nGy\nS6(T\u00001\n2(i))GS6(i)\u0011\u0000x\n12=\u0011S6T1I (s=w):(B27)\nDeﬁningGS6(0;0;s) =gS6(s), we ﬁnd that\nGS6(n;\u0000n;s) =\u0011n(n\u00001)=2\n12gS6(s) (B28)\nGS6(x;y;s ) =\u0011x(x\u00001)=2+y+xy\n12 \u0011x+y\nS6T1gS6(s) (s=u;v)\n(B29)\nGS6(x;y;s ) =\u0011x(x\u00001)=2+xy\n12 \u0011x+y\nS6T1gS6(s) (s=w):\n(B30)\nUsing the commutation relation between the \u001bandS6gauge\ntransformations, we ﬁnd that\n\u0011\u001bS6I=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1gy\n\u001b(v)gS6(v)g\u001b(u)gS6(u) (B31)\n=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1\u0011\u001bg\u001b(w)gS6(v)g\u001b(u)gS6(u)(B32)\ngiving us that \u0011\u001bT1\u001112\u0011S6T1 = 1 and\ng\u001b(u)gS6(u)g\u001b(w)gS6(v) =\u0011\u001bS6\u0011\u001bI. A similar calcu-\nlation on a different sublattice gives us\n\u0011\u001bS6I=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1gy\n\u001b(w)gS6(w)g\u001b(v)gS6(w) (B33)\n=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1\u0011\u001bg\u001b(v)gS6(w)g\u001b(v)gS6(w)(B34)10\nTABLE II: We list the solutions of Eq. B43 - B54, along with a set of gauge transformations which realize these solutions.\nNo.\u0011T\u0011\u001bT\u0011S6T\u0011\u001b\u0011\u001bS6\u0011S6\u001112g\u001b(u)g\u001b(v)g\u001b(w)gS6(u)gS6(v)gS6(w)\n1,2 -1 1 1 1 1 \u00061\u00061 \u00000\u00000\u00000\u00000\u00000\u00000\n3,4 -1 1 1 1 -1 \u00071\u00061 \u00000\u00000\u00000\u00000-\u00000i\u00001\n5,6 -1 1 -1 1 -1 \u00071\u00061 \u00000\u00000\u00000i\u00003i\u00003i\u00003\n7,8 -1 1 1 -1 -1 \u00071\u00061i\u00001\u00000-\u00000\u00000i\u00001\u00000\n9,10 -1 1 1 -1 1 \u00061\u00061i\u00001\u00000-\u00000\u00000-i\u00001i\u00001\n11,12 -1 1 -1 -1 1 \u00071\u00061i\u00001\u00000-\u00000i\u00003-i\u00002i\u00003\n13,14 -1 -1 -1 -1 -1 \u00071\u00061i\u00003i\u00003i\u00003i\u00003i\u00003i\u00003\n15,16 -1 -1 1 -1 1 \u00061\u00061i\u00003i\u00003i\u00003\u00000\u00000\u00000\n17,18 -1 -1 1 -1 1 \u00071\u00061i\u00003i\u00003i\u00003\u00000\u00000i\u00001\n19,20 -1 -1 -1 -1 1 \u00071\u00061i\u00003i\u00003i\u00003i\u00003-i\u00003i\u00003\n21,22 1 1 1 1 1 \u00061\u00061 \u00000\u00000\u00000\u00000\u00000\u00000\n23,24 1 1 1 1 -1 \u00071\u00061 \u00000\u00000\u00000\u00000-\u00000i\u00003\n25,26 1 1 1 -1 -1 \u00071\u00061i\u00003\u00000-\u00000\u00000i\u00003\u00000\n27,28 1 1 1 -1 1 \u00071\u00061i\u00003\u00000-\u00000\u00000-i\u00003i\u00001\n29,30 1 1 1 -1 1 \u00061\u00061i\u00003\u00000-\u00000\u00000-i\u00003i\u00003\ngiving us (g\u001b(v)gS6(w))2=\u0011\u001bS6\u0011\u001bI. AZ2(IGG) gauge\ntransformation of the form W(x;y;s ) =\u0011y\n\u001bT1changes\u0011\u001bT1\nto 1. Using the cyclic relation of the gauge transformations\nrelated to the S6operators, we ﬁnd\n\u0011S6I=\u001112(gS6(w)gS6(v)gS6(u))2(B35)\ngiving us that\n[gS6(w)gS6(v)gS6(u)]2=\u0011S6\u001112I: (B36)\nNext we turn to the time reversal symmetry. Similar meth-\nods to the above give us that\n[GT(i)]2=\u0011TI (B37)\nGy\nT(i)GT(i+ ^x) =\u0011T1TI (B38)\nGy\nT(i)GT(i+ ^y) =\u0011T2TI: (B39)\nThe ﬁrst of these relations tells us that GT(i)is either the\nidentity (for \u0011T= 1) ori~ a\u0001~ \u0014(for\u0011T=\u00001, wherej~ aj= 1.\nDeﬁningGT(0;0;s) =gT(s),\nGT(x;y;s ) =\u0011x\nT1T\u0011y\nT2TgT(s) (B40)\nand further, using the commutation relations between the \u001b\nandTgauge transformations and the S6andTgauge trans-\nformations,\ngy\n\u001b(s)gy\nT(s)g\u001b(s)gT(\u001b(s))\u0011x+y\nT1T\u0011x+y\nT2T=\u0011\u001bTI(B41)\ngy\nS6(s)gy\nT(s)gS6(s)gT(S\u00001\n6(s))\u0011f1(i)\nT1T\u0011f2(i)\nT2T=\u0011S6TI:(B42)Because this is true for all x and y, and f1(i)is not equal to\nf2(i),\u0011T1T=\u0011T2T= 1. IfGT(i) =i~ a\u0001~ \u0014, we perform a\ngauge transformation W on GT(i)such thatWyGT(i)W=\ni\u00141(as this is the same on all sites, it does not affect our gauge\nﬁxing for the translation gauge transformations). Collecting\nthe necessary results for further use,\nGT1(x;y;s ) =\u0011y\n12I (B43)\nGT2(x;y;s ) =I (B44)\nG\u001b(x;y;s ) =\u0011xy\n12g\u001b(s) (B45)\nGS6(x;y;s ) =\u0011xy+(x+1)x=2\n12 gS6(s)s=u;v (B46)\nGS6(x;y;s ) =\u0011xy+x+y+(x+1)x=2\n12 gS6(s)s=w(B47)\nGT(s) =I=gT(s)\u0011T= 1 (B48)\nGT(s) =i\u00001=gT(s)\u0011T=\u00001 (B49)\ng\u001b(\u001b(s))g\u001b(s) =\u0011\u001bI (B50)\ng\u001b(u)gS6(u)g\u001b(w)gS6(v) = (g\u001b(v)gS6(w))2=\u0011\u001bS6\u0011\u001bI\n(B51)\n(gS6(w)gS6(v)gS6(u))2=\u0011S6\u001112I (B52)\ng\u001b(s)gT(\u001b(s)) =\u0011\u001bTgT(s)g\u001b(s) (B53)\ngS6(s)gT(S\u00001\n6(s)) =\u0011S6TgT(s)gS6(s): (B54)\nWe also have the gauge freedom left to perform a gauge ro-\ntation arbitrarily at all positions for \u0011T= 1 or an arbitrary\ngauge rotation about the x axis for \u0011T=\u00001.\nThe solution to the above equations is derived in detail by\nLuet al.15and as such we simply list the results in table II.\nThe basic method of obtaining these solutions is as follows:11\nfor each choice of Z2parameter set, we determine whether\nthere is a choice of gauge matrices fgSgwhich satisfy the\nequations B43 - B54. In order to do so, we determine the\nallowed forms of the gSmatrices from the equations, then use\nthe gauge freedom on each site to ﬁx the form of these. Of\nparticular not is the fact that in the consistency equations for\nthegmatrices, the terms \u001112and\u0011S6only appear multiplied\ntogether, meaning that for any choice of the gauge matrices\ngSwe can choose \u001112=\u00061, which ﬁxes the form of \u0011S6.\nAppendix C: Relation among the mean-ﬁeld paramters\nThe relation among the different singlet and triplet param-\neters in terms of \u0018ijis given by\n\u001fij=\u001800\nij+\u001803\nij;\u0011ij=\u0000\u001801\nij+i\u001802\nij;\nE1\nij=\u001810\nij+\u001813\nij;E2\nij=\u001820\nij+\u001823\nij;E3\nij=\u001830\nij+\u001833\nij\nD1\nij=\u0000\u001811\nij+i\u001812\nij;D2\nij=\u0000\u001821\nij+i\u001822\nij;D3\nij=\u0000\u001831\nij+i\u001832\nij\n(C1)\nUsing these, we can derive the form of the bond nematic\norder parameter and vector chirality order parameters, which\nare given in terms of the mean ﬁeld parameters21as\nQ\u0016;\u0017\nij=\u00001\n2\u0000\nE\u0016\nijE\u0003\u0017\nij\u00001\n3\u000e\u0016;\u0017j~Eijj2\u0001\n+h:c:\n\u00001\n2\u0000\nD\u0016\nijD\u0003\u0017\nij\u00001\n3\u000e\u0016;\u0017j~Dijj2\u0001\n+h:c:\nJ\u0015\nij=i\n2\u0000\n\u001fijE\u0003\u0015\nij\u0000\u001f\u0003\nijE\u0015\nij\u0001\n+i\n2\u0000\n\u0011ijD\u0003\u0015\nij\u0000\u0011\u0003\nijD\u0015\nij\u0001\n(C2)where our deﬁnition of \u0011ijdiffers by a factor of (-1) from that\nof the cited work. We rewrite this in terms of our variables,\nﬁnding\nQ\u0016\u0017\nij=\u0000\u0018\u00160\nij\u0018\u00170\nij+X\na\u0018\u0016a\nij\u0018\u0017a\nij\n+\u000e\u0016\u0017\n3X\nb\u0000\n(\u0018b0\nij)2\u0000X\na(\u0018ba\nij)2)\nJ\u0015\nij=i(\u001800\nij\u0018\u00150\nij\u0000X\na\u00180a\nij\u0018\u0015a\nij) (C3)\nIn particular, we ﬁnd that J1,J2,Q13andQ23must be zero\nfor all non-Kramers spin liquids, as the terms allowed by sym-\nmetry in Eq. 32 and 33 do not allow non-zero values for these\norder parameters.\n1P. Fazekas, Lecture Notes on Electron Correlation and Magnetism\n(Series in Modern Condensed Matter Physics) , World Scientiﬁc\nPub Co Inc (1999).\n2K. Yosida, Theory of Magnetism , Springer (2001).\n3B. Bleaney and H. E. D. Scovil, Phil. Mag. 43, 999 (1952).\n4Shigeki Onoda and Yoichi Tanaka, Phys. Rev. B 83, 094411(R)\n(2011).\n5K. Matsuhira, Y . Hinatsu, K. Tenya1, H. Amitsuka and T. Sakak-\nibara, J. Phys. Soc. Jpn. 71, 1576 (2002).\n6K. Matsuhira, C. Sekine, C. Paulsen, M. Wakeshima, Y . Hinatsu,\nT. Kitazawa, Y . Kiuchi, Z. Hiroi, and S. Takagi, J. Phys. Conf. Ser.\n145, 012031 (2009).\n7S. Nakatsuji, Y . Machida, Y . Maeno, T. Tayama, T. Sakakibara, J.\nvan Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and J. Y .\nChan, Phys. Rev. Lett. 96, 087204 (2006).\n8SB. Lee, S. Onoda, and L. Balents, Phys. Rev. B 86, 104412\n(2012).\n9R. Flint, and T. Senthil, arXiv:1301.0815 (2013).\n10P. Chandra, P. Coleman, and R. Flint, Nature 493, 621 (2013).\n11O. Suzuki, H. S. Suzuki, H. Kitazawa, G. Kido, T. Ueno, T. Ya-maguchi, Y . Nemoto, T. Goto, J. Phys. Soc. Japan 75, 013704\n(2006).\n12H. A. Kramers, Proc. Amsterdam Acad. 33, 959 (1930).\n13X. G. Wen, Phys. Rev. B 65, 165113 (2002); X.-G. Wen, Quantum\nField Theory of Many-Body Systems. Oxford University Press,\nOxford (2004).\n14P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett.\n58, 2790 (1987).\n15Yuan-Ming Lu, Ying Ran, and Patrick A. Lee, Phys. Rev. B 83,\n224412 (2011).\n16T. Dodds, S. Bhattacharjee, and Y . B. Kim, arXiv:1303.1154\n17R. Schaffer, S. Bhattacharjee, and Y . B. Kim, Phys. Rev. B 86,\n224417 (2012).\n18I. Afﬂeck, and J. B. Marston, Phys. Rev. B 37, 3774 (1988).\n19P. A. Lee, and N. Nagaosa, Phys. Rev. B 46, 5621 (1992).\n20P. A. Lee, and N. Nagaosa, Phys. Rev. B 87, 064423 (2013).\n21R. Shindou, and T. Momoi, Phys. Rev. B 80, 064410 (2009).\n22S. Bhattacharjee, Y . B. Kim, S.-S. Lee, and D.-H. Lee, Phys. Rev.\nB 85, 224428 (2012)."    },    {        "title": "1902.09784v1.Dynamic_spin_charge_coupling__spin_Hall_magnetoresistance_in_non_magnetic_conductors.pdf",        "content": "arXiv:1902.09784v1  [cond-mat.mes-hall]  26 Feb 2019Dynamic spin-charge coupling:\nacspin Hall magnetoresistance in non-magnetic conductors\nP.S. Alekseev1and M.I. Dyakonov2\n1Ioﬀe Institute,194021, St. Petersburg, Russia\n2Laboratoire Charles Coulomb,\nUniversit´ e Montpellier, CNRS, France\nThe dynamic couplingbetween spinand charge currents innon -magnetic conductors is considered.\nAs a consequence of this coupling, the spin dynamics is direc tly reﬂected in the electrical impedance\nof the sample, with a relevant frequency scale deﬁned by spin relaxation and spin diﬀusion. This\nallows the observation of the electron spin resonance by pur ely electrical measurements.\nPACS numbers: 72.25.-b, 71.70.Ej, 72.20.Dp\n1. Introduction. It was predicted nearly half a cen-\ntury ago [1, 2] that spin-orbit interaction results in the\ninterconnection between electrical and spin currents: an\nelectrical current produces a transverse spin current and\nvice versa . This leads respectively to the direct and in-\nverse spin Hall eﬀects. Following the proposal in Ref. [3],\nthe inverse spin Hall eﬀect was observed experimentally\nby Bakun et al.[4] in 1984, without causing much excite-\nment at that time.\nTwenty years later, after the ﬁrst experimental obser-\nvations of the (direct) spin Hall eﬀect [5, 6] this topic has\nbecome a subject of considerable interest with thousands\nof publications, see for example a review in Ref. [7].\nBecause of the interconnection between the spin and\ncharge currents, anything that happens with spins will\ninﬂuence the charge current, i.e. result in corresponding\nchanges of the electrical resistance, which can be mea-\nsured with a very high precision. An example of this link\nis provided by the spin Hall magnetoresistance [8], the\nreason for which is the depolarization of spins accumu-\nlated at the sample boundaries by a transverse magnetic\nﬁeld and the resulting decrease of the driving electric\ncurrent (for a given voltage) [9]. This eﬀect was experi-\nmentally demonstrated in platinum by V´ elez et al.[10].\nEarlier, a similar eﬀect was discovered and studied by\nNakayama et al.[11] in layered structures ferromagnet-\nnormalmetal. The magnetizationin the ferromagnetcan\nbe rotated by an applied magnetic ﬁeld which results in\na change in the normal metal resistivity.\nIn recent years, the acspin Hall eﬀect in ferromagnet-\nnormal metal structures has also been studied both ex-\nperimentally [12–15] and theoretically [16–18]. The pre-\ncession of the magnetization in a ferromagnet leads to a\ntime-varyinginjection ofspin intothenormalmetal. Due\nto the inverse spin Hall eﬀect, the resulting spin current\nin the normal metal generates the acelectric current.\nIn particular, the observed acvoltage resonantly de-\npends on the Larmor frequency in the ferromagnet and\nthe frequency of the external acmagnetic ﬁeld, which ex-\ncites the precession of magnetization. In this way, with\nthe aid of the spin Hall eﬀect in a normal metal, theferromagnetic resonance was observed by electric mea-\nsurements.\nWhile these studies are quite important for achieving\nthe ultimate goal of storing and manipulating informa-\ntion by the use of spin Hall eﬀect for switching mag-\nnetic domains in magnetics (see the reviews [19, 20]), the\nphysics of the layered magnetic structures is quite com-\nplicated, and this makes the exact theory and the quan-\ntitative analysis of experimental data rather diﬃcult.\nHere, following Ref. [8], we develop a much more sim-\nple theory of acelectron magnetotransport controlled by\nthe direct and inverse spin Hall eﬀects in non-magnetic\nmaterials, semiconductorsormetals. The theory is based\non the phenomenological transport equations [1, 2, 7, 8]\ndescribing the interconnection between spin and charge\ncurrents. We show that spin resonance in non-magnetic\nmaterials can be observed by purely transport measure-\nments.\n2. Transport equations. Consider a conductor in an\nexternal acelectric ﬁeld E(t)∼cos(ωt) and a magnetic\nﬁeldB, see Fig. 1. We assume that the acfrequency ω\nis much lower than the cyclotron frequency ωc, and that\nωτ≪1, where τisthemomentumrelaxationtime. How-\nW\nLxyz\nI(t)\nBE(t)  ~   cos    ωt\nFIG. 1: A metal or semiconductor sample in a magnetic\nﬁeldBandacelectric ﬁeld E(t). The length of the sample,\nL, is much greater than its width, W. It is assumed that the\nacelectric ﬁeld penetrates everywhere into the sample, i.e.\nthat the electron system is either two-dimensional, or thre e-\ndimensional, but with thickness less than the skin depth.2\neverthe spin Larmorfrequency Ωandthe spin relaxation\ntimeτs≫τare such that ω∼Ω∼1/τs.\nIn this frequency range, the basic phenomenological\nequations for the electron ﬂow density q=j/e, the spin\ncurrent density tensor qij, and the spin density vector P\nare [7, 21]:\nq=µnE+γDrotP, (1)\nqij=−D∂Pj\n∂xi+γµnǫijkEk, (2)\n∂Pj\n∂t+∂qij\n∂xi+(Ω×P)j+Pj\nτs= 0,(3)\nwherenis the electron density, µis the electron mobility,\nDis the diﬀusion coeﬃcient, γ≪1 is the dimensionless\nparameter proportional to the strength of the spin-orbit\ninteraction and describing the interconnection between\nthe particle and the spin currents, ǫijkis the unit an-\ntisymmetric tensor, the vector Ωis directed along the\napplied magnetic ﬁeld, Ωbeing the Larmor frequency\nfor electron spins, and τsis the spin relaxation time.\nThe ﬁrst term in Eq. (1) is the usual Drude contribu-\ntion, while the second term expresses the interconnection\n(caused by spin-orbit interaction) between particle cur-\nrent and the spin current caused by the inhomogeneity\nof the spin density.\nEq. (2) describes two contributions to the spin cur-\nrent density qij: the ﬁrst one is due to diﬀusion of spin-\npolarizedelectronsandthesecondoneisduetothetrans-\nformation of particle current into spin current.\nEq. (3) is the continuity equation for the spin den-\nsity, taking into account spin diﬀusion, rotation of spin\nin magnetic ﬁeld, and spin relaxation.\nIn the geometry of Fig. 1, both the particle and the\nspin ﬂows depend on the ycoordinate only. We consider\nthat the acelectric ﬁeld, as well as the magnetic ﬁeld,\nare directed along the xdirection.\nAs it followsfrom Eq.(2), the twononzerocomponents\nof the spin current density tensor are:\nqyy=−DdPy\ndy, qyz=−DdPz\ndy+γµnEcosωt.(4)\nThe absence of spin currents across the sample\nboudariesis describedbythe boundaryconditions: qyy=\n0 andqyz= 0 aty=±W/2.\n3. Solution of the transport equations. The nonzero\ncomponents of the spin density vector Pcan be written\nin a complex form: Py(y,t) = [Py(y)e−iωt+c.c.]/2, and\nsimilarly for Pz(y,t). Then, from Eqs. (3) and (4) we\nobtainasystemofcoupledequationsfor Py(y)andPz(y):\nDd2Py\ndy2=/parenleftBig\n−iω+1\nτs/parenrightBig\nPy+ΩPz,\nDd2Pz\ndy2=/parenleftBig\n−iω+1\nτs/parenrightBig\nPz−ΩPy,(5)with the boundary conditions:\ndPy\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=±W/2= 0,dPz\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=±W/2=γµnE\nD.(6)\nThe solution of Eqs. (5) with the boundary conditions\n(6) yields the spin density proﬁle:\nPy(y) =−iγµnE\n2D[F+(y)−F−(y)],\nPz(y) =γµnE\n2D[F+(y)+F−(y)],(7)\nwhere\nF±(y) =sinh(λ±y)\nλ±cosh(λ±W/2), (8)\nλ±=/radicalbig\n1+i(−ω±Ω)τs\nLs, (9)\nandLs=√Dτsis the spin diﬀusion length.\nThus for narrow samples, |λ±|W≪1, the spin density\nPdepends linearly on the coordinate y, while for wide\nsamples, |λ±|W≫1, the spin density Pis concentrated\nnear the sample edges. In the last case, spin density ex-\nhibits spin resonance at ω=±Ωprovided that ωτs/greaterorsimilar1.\nThesigns ±correspondtothecontributionto Pfromthe\ncomponents of E(t) with the right and the left circular\npolarizations, respectively.\nThe current density jx=eqxcan now be calculated\nusingEq. (1): jx=eµnEcosωt+∆j(y,t),wheretheﬁrst\nterm is the normal Drude contribution (in the assumed\nlimitωτ≪1), while the second term is a correction\nwhich is of second order in the spin-orbit interaction:\n∆j(y,t) = [∆j(y)e−iωt+c.c.]/2, where\n∆j(y) =γ2eµnE\n2/bracketleftBigdF+\ndy+dF−\ndy/bracketrightBig\n.(10)\nFor wide samples, |λ±|W≫1, this correction to the ac\ncurrentdensity, likethe spin density, is concentratednear\nthe sample edges.\nThe spin-orbit correction ∆ Ito the main Drude part,\nI0=eµnEW, of the total current can be calculated from\nEq. (10):\n∆I=/integraldisplayW/2\n−W/2∆j(y)dy . (11)\nThus we obtain the ﬁnal result for the correction to the\nsample impedance, ∆ Z= ∆Z(ω,Ω), caused by spin-\norbit interaction:\n∆Z\nZ0=−γ2/bracketleftBigtanh(λ+W/2)\nλ+W+tanh(λ−W/2)\nλ−W/bracketrightBig\n,(12)\nwhereZ0=L/(eµnW) andλ±are deﬁned by Eq. (9).3\nωτ ρ  \n00.20.40.6\n0 20 0.8\n10 1\n*\nFIG. 2: The ratio ̺of the real part of the spin-orbit correc-\ntion to impedance ∆ Zto its value at zero frequency ∆ Z0as\na function of ac frequency ωin the absence of magnetic ﬁeld.\nThin curves correspond to W/Ls= 0.2,0.8,1.3,2, while the\nthick curve corresponds to W/Ls=∞. It is seen that all\ncurves practically coincide.\n4. Results and discussion. We now analyze our results\ngiven by Eq. (12) for some special cases.\n(i) Low frequencies: ωτs≪1.The results coincide\nwith those of Ref. [8] for the dcspin Hall magnetoresis-\ntance.\n(ii) Zero magnetic ﬁeld, B= 0.In Fig. 2 we plot\nthe ratio ̺(ω) = Re∆Z(ω,0)/∆Z0of the real part of the\nspin-orbitcorrection∆ Z[Eq.(12)]intheabsenceofmag-\nnetic ﬁeld toits value atzerofrequency, ∆ Z0= ∆Z(0,0).\nIt is seen that ̺(ω) has a quasi-universal behavior as a\nfunction of the parameter ωτ∗. Here 1/τ∗= 1/τs+1/τd\nis the eﬀective total relaxation rate which is the sum\nof the bulk spin relaxation rate 1 /τsand the diﬀusion\nrate for space inhomogeneity in the spin distribution\n1/τs= 4D/W2.\nThus there are the two relaxation processes: the bulk\nspin relaxation with the rate 1 /τsand the decay of spin\ninhomogeniety due to diﬀusion of spin-polarized elec-\ntrons. The quasi-universal results in Fig. 2 are quite\nsimilar to those obtained in Ref. [8] for the dcspin Hall\nmagnetoresistance.\nIndeed, from Eq. (12) one can obtain the relation be-\ntween the corrections to the acimpedance in zero mag-\nnetic ﬁeld and to the dcmagnetoresistance:\nRe∆Z(ω,0) = ∆Z(0,Ω=−ω). (13)\nHere ∆Z(0,Ω) is, in fact, the spin Hall magnetoresis-\ntance calculated in Ref. [8] and denoted therein as\n∆R(Ω).\n(iii) High frequencies: ωτs≫1.For narrow samples\n(W≪Ls/√ωτs) the correction ∆ Zdepends neither on\nfrequency, nor on magnetic ﬁeld in the main order by theΩ/ωωτ  =50 S\nζ1\n00.20.40.6\n0 1 22\n3\n4\n5\nFIG. 3: The real part of the correction to the acimpedance ζ\nas a function of the Larmor frequency Ωat a ﬁxed acfre-\nquencyωfor medium and large sample widths. For curves 1,\n2, 3, 4, 5 the parameter W/Lsis equal to 0.4, 0.6, 1, 2, 6,\nrespectively.\nparameter W/Ls≪1 (at not too high magnetic ﬁelds\nwhenΩ/lessorsimilarω). With the small correction on the order of\n(W/Ls)2included, we obtain:\n∆Z\nZ0=−γ2/parenleftBig\n1−1−iωτs\n24W2\nL2s/parenrightBig\n.(14)\nFor wide samples ( W≫Ls) Eq. (12) leads to the\nformula:\n∆Z\nZ0=−γ2Ls\nW/summationdisplay\n±1/radicalbig\n1+i(−ω±Ω)τs,(15)\ndisplaying spin resonance at ω=±Ω.\nThe general formula (12) is needed for medium sample\nwidths (Ls/√ωτs≪W≪Ls). In this case Eq. (12)\ndescribes the crossover between the resonant dependence\nof ∆ZonΩfor wide samples [Eq. (15)] and the non-\nresonant dependence of ∆ ZonΩfor narrow samples\n[Eq. (14)].\nIn Fig. 3 we plot the ratio ζ= Re∆Z/(−γ2Z0) of the\nreal part of the spin-orbit correction ∆ Z[Eq. (12)] to its\nvalue for narrow samples [Eq. (14)] as a function of Ωat\na ﬁxedωfor diﬀerent sample widths W. The transition\nfrom the non-resonant to the resonant behavior of ζ(Ω)\nwith the increaseof Wisclearlyseen. Eq. (15) and Fig. 3\nshow that the wider is the sample, the smaller are both\nthe amplitude and the width of the resonance peak.\nItisinterestingtostudythebehaviorofthenormalized\nacmagnetoresistance\n̺(ω,Ω) =Re∆Z(ω,Ω)\nRe∆Z(ω,Ω=ω). (16)\nAn analysis similar to that performed above for the de-\npendence ̺(ω) in zero magnetic ﬁeld, shows that ̺(ω,Ω)4\nat a ﬁxed ωhas a quasi-universal behavior as a function\nof the parameter ( Ω−ω)τ∗at|Ω|> ω, similar to the\nbehaviorof ̺(ω) displayedatFig. 2. However,the depen-\ndencies of ̺on Ω at|Ω|< ω, as it is seen from Fig. 3, are\nqualitatively diﬀerent for wide and for narrow samples.\n4. Conclusion. We have shown that the combination\nof the direct and inverse spin Hall eﬀects in nonmagnetic\nmetals and semiconductors oﬀers an interesting possibil-\nity to study high-frequency spin phenomena, including\nspin resonance, by purely electrical measurements. The\ncorresponding corrections to the sample impedance are\nof second order in the spin-orbit coupling parameter, γ.\nInthe absenceofexternalmagneticﬁeld, the frequency\ndependence of the electrical impedance is deﬁned by the\nsum of the bulk spin relaxation rate and the spin diﬀu-\nsion rate. The interplay between the two corresponding\nrelaxationtimes deﬁnes alsothe width and the amplitude\nof the electrically measured spin resonance.\n[1] M.I. Dyakonovand V.I. Perel, JETP Lett. 13, 467 (1971)\n[2] M.I. Dyakonov and V.I. Perel, Phys. Lett. A35, 459\n(1971)\n[3] N.S. Averkiev and M.I. Dyakonov, Sov. Phys. Semicond.\n17, 393 (1983)\n[4] A.A. Bakun, B.P. Zakharchenya, A.A. Rogachev,\nM.N. Tkachuk, and V.G. Fleisher, JETP Lett. 40, 1293\n(1984)\n[5] Y.K. Kato, R.C. Myers, A.C. Gossard, and\nD.D. Awschalom, Science 306, 1910 (2004)\n[6] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,\nPhys. Rev. Lett. 94, 047204 (2005)\n[7] M.I. Dyakonov and A.V. Khaetskii, In: Spin Physics\nin Semiconductors , M.I. Dyakonov, editor, 2nd edition,\nSpringer (2017), Ch. 8[8] M.I. Dyakonov, Phys. Rev. Lett. 99, 126601 (2007)\n[9] In Ref. [8] a 2D conductor in the xyplane was consid-\nered, which can be also viewed as a cross-section of a 3D\nsample. Since there is no dependence on the zcoordinate,\nthe results are equally applicable for a 3D conductor.\n[10] S. Velez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa,\nE. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, F. S.\nBergeret, and F. Casanova, Phys. Rev. Lett. 116, 016603\n(2016)\n[11] H. Nakayama, M. Althammer, Y.-T. Chen, K.Uchida, Y.\nKajiwara, D. Kikuchi, T. Ohtani, S. Geprags, M. Opel,\nS. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goen-\nnenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601\n(2013)\n[12] D. Wei, M. Obstbaum, M. Ribow, C. H. Back, and G.\nWoltersdorf, Nat. Comm. 5, 3768 (2014)\n[13] P. Hyde, Lihui Bai, D. M. J. Kumar, B. W. Southern,\nC.-M. Hu, S. Y. Huang, B. F. Miao, and C. L. Chien,\nPhys. Rev. B 89, 180404 (2014)\n[14] M. Weiler, J. M. Shaw, H. T. Nembach, T. J. Silva, Phys.\nRev. Lett. 113, 157204 (2014)\n[15] C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Nale-\ntov, J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013)\n[16] T. Chiba, G. E. W. Bauer, and S. Takahashi, Phys. Rev.\nApplied 2, 034003 (2014)\n[17] W. Chen, M. Sigrist, J. Sinova, D. Manske, Phys. Rev.\nLett.115, 217203 (2015)\n[18] C. Ulloa and R. A. Duine, Phys. Rev. Lett. 120, 177202\n(2018)\n[19] A. Hoﬀmann, IEEE Transactions on Magnetics 49, 5172\n(2013)\n[20] M.B. Jungﬂeisch, W. Zhang, R. Winkler, and A.\nHoﬀmann, In: Spin Physics in Semiconductors , M.I.\nDyakonov, editor, 2nd edition, Springer (2017), Ch. 11\n[21] In the absence of inversion symmetry there may be addi-\ntional terms describing this coupling. In particular, ther e\nis a spin current induced by a non-equilibrium spin po-\nlarization and a uniform spin polarization generated by\nelectric current. Here, such eﬀects are not considered."    },    {        "title": "1805.10328v2.Generation_of_Spin_Current_from_Lattice_Distortion_Dynamics__Spin_Orbit_Routes.pdf",        "content": "arXiv:1805.10328v2  [cond-mat.mes-hall]  20 Jul 2018Journal of the Physical Society of Japan LETTERS\nGeneration of Spin Current from Lattice Distortion Dynamic s: Spin-Orbit Routes\nTakumi Funato and Hiroshi Kohno\nDepartment of Physics, Nagoya University, Nagoya 464-8602 , Japan\nGeneration of spin current from lattice distortion dynamic s in metals is studied with special attention on the e ﬀect\nof spin-orbit coupling. Treating the lattice distortion by local coordinate transformation, we calculate spin curren t and\nspin accumulation with the linear response theory. It is fou nd that there are two routes to the spin-current generation:\none via the spin Hall e ﬀect and the other via the spin accumulation. The present e ﬀect due to spin-orbit coupling can\nbe comparable to, or even larger than, the one based on the spi n-vorticity coupling in systems with strong spin-orbit\ncoupling.\nIn the ﬁeld of spintronics, spin current occupies a central\nposition for the development of new devices or the discov-\nery of novel physical phenomena. To date we know several\nmethods available to generate spin currents, which include\nspin pumping,1–3)spin Hall eﬀect,4)spin accumulation at the\nferromagnet/nonmagnet interface,5)and spin Seebeck eﬀect.6)\nThese are classiﬁed as magnetic, electrical, magnetoelect ric,\nand thermal means, respectively.\nRecently, there has also been interest in generating spin\ncurrents by mechanical means, namely, by converting angu-\nlar momentum associated with mechanical motion, such as\nthe rigid rotation of a solid or vorticity of a ﬂuid, into spin an-\ngular momentum of electrons. In the experiments reported so\nfar, two mechanisms have been considered. One is the acous-\ntic spin pumping by Uchida et al. ,7, 8)which is based on the\nmagnon-phonon coupling. They succeeded in generating spin\ncurrent by injecting acoustic waves into yttrium iron garne t\n(YIG) from the attached piezoelectric element. Theoretica l\nanalyses were given by Adachi and Maekawa,9, 10)Keshtgar\net al. ,11)and Deymier et al.12)Another mechanism, proposed\nby Matsuo et al. ,13, 14)is based on the spin-rotation coupling\nor the spin-vorticity coupling (SVC). This is the coupling o f\nthe spin to the eﬀective magnetic ﬁeld that emerges in a ro-\ntating (non-inertial) frame of reference locally ﬁxed on th e\nmaterial that is in motion. The ﬁrst experiment for the SVC\nmechanism was conducted on liquid metals.15, 16)To realize\nthe SVC mechanism in solids, it was proposed to use sur-\nface acoustic waves.17, 18)Nozaki et al. used Py/Cu bilayer\nand injected surface acoustic waves into Cu from the attache d\nLiNbO 3(surface acoustic wave ﬁlter).19)The generated AC\nspin current was detected via the spin-torque ferromagneti c\nresonance.\nOne of the reasons that the mechanical generation of spin\ncurrent has attracted attention is that it does not rely on sp in-\norbit interaction (SOI). Therefore, previous works did not pay\nattention to the eﬀects of SOI. However, it is well expected\nthat SOI plays certain roles in the mechanical processes of\nspin-current generation. For example, the previous experi -\nments7, 8)were conducted on systems with an interface, which\npotentially possesses Rashba SOI. Furthermore, the mechan i-\ncal generation method may be used in combination with other\n“conventional” mechanisms that utilize SOI, and thereby en -\nhance spin current.\nIn this paper, we study a mechanical generation of spin cur-rent by focusing on the e ﬀects of SOI. As a mechanical pro-\ncess, we consider dynamical lattice deformations of a solid\nwith metallic electrons and with SOI. To treat lattice defor -\nmations analytically, we use the method of Tsuneto develope d\nin the context of ultrasonic attenuation in superconductor s,20)\nwhich employs a local coordinate transformation. By calcu-\nlating spin current and spin accumulation induced by dynam-\nical lattice deformations, we found two routes to spin-curr ent\ngeneration: one via the spin accumulation and the other via\nthe spin Hall eﬀect. As a related work, Wang et al.21)derived\nthe Hamiltonian that includes SOI in a general coordinate sy s-\ntem starting from the general relativistic Dirac equation, but\nthey did not give an explicit analysis of spin-current gener a-\ntion.\nModel: We consider a free-electron system in the presence\nof random impurities and the associated SOI. The Hamilto-\nnian is given by\nH=−∇′2\n2m+Vimp(r′)+iλso{[∇′Vimp(r′)]×σ}·∇′. (1)\nThe second term represents the impurity potential, Vimp(r′)=\nui/summationtext\njδ(r′−Rj), with strength uiand at position Rj(for jth\nimpurity), and the third term is the SOI associated with Vimp,\nwith strength λsoand the Pauli matrices σ=(σx,σy,σz).\nWhen the lattice is deformed, e.g., by sound waves, the\nHamiltonian becomes\nHlab=−∇′2\n2m+Vimp(r′−δR(r′,t))\n+iλso/braceleftBig/bracketleftBig\n∇′Vimp/parenleftbigr′−δR(r′,t)/parenrightbig/bracketrightBig\n×σ/bracerightBig\n·∇′,(2)\nwhereδR(r′,t) is the displacement vector of the lattice from\ntheir equilibrium position r′.\nFollowing Tsuneto,20)we make a local coordinate transfor-\nmation, r=r′−δR(r′,t), from the laboratory (Lab) frame\n(with coordinate r′) to a “material frame” (with coordinate r)\nwhich is ﬁxed to the ‘atoms’ in a deformable lattice. At the\nsame time, the wave function needs to be redeﬁned to keep\nthe normalization condition,\nψ(r,t)=[1+∇·δR]1/2ψ′(r′,t)+O(δR2), (3)\nwhereψ′(r′,t) is the wave function in the Lab frame, and\nψ(r,t) is the one in the material frame. Up to the ﬁrst order\n1J. Phys. Soc. Jpn. LETTERS\ninδR, the Hamiltonian for ψ(r,t) is given by\nHmat=H+H′\nK+H′\nso, (4)\nwhere H=HK+Himp+Hsois the unperturbed Hamiltonian de-\nﬁned by HlabwithδR=0. Here, HK=/summationtext\nk(k2/2m)ψ†\nkψkis the\nkinetic energy, with ψk(ψ†\nk) being the electron annihilation\n(creation) operator. HimpandHsodescribe the impurity poten-\ntial and impurity SOI, respectively, Himp=/summationtext\nk,k′Vk′−kψ†\nk′ψk,\nHso=iλso/summationtext\nk,k′Vk′−k(k′×k)·ψ†\nk′σψk, where Vk′−kis the\nFourier component of Vimp(r). Assuming a uniformly ran-\ndom distribution, we average over the impurity positions as\n/an}b∇acketle{tVkVk′/an}b∇acket∇i}htav=niu2\niδk+k′,0, and/an}b∇acketle{tVkVk′Vk′′/an}b∇acket∇i}htav=niu3\niδk+k′+k′′,0,\nwhere niis the impurity concentration. The impurity-averaged\nretarded/advanced Green function is given by GR/A\nk(ε)=(ε+\nµ−k2/2m±iγ)−1, whereγ=πniu2\niN(µ)(1+2\n3λ2\nsok4\nF) is the\ndamping rate. Here, N(µ) is the Fermi-level density of states\n(per spin), and kFis the Fermi wave number. In this work, we\nconsider the eﬀects of SOI up to the second order.\nThe eﬀects of lattice distortion are contained in H′\nKandH′\nso,\nwhich come from HKandHso, respectively. In the ﬁrst order\ninδR, they are given by\nH′\nK=/summationdisplay\nkWK\nn(k)un\nq,ωψ†\nk+q\n2ψk−q\n2, (5)\nH′\nso=/summationdisplay\nk,k′Vk′−kWso\nln(k,k′)un\nq,ωψ†\nk′+q\n2σlψk−q\n2. (6)\nHere, uq,ωis the Fourier component of the lattice velocity\nﬁeld, u(r,t)=∂tδR(r,t), and we deﬁned (see Fig. 1),\nWK\nn(k)=/bracketleftBigq·k\nmω−1/bracketrightBig\nkn, (7)\nWso\nln(k,k′)=λso\niω/bracketleftBig\n(k×q)lk′\nn−(k′×q)lkn/bracketrightBig\n. (8)\nThe ﬁrst term in WK\nndescribes the coupling of the strain ∂iδRn\nto the stress tensor ∼/summationtext\nkkiknc†\nkckof electrons, and modiﬁes\nthe eﬀective mass tensor. Throughout this report, qrepresents\nthe wave vector of the lattice deformation and ωis its fre-\nquency. We assume that the spatial and temporal variations o f\nδRare slow and satisfy the conditions q≪ℓ−1andω≪γ,\nwhereℓis the mean free path.\nSpin and spin-current density operators are given by\nˆjα\ns,0(q)=ˆσα(q)=/summationdisplay\nkψ†\nk−q\n2σαv0ψk+q\n2, (9)\nˆjα\ns,i(q)=/summationdisplay\nkψ†\nk−q\n2σαviψk+q\n2+ˆja,α\ns,i(q), (10)\nwhereα=x,y,zspeciﬁes the spin direction, i=x,y,zthe\ncurrent direction, and v0=1. Here,\nˆja,α\ns,i(q)=−iλsoǫαi j/summationdisplay\nk,k′Vk′−k(k′\nj−kj)ψ†\nk′−q\n2ψk+q\n2, (11)\nis the ‘anomalous’ part of the spin-current density, with ǫαi j\nbeing the Levi-Civita symbol. We calculate ˆjα\ns,µin (linear) re-\nsponse22)tou,\n/an}b∇acketle{tˆjα\ns,µ(q)/an}b∇acket∇i}htω=−/bracketleftBig\nKss,α\nµn+Ksj,α\nµn+Kso,α\nµn/bracketrightBig\nq,ωun, (12)\nwhere Kss,α\nµn(Ksj,α\nµn) is the skew-scattering (side-jump) type\ncontribution in response to H′\nK, and Kso,α\nµndescribes the re-\nsponse to H′\nso.\nFig. 1. Two types of vertices associated with the coupling to the lat tice\ndisplacement δR, or the velocity ﬁeld u=dδR/dt.\nFig. 2. Skew-scattering type contributions to the spin current ( µ=x,y,z)\nand/or spin accumulation ( µ=0). The black (white) circles represent spin-\nﬂip (spin non-ﬂip) vertices. The cross and the dashed line re present an impu-\nrity and the impurity potential, respectively. The shaded p art represents the\nimpurity ladder vertex corrections. The upside-down diagr ams are also con-\nsidered in the calculation.\nSkew-scattering process: The skew-scattering contribution\nwithout ladder vertex corrections, shown in Fig. 2, is given by\nKss,α\nµν(ω)=iλsoniu3\ni/summationdisplay\nk1,k2(k1×k2)αv1µWK\nν(k2)\n×ω\niπ/summationdisplay\np/bracketleftbigg\nGR\np/parenleftbiggω\n2/parenrightbigg\n−GA\np/parenleftbigg\n−ω\n2/parenrightbigg/bracketrightbigg\n×GR\nk1+GA\nk1−GR\nk2+GA\nk2−, (13)\nwith GR/A\nk±=GR/A\nk±q\n2(±ω\n2). By including ladder vertex correc-\ntions, spin accumulation and spin-current density are calc u-\nlated as23)\n/an}b∇acketle{tσα/an}b∇acket∇i}htss=αss\nSHneτ/parenleftBigg3\n5Dq2−iω/parenrightBigg(iq×u)α\nDq2−iω+τ−1\nsf, (14)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htss=−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htss+αss\nSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht, (15)\nwhereαss\nSH=2π\n3k2\nFλsoN(µ)uiis the spin Hall angle due to skew\nscattering,24)ne=2\n3mk2\nFN(µ) is the electron number density,\nτ=(2γ)−1is the scattering time, D=1\n3v2\nFτis the diﬀusion\nconstant, and τ−1\nsf=(4λ2\nsok4\nF/3)τ−1is the spin relaxation rate\ndue to SOI. In Eq. (15), /an}b∇acketle{tjm/an}b∇acket∇i}htis the charge current,\n/an}b∇acketle{tjm/an}b∇acket∇i}ht=neτ/braceleftbigg\n−/parenleftBigg3\n5Dq2−iω/parenrightBigg\num\n+/parenleftBigg6\n5+1\nτ(Dq2−iω)/parenrightBigg\nDiq m(iq·u)/bracerightbigg\n, (16)\ngenerated by u.20)Here, in the ﬁrst line, the term ∼Dq2um\n(the term∼iωum) is induced by the ﬁrst (second) term in WK\nn,\nEq. (7), via the spatio-temporal variation of the strain ten sor\n∂iδRm(temporal variation of the velocity ﬁeld um). The last\nterm is the diﬀusion current. We see that a spin accumulation\n(14) is induced by the vorticity of the lattice velocity ﬁeld u.\nThe ﬁrst term and the second terms in Eq. (15) are written\nwith the spin accumulation (Eq. (14)) and the charge current\n(Eq. (16)), respectively.\nSide-jump process: The side-jump contributions are ob-\n2J. Phys. Soc. Jpn. LETTERS\nFig. 3. Side-jump type contribution to the spin-current density ( i,µ=\nx,y,z) and/or spin accumulation ( µ=0). The diagrams in (a) come from the\nanomalous velocity, Eq. (11); hence they contribute only to the spin current.\nThe diagrams in (b) can be nonvanishing only when the lattice deformation is\nnonuniform. The upside-down diagrams are also included in t he calculation.\nFig. 4. Response to H′\nso, which turned out to vanish.\ntained from the two types of diagrams in Fig. 3. They give\nKsj (a),α\nin(ω)=iλsoniu2\niǫαi j/summationdisplay\nk1,k′\n1,k2(k′\n1,j−k1,j)WK\nn(k2)\n×ω\niπ/bracketleftBig\nδk′\n1k2GR\nk1++δk1k2GA\nk1′−/bracketrightBig\nGR\nk2+GA\nk2−, (17)\nKsj (b),α\nµν (ω)=λsoniu2\ni/summationdisplay\nk1,k2[(k1−k2)×iq]αv1µWK\nν(k2)\n×ω\niπGR\nk1+GA\nk1−GR\nk2+GA\nk2−, (18)\ncorresponding to the diagrams in Fig. 3 (a) and (b), respec-\ntively. With the ladder vertex corrections included, spin a ccu-\nmulation and spin-current density are calculated as23)\n/an}b∇acketle{tσα/an}b∇acket∇i}htsj=αsj\nSHneτ/parenleftBigg3\n5Dq2−iω/parenrightBigg(iq×u)α\nDq2−iω+τ−1\nsf, (19)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htsj (a)=αsj\nSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht, (20)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htsj (b)=αsj\nSHneτǫαimDiq m\nDq2−iωiq·u−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htsj. (21)\nwhereαsj\nSH=−λsom/τis the spin Hall angle due to side-jump\nprocesses.24)The diagrams in Fig. 3 (a) give only the spin cur-\nrent (Eq. (20)) since the left vertices come from the anoma-\nlous velocity. This contribution is also written with the ch arge\ncurrent/an}b∇acketle{tjm/an}b∇acket∇i}htgiven by Eq. (16). On the other hand, spin ac-\ncumulation coming from Fig. 3 (b) is again proportional to\nthe vorticity of the velocity ﬁeld u. In Eq. (21), the ﬁrst term\nis proportional to the di ﬀusion part of the charge current [the\nlast term in Eq (16)], and the second term is the di ﬀusion spin\ncurrent. We note that these contributions, coming from the d i-\nagrams of Fig. 3 (b), vanish when the external perturbation i s\nuniform, i.e., q=0.\nFinally, the response to H′\nso, shown in Fig. 4, turned out to\nvanish, Kso,α\nµn=0. This is also the case when the ladder vertex\ncorrections are included.\nResult: Taken together, the total spin accumulation and\nFig. 5. (Color online) Two routes to the generation of spin current f rom\nlattice distortion dynamics. The thick arrows indicate the processes governed\nby SOI. ‘AE’ means acousto-electric e ﬀect.\nspin-current density arising from the dynamical lattice di stor-\ntion via SOI have been obtained as\n/an}b∇acketle{tσα/an}b∇acket∇i}htSOI=αSHneτ/parenleftBigg3\n5Dq2−iω/parenrightBigg(iq×u)α\nDq2−iω+τ−1\nsf, (22)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSOI=−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htSOI+αSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht\n+αsj\nSHneτǫαimDiq m\nDq2−iωiq·u, (23)\nwhereαSH=αss\nSH+αsj\nSHis the ‘total’ spin Hall angle. As seen\nfrom Eq. (22), spin accumulation is induced by the vorticity of\nthe lattice velocity ﬁeld via SOI. The resulting di ﬀusion spin\ncurrent contributes to Eq. (23) as the ﬁrst term. In addition ,\ndynamical lattice distortion generates a charge current as well\n(known as the acousto-electric e ﬀect25)), which is then con-\nverted to a spin Hall current (in the transverse direction) v ia\nSOI, as expressed by the second and third terms in Eq. (23).\nTherefore, there are two routes to the spin-current generat ion\nin the present mechanism; one is the “di ﬀusion route” caused\nby the spin accumulation and the other is the “spin Hall route ”\nthat follows the acousto-electric e ﬀect.26)This is illustrated in\nFig. 5. In the latter (spin Hall) route, the longitudinal com -\nponent of ualso induces spin current via the generation of\ncharge current. Finally, we note that the induced spin accu-\nmulation (22) and the spin-current density (23) satisfy the spin\ncontinuity equation,\n∂t/an}b∇acketle{tσα/an}b∇acket∇i}htSOI+∇·/an}b∇acketle{tjα\ns/an}b∇acket∇i}htSOI=−/an}b∇acketle{tσα/an}b∇acket∇i}htSOI\nτsf. (24)\nThe term on the right-hand side represents spin relaxation d ue\nto SOI.\nThe above result does not include the e ﬀects of lattice dis-\ntortion on the spinorial character of the electron wave func -\ntion. Such eﬀects are derived from the spin connection in the\ngeneral relativistic Dirac equation.13)The total spin current\nand spin accumulation are given by the sum of the contri-\nbutions from the SVC (previous work13)) and SOI (present\nwork). Next, we study the contribution from SVC, an e ﬀect\noriginating from the spin connection.\nSpin-rotation coupling: For comparing the present result\nwith the previous one that is based on the spin-vorticity cou -\npling (SVC),16)we also calculate the spin accumulation and\nspin-current density in response to the vorticity of the lat tice\nvelocity ﬁeld,ω=∇× u.27)By treating the SVC Hamil-\ntonian HSV=−1\n4σ·ω(q,ω) as a perturbation, one has\n/an}b∇acketle{tjα\ns,µ/an}b∇acket∇i}htω=χα\nµβ(q,ω)ωβ(q,ω), whereχα\nµβ(q,ω) is the response\nfunction. The response function (without vertex correctio ns)\nis given as χα\nµβ(q,ω)=1\n2N(µ)δαβδµ0+iω\n4πδαβ/summationtext\nkvµGR\nk+GA\nk−.\nWith ladder vertex corrections, spin accumulation and spin -\n3J. Phys. Soc. Jpn. LETTERS\ncurrent density are obtained as\n/an}b∇acketle{tσα/an}b∇acket∇i}htSV=N(µ)\n2Dq2+τ−1\nsf\nDq2−iω+τ−1\nsfωα, (25)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSV=−iωN(µ)\n2Diq i\nDq2−iω+τ−1\nsfωα. (26)\nThey satisfy the spin continuity equation,\n(∂t+τ−1\nsf)/an}b∇acketle{tσα/an}b∇acket∇i}htSV+∇·/an}b∇acketle{tjα\ns/an}b∇acket∇i}htSV=N(µ)\n2τsfωα, (27)\nwith a source term ( ∼ω) on the right-hand side. Alterna-\ntively, one may deﬁne the “spin accumulation” δµα=µ↑−µ↓\nby/an}b∇acketle{tσα/an}b∇acket∇i}htSV=n↑−n↓=N(µ)(δµα+/planckover2pi1ωα/2),17)where the\nspin quantization axis has been taken along the ˆ αaxis. Then,\nEq. (25) leads to\n(∂t−D∇2+τ−1\nsf)δµα=−/planckover2pi1\n2˙ωα. (28)\nThis is the basic equation used in Ref. 16 to study spin-curre nt\ngeneration. Therefore, in the SVC mechanism, only the trans -\nverse acoustic waves generate spin current, and the generat ed\nspin current is purely of di ﬀusion origin. These are in stark\ncontrast with the SOI-induced mechanism.\nComparison: To see the magnitude of the present e ﬀect,\nwe estimate the di ﬀusion spin current generated via SOI,\nEq. (22), relative to the one due to SVC, Eq. (25),\nRdiﬀ(f)≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htdiﬀ\nSOI\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSV/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=8\n3αSHεFτ\n/planckover2pi1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+6πi\n5D f\nv2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (29)\nwhere f=ω/2πis the frequency and va=ω/qis the (phase)\nvelocity of acoustic waves, εF=/planckover2pi12k2\nF/2mis the Fermi en-\nergy, and/planckover2pi1has been recovered. This ratio is larger for higher\nfrequency f, and for materials with stronger SOI.\nFor CuIr, the spin Hall angle is 2 αSH=2.1±0.6%, indepen-\ndent of impurity concentration, which is dominated by the ex -\ntrinsic, skew-scattering process.28)In the nearly free electron\napproximation with the Fermi wave number kF=1.36×1010\nm−1, Fermi velocity vF=1.57×106m/s, eﬀective mass\nm∗=8.66×10−31kg,29)and resistivity ρimp=7.5µΩcm (for\n3% Ir), we estimate the scattering time as τimp=5.30×10−15\ns, and the diﬀusion constant as Dimp=4.35×10−3m2/s, due to\nimpurities. With the speed of the Rayleigh type surface acou s-\ntic wave, va=3.80×103m/s, on a single crystal of LiNbO 3,30)\nwe obtain\nRCuIr\ndiﬀ(f)=1.51/radicalBig\n1+(1.14×f)2, (30)\nwhere fis expressed in GHz. Therefore the di ﬀusion spin\ncurrent/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htdiﬀ\nSOIvia SOI is comparable to, or even larger than,\nthat from SVC in metals with strong SOI. It is thus expected\nthat the total contribution /an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSOI, which includes both the\ndiﬀusion spin current and the spin Hall current, can be larger\nthan/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSV. The magnitude itself is, however, small; /an}b∇acketle{tjx\ns,z/an}b∇acket∇i}ht=\n1020∼1024m−2s−1=10∼105A/m2forτ−1\nsf=0∼5×\n1013s−1,δR=1Å, and f=3.8 GHz, as in the case of the\nSVC mechanism.14)\nTo summarize, we studied the generation of spin current\nand spin accumulation by dynamical lattice distortion in me t-\nals with SOI at the impurities. We identiﬁed two routes to\nthe spin-current generation, namely, the “spin Hall route” andthe “spin diﬀusion route.” In the former route, a charge cur-\nrent is ﬁrst induced by dynamical lattice distortion, which is\nthen converted into a spin Hall current. In the latter route, a\nspin accumulation is ﬁrst induced from the vorticity of the\nlattice velocity ﬁeld, which then induces a di ﬀusion spin cur-\nrent. The result suggests that the spin accumulation (hence\nthe associated diﬀusion spin current) generated via SOI is\nlarger than that due to SVC for systems with strong SOI. Sim-\nilar eﬀects are expected in systems with other types of SOI,\nsuch as Rashba, Weyl, etc., and such studies will be reported\nelsewhere. In this connection, we note that Xu et al. recently\nreported an experiment on the mechanical spin-current gen-\neration (due to magnon-phonon coupling) in a system with\nRashba SOI.31)\nAcknowledgment We would like to thank K. Kondou, J. Puebla, and M.\nXu for the valuable and informative discussion, and J. Ieda, S. Maekawa,\nM. Matsuo, M. Mori, and K. Yamamoto for their valuable advice and com-\nments. We also thank A. Yamakage, K. Nakazawa, T. Yamaguchi, Y . Imai,\nand J. Nakane for the daily discussions. This work is support ed by JSPS\nKAKENHI Grant Numbers 25400339, 15H05702 and 17H02929. TF i s sup-\nported by a Program for Leading Graduate Schools: “Integrat ive Graduate\nEducation and Research in Green Natural Sciences.”\n1) S. Mizukami, Y . Ando, and T. Miyazaki, J. Magn. Magn. Mater .226-\n230, 1640 (2001); Phys. Rev. B 66, 104413 (2002).\n2) Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. L ett.88,\n117601 (2002); Phys. Rev. B 66, 224403 (2002).\n3) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88,\n182509 (2006).\n4) J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T . Jungwirth,\nRev. Mod. Phys. 87, 1213 (2015).\n5) M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1709 (1985).\n6) K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S.\nMaekawa, and E. Saitoh Nature 455, 778 (2008).\n7) K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands , S.\nMaekawa, and E. Saitoh, Nature Materials 10, 737 (2011).\n8) K. Uchida, T. An, Y . Kajiwara, M. Toda, and E. Saitoh, Appl. Phys. Lett.\n99, 212501 (2011).\n9) H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76,\n036501 (2013).\n10) H. Adachi, S. Maekawa, Solid State Comm. 198, 22 (2014).\n11) H. Keshtgar, M. Zareyan, and G. E. W. Bauer, Solid State Co mm. 198,\n30 (2014).\n12) P. A. Deymier, J. O. Vasseur, K. Runge, A. Manchon, and O. B ou-Matar,\nPhys. Rev. B 90, 224421 (2014).\n13) M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Phys. Rev. L ett.106,\n076601 (2011).\n14) M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Appl. Phys. Lett. 98,\n242501 (2011).\n15) R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo, S. Oka yasu, J.\nIeda, S. Takahashi, S. Maekawa and E. Saitoh, Nature Physics 12, 52\n(2016).\n16) M. Matsuo, Y . Ohnuma, and S. Maekawa, Phys. Rev. B 96, 020401(R)\n(2017).\n17) M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Ph ys. Rev. B\n87, 180402(R) (2013).\n18) J. Ieda, M. Matsuo, and S. Maekawa, Solid State Comm. 198, 52-56\n(2014).\n19) D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S. Maek awa, E.\nSaitoh, and Y . Nozaki, Phys. Rev. Lett. 119, 077202 (2017).\n20) T. Tsuneto, Phys. Rev. 121, 402 (1961).\n21) J. Wang, K. Ma, K. Li, and H. Fan, Ann. Phys. 362, 327 (2015).\n22) R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).\n23) (Supplemental Material) Some details of the calculatio n of Eqs. (14),\n(15), (19), (20) and (21), together with all diagrams, are pr ovided on-\nline.\n24) We deﬁne the spin Hall angle as half of that of Ref.4)in order to be\nconsistent with the deﬁnition of spin-current operators.\n4J. Phys. Soc. Jpn. LETTERS\n25) R. H. Parmenter, Phys. Rev. 89, 990 (1953).\n26) These two routes were also identiﬁed in the electrical ge neration of spin\ncurrent in, K. Hosono, A. Yamaguchi, Y . Nozaki, and G. Tatara , Phys.\nRev. B 83, 144428 (2011).\n27) While the same Greek letter is used for the external frequ ency (ω) and\nthe vorticity (ωorωα), we hope no confusion will arise since the latter\nis denoted as a vector.\n28) Y . Niimi, M. Morota, D.H. Wei, C. Deranlot, M. Basletic, A . Hamzic,A. Fert, and Y . Otani, Phys. Rev. Lett. 106, 126601 (2011).\n29) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart\nand Winston, New York, 1981).\n30) S. Ashida, Bulletin of the Japan Institute of Metals 17, 1004 (1978) (in\nJapanese).\n31) M. Xu, J. Puebla, F. Auvray, B. Rana, K. Kondou, and Y . Otan i, Phys.\nRev. B 97, 180301(R) (2018).\n5"    },    {        "title": "0803.0915v1.Effective_one_body_approach_to_the_dynamics_of_two_spinning_black_holes_with_next_to_leading_order_spin_orbit_coupling.pdf",        "content": "arXiv:0803.0915v1  [gr-qc]  6 Mar 2008Eﬀective one body approach to the dynamics of two spinning bl ack holes\nwith next-to-leading order spin-orbit coupling\nThibault Damour∗\nInstitut des Hautes ´Etudes Scientiﬁques, 91440 Bures-sur-Yvette, France\nPiotr Jaranowski†\nFaculty of Physics, University of Bia/suppress lystok, Lipowa 41, 15 –424 Bia/suppress lystok, Poland\nGerhard Sch¨ afer‡\nTheoretisch-Physikalisches Institut, Friedrich-Schill er-Universit¨ at, Max-Wien-Pl. 1, 07743 Jena, Germany\n(Dated: October 31, 2018)\nUsing a recent, novel Hamiltonian formulation of the gravit ational interaction of spinningbinaries,\nwe extend the Eﬀective One Body (EOB) description of the dyna mics of two spinning black holes to\nnext-to-leading order (NLO) in the spin-orbit interaction . The spin-dependent EOB Hamiltonian is\nconstructed from four main ingredients: (i) a transformati on between the “eﬀective” Hamiltonian\nand the “real” one, (ii) a generalized eﬀective Hamilton-Ja cobi equation involving higher powers of\nthe momenta, (iii) a Kerr-type eﬀective metric (with Pad´ e- resummed coeﬃcients) which depends\non the choice of some basic “eﬀective spin vector” Seﬀ, and which is deformed by comparable-mass\neﬀects, and (iv) an additional eﬀective spin-orbit interac tion term involving another spin vector σ.\nAs a ﬁrst application of the new, NLO spin-dependent EOB Hami ltonian, we compute the binding\nenergy of circular orbits (for parallel spins) as a function of the orbital frequency, and of the spin\nparameters. We also studythe characteristics of the last st able circular orbit: bindingenergy, orbital\nfrequency, and the corresponding dimensionless spin param eter ˆaLSO≡cJLSO/(G(HLSO/c2)2). We\nﬁnd that the inclusion of NLO spin-orbit terms has a signiﬁca nt “moderating” eﬀect on the dynam-\nical characteristics of the circular orbits for large and pa rallel spins.\nPACS numbers: 04.25.-g, 04.25.Nx\nI. INTRODUCTION\nCoalescing black hole binaries are among the most promising sources f or the currently operating ground-based\nnetwork of interferometric detectors of gravitational waves. I t is plausible that the ﬁrst detections concern binary\nsystems made of spinning black holes, because (as emphasized in [1]) the spin-orbit interaction can increase the\nbinding energy of the last stable orbit, and thereby lead to larger gr avitational wave emission. This makes it urgent to\nhave template waveforms accurately describing the gravitational wave emission of spinning binary black holes. These\nwaveformswill be functions of at least eight intrinsic real paramete rs: the two masses m1,m2and the two spin vectors\nS1,S2. Due to the multi-dimensionality of the parameter space, it seems imp ossible for state-of-the-art numerical\nsimulations to densely sample this parameter space. This gives a clear motivation for developing analytical methods\nfor computing the needed, densely spaced, bank of accurate tem plate waveforms.\nAmong existing analytical methods for computing the motion and rad iation of binary black hole systems, the most\ncomplete, and the most promising one, is the Eﬀective One Body (EOB ) approach [1, 2, 3, 4]. This method was the\nﬁrst to provide estimates of the complete waveform (covering insp iral, plunge, merger, and ring-down) of a coalescing\nblack hole binary, both for non-spinning systems [3], and for spinning ones [5]. Several recent works [6, 7, 8, 9, 10]\nhave shown that there was an excellent agreement1between the EOB waveforms (for non-spinning systems) and the\nresults of recent numerical simulations (see [11] for references a nd a review of the recent breakthroughs in numerical\nrelativity). In addition, theEOBmethod predicted, beforethe ava ilabilityofreliablenumericalrelativity(NR) results,\na value for the ﬁnal spin parameter ˆ aﬁnof a coalescing black hole binary [3, 5] which agrees within ∼10% with the\n∗Electronic address: damour@ihes.fr\n†Electronic address: pio@alpha.uwb.edu.pl\n‡Electronic address: gos@tpi.uni-jena.de\n1For instance, Ref. [9] ﬁnds a maximal dephasing of ±0.005 gravitational wave cycles between EOB and numerical rel ativity waveforms\ndescribing 12 gravitational wave cycles corresponding to t he end of the inspiral, the plunge, the merger and the beginni ng of the ringdown\nof an equal-mass coalescing binary black hole.2\nresults of recent numerical simulations (see [11] for a review and r eferences). Recently, it has been shown that the\nintroduction of some reﬁnements in the EOB approach, led to an EOB /NR agreement for ˆ aﬁnat the 2% level [12].\nIn a previous paper [1] the EOB method (originally developed for non- spinning systems) has been generalized to\nthe case of spinning black holes. It was shown there that one could m ap the third post-Newtonian (3PN) orbital\ndynamics, together with the leading order (LO) spin-orbit and spin-spin dynamical eﬀects of a binary system o nto an\n“eﬀective test particle” moving in a Kerr-type metric . In the present paper, we extend and reﬁne the EOB description\nof spinning binaries by using a recently derived [13] Hamiltonian description of the spin-orbit interaction valid at the\nnext to leading order (NLO) in the PN expansion. (The NLO spin-orbit eﬀects in the harmon ic-gauge equations of\nmotion were ﬁrst obtained in [14, 15].) Let us recall that LO spin-orbit eﬀects are proportional to G/c2, while NLO\nones contain two sorts of contributions: ∝G/c4and∝G2/c4. Regarding the spin-spin coupling terms, we shall use\nhere only the LO results which are made of two diﬀerent contribution s: the LO S1S2terms [16] (which have been\nrecently extended to NLO in [17]), and the LO S2\n1andS2\n2terms. The latter are speciﬁc to Kerr black holes, being\nrelated to the quadrupole gravitational moment of a rotating black hole.2It was shown in [1] that the complete LO\nspin-spin terms (the sum of S1S2,S2\n1, andS2\n2terms) admitted a remarkable rewriting involving a particular linear\ncombination S0, deﬁned below, of the two spin vectors. This fact, together with t he more complicated structure of\nspin-orbit terms at the NLO, will lead us below to deﬁne a particular, im proved EOB description of spinning binaries.\nThe present paper consists of two parts: In the ﬁrst part (Sect ions 2 and 3) we shall develop the formalism needed\nto ﬁnally deﬁne (in Section 4) our improved EOB description of spinning binaries. In the second part (Section 5), we\nshall consider one of the simplest “applications” of our EOB Hamiltonia n: a discussion of the energetics of circular,\nequatorialorbits for systems with parallel spins. In this section, w e shall make contact with previous related analytical\ninvestigations, notably [15], and prepare the ground for making co ntact with numerical data.\nA few words about our notation: We use the letters a,b= 1,2 as particle labels. Then, ma,xa= (xi\na),pa= (pai),\nandSa= (Sai) denote, respectively, the mass, the position vector, the linear m omentum vector, and the spin vector\nof theath body; for a∝ne}ationslash=bwe also deﬁne rab≡xa−xb,rab≡ |rab|,nab≡rab/rab,|·|stands here for the Euclidean\nlength of a 3-vector.\nII. PN-EXPANDED HAMILTONIAN\nOur starting point is the PN-expanded (or “Taylor-expanded”) tw o-body Hamiltonian Hwhich can be decomposed\nas the sum of: (i) an orbital part Ho, (ii) a spin-orbit part Hso(linear in the spins), and (iii) a spin-spin term Hss\n(quadratic in the spins),\nH(xa,pa,Sa) =Ho(xa,pa)+Hso(xa,pa,Sa)\n+Hss(xa,pa,Sa). (2.1)\nThe orbital Hamiltonian Hoincludes the rest-mass contribution and is explicitly known (in ADM-like coordinates)\nup to the 3PN order [18, 19]. Its structure is\nHo(xa,pa) =/summationdisplay\namac2+HoN(xa,pa)\n+1\nc2Ho1PN(xa,pa)+1\nc4Ho2PN(xa,pa)\n+1\nc6Ho3PN(xa,pa)+O/parenleftbigg1\nc8/parenrightbigg\n. (2.2)\nThe spin-orbit Hamiltonian Hsocan be written as\nHso(xa,pa,Sa) =/summationdisplay\naΩa(xb,pb)·Sa, (2.3)\nHere, the quantity Ωais the sum of a LO contribution ( ∝1/c2) and a NLO one ( ∝1/c4),\nΩa(xb,pb) =ΩLO\na(xb,pb)+ΩNLO\na(xb,pb). (2.4)\n2Note in passing that, if one wishes to describe the dynamics o f, say, neutron-star binaries with the EOB formalism, one sh ould add\n“correcting” S2\n1andS2\n2terms.3\nThe 3-vectors ΩLO\naandΩNLO\nawere explicitly computed in Ref. [13]. They are given, for the particle labela= 1, by\nΩLO\n1=G\nc2r2\n12/parenleftbigg3m2\n2m1n12×p1−2n12×p2/parenrightbigg\n, (2.5a)\nΩNLO\n1=G2\nc4r3\n12/parenleftBigg/parenleftbigg\n−11\n2m2−5m2\n2\nm1/parenrightbigg\nn12×p1+/parenleftbigg\n6m1+15\n2m2/parenrightbigg\nn12×p2/parenrightBigg\n+G\nc4r2\n12/parenleftBigg/parenleftbigg\n−5m2p2\n1\n8m3\n1−3(p1·p2)\n4m2\n1+3p2\n2\n4m1m2−3(n12·p1)(n12·p2)\n4m2\n1−3(n12·p2)2\n2m1m2/parenrightbigg\nn12×p1\n+/parenleftbigg(p1·p2)\nm1m2+3(n12·p1)(n12·p2)\nm1m2/parenrightbigg\nn12×p2+/parenleftbigg3(n12·p1)\n4m2\n1−2(n12·p2)\nm1m2/parenrightbigg\np1×p2/parenrightBigg\n.(2.5b)\nThe expressionsfor ΩLO\n2andΩNLO\n2canbe obtained from the aboveformulasby exchangingthe particle labels 1and 2.\nLet us now focus our attention on the dynamics of the relative motion of the two-body system in the center-of-mass\nframe, which is deﬁned by the requirement p1+p2=0. It will be convenient in the following to work with suitably\nrescaled variables. We rescale the phase-space variables R≡x1−x2andP≡p1=−p2of the relative motion as\nfollows\nr≡R\nGM≡x1−x2\nGM,p≡P\nµ≡p1\nµ=−p2\nµ, (2.6)\nwhereM≡m1+m2andµ≡m1m2/M. Note that this change of variables corresponds to rescaling the a ction by\na factor 1 /(GMµ). It is also convenient to rescale the original time variable Tand any part of the Hamiltonian\naccording to\nt≡T\nGM,ˆHNR≡HNR\nµ, (2.7)\nwhereHNR≡H−Mc2denotes the “non relativistic” version of the Hamiltonian, i.e. the Ham iltonian without the\nrest-mass contribution. It has the structure ˆHNR=1\n2p2−1\nr+O/parenleftbig1\nc2/parenrightbig\n.\nIt will be convenient in the following to work with the following two basic c ombinations of the spin vectors:\nS≡S1+S2=m1ca1+m2ca2, (2.8a)\nS∗≡m2\nm1S1+m1\nm2S2=m2ca1+m1ca2, (2.8b)\nwhere we have introduced (as is usually done in the general relativist ic literature) the Kerr parameters3of the\nindividual black holes, a1≡S1/(m1c) anda2≡S2/(m2c). Note that, in the “spinning test mass limit” where, say,\nm2→0 andS2→0, while keeping a2=S2/(m2c) ﬁxed, we have a “background mass” M≃m1, a “background\nspin”Sbckgd≡Mcabckgd≃S1=m1ca1, a “test mass” µ≃m2, and a “test spin” Stest=S2=m2ca2≃µcatest\n[withatest≡Stest/(µc)]. Then, in this limit the combination S≃S1=m1ca1≃Mcabckgd=Sbckgdmeasures\nthe background spin, while the other combination, S∗≃m1ca2≃Mcatest=MStest/µmeasures the (speciﬁc) test\nspinatest=Stest/(µc). The quantities SandS∗are the two simplest symmetric (under the permutation 1 ↔2)\ncombinations of the two spin vectors which have these properties.\nIn view of the rescaling of the action by a factor 1 /(GMµ), corresponding to the rescaled phase-space variables\nabove, it will be natural to work with correspondingly rescaled spin v ariables4\n¯SX≡SX\nGMµ, (2.9)\n3Note that we use here the usual deﬁnition where the Kerr param etera≡S/(Mc) has the dimension of length. We denote the associated\ndimensionless rotational parameter with an overhat: ˆ a≡ac2/(GM) =cS/(GM2).\n4We recall that (orbital and spin) angular momenta have the sa me dimension as the action.4\nfor any label X (X = 1 ,2,∗,···).\nMaking use of the deﬁnitions (2.6)–(2.9) one easily gets from Eqs. (2 .3)–(2.5) the center-of-mass spin-orbit Hamil-\ntonian (divided by µ) expressed in terms of the rescaled variables:\nˆHso(r,p,¯S,¯S∗) =Hso(r,p,¯S,¯S∗)\nµ\n=1\nc2ˆHso\nLO(r,p,¯S,¯S∗)\n+1\nc4ˆHso\nNLO(r,p,¯S,¯S∗)+O/parenleftbigg1\nc6/parenrightbigg\n, (2.10)\nwhere (here n≡r/|r|)5\nˆHso\nLO(r,p,¯S,¯S∗) =ν\nr2/braceleftBigg\n2/parenleftbig¯S,n,p/parenrightbig\n+3\n2/parenleftbig¯S∗,n,p/parenrightbig/bracerightBigg\n, (2.11a)\nˆHso\nNLO(r,p,¯S,¯S∗) =ν\nr3/braceleftBigg\n−(6+2ν)/parenleftbig¯S,n,p/parenrightbig\n−(5+2ν)/parenleftbig¯S∗,n,p/parenrightbig/bracerightBigg\n+ν\nr2/braceleftBigg/parenleftbigg /parenleftbigg/parenleftbigg19\n8νp2+3\n2ν(n·p)2/parenrightbigg /parenrightbigg/parenrightbigg/parenleftbig¯S,n,p/parenrightbig\n+/parenleftbigg /parenleftbigg/parenleftbigg/parenleftbigg\n−5\n8+2ν/parenrightbigg\np2+3\n4ν(n·p)2/parenrightbigg /parenrightbigg/parenrightbigg/parenleftbig¯S∗,n,p/parenrightbig/bracerightBigg\n, (2.11b)\nwithν≡µ/Mranging from 0 (test-body limit) to 1/4 (equal-mass case).\nNote that the structure of the rescaled spin-orbit Hamiltonian is\nˆHso(r,p,¯S,¯S∗) =ν\nc2r2/parenleftBig\ngADM\nS/parenleftbig¯S,n,p/parenrightbig\n+gADM\nS∗/parenleftbig¯S∗,n,p/parenrightbig/parenrightBig\n. (2.12)\nThis corresponds to an unrescaled spin-orbit Hamiltonian of the for m\nHso=G\nc2L\nR3·/parenleftBig\ngADM\nSS+gADM\nS∗S∗/parenrightBig\n, (2.13)\nwhereR=GMris the unrescaled relative distance (in ADM coordinates), L≡R×P=GMµr×pthe relative\norbital angular momentum, and where we have introduced two dimen sionless coeﬃcients which might be called the\n“gyro-gravitomagnetic ratios”, because they parametrize the c oupling between the spin vectors and the “apparent”\ngravitomagnetic ﬁeld\nv×∇GM\nc2R∝R×P\nR3\nseen in the rest-frame of a moving particle (see, e.g., Refs. [20, 21] for a discussion of the expression of the “grav-\nitomagnetic ﬁeld” in the rest-frame of a moving body). The explicit ex pressions of these two gyro-gravitomagnetic\nratios are\ngADM\nS= 2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg19\n8νp2+3\n2ν(n·p)2−/parenleftBig\n6+2ν/parenrightBig1\nr/parenrightbigg /parenrightbigg/parenrightbigg\n, (2.14a)\ngADM\nS∗=3\n2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg/parenleftBig\n−5\n8+2ν/parenrightBig\np2+3\n4ν(n·p)2−/parenleftBig\n5+2ν/parenrightBig1\nr/parenrightbigg /parenrightbigg/parenrightbigg\n. (2.14b)\n5We introduce the following notation for the Euclidean mixed product of 3-vectors: ( V1,V2,V3)≡V1·(V2×V3) =εijkVi\n1Vj\n2Vk\n3.5\nIn the following we shall introduce two related “eﬀective” “gyro-gr avitomagnetic ratios”, that enter the eﬀective\nEOBHamiltonian (in eﬀective coordinates). The label “ADM” on the gy ro-gravitomagneticratios(2.14) is a reminder\nof the fact that the NLO value of these ratios depend on the precis e deﬁnition of the radial distance R(which is\ncoordinate dependent). Let us, however, brieﬂy discuss the orig in of the (coordinate-independent) LO values of these\nratios, namely\ngLO\nS= 2, gLO\nS∗=3\n2= 2−1\n2. (2.15)\nHere the basic ratio 2 which enters both gLO\nSandgLO\nS∗comes from the leading interaction, predicted by the Kerr\nmetric, between the orbital angular momentum of a test particle an d the background spin. See Eq. (4.17) below. As\nfor the−1\n2“correction” in the coupling of the “test mass” spin combination S∗it can be seen (e.g. from Eq. (3.6b)\nof [22]) to come from the famous1\n2factor in the Thomas precession (which is a universal, special relativ istic eﬀect,\nseparate from the eﬀects which are speciﬁc to the gravitational in teraction, see Eqs. (3.2) and (3.3) in [22]).\nTo complete this Section, let us recall the remarkable form [found in R ef. [1], see Eq. (2.54) there] of the leading-\norder spin-spin Hamiltonian Hss(including S2\n1,S2\n2as well as S1S2terms). The unrescaled form of the spin-spin\nHamiltonian reads\nHss(R,S0) =ν\n2G\nc2Si\n0Sj\n0∂ij1\nR, (2.16)\nwhile its rescaled version reads\nˆHss(r,¯S0)≡Hss(R,S0)\nµ\n=1\n2ν2\nc2¯Si\n0¯Sj\n0∂ij1\nr=1\n2ν2\nc23(n·¯S0)2−¯S2\n0\nr3. (2.17)\nThe remarkable fact about this result is that it is entirely expressible in terms of the speciﬁc combination of spins\nS0≡GMµ¯S0deﬁned as:\nS0≡S+S∗=/parenleftBig\n1+m2\nm1/parenrightBig\nS1+/parenleftBig\n1+m1\nm2/parenrightBig\nS2. (2.18)\nWe shall come back below to the remarkable properties of the combin ationS0, which will play a central role in our\nEOB construction.\nIII. EFFECTIVE HAMILTONIAN AND “EFFECTIVE GYRO-GRAVITOMA GNETIC” RATIOS\nWe have obtained in the previous Section the expression of the full c enter-of-mass-frame Hamiltonian (2.1), in PN-\nexpanded form. In order to transform this Hamiltonian into a forma t which can be resummed in a manner compatible\nwith previous work on the EOB formalism, we need to perform two ope rations on the Hamiltonian (2.1). First, we\nneed to transform the phase-space coordinates ( xa,pa,Sa) by a canonical transformation compatible with the one\nused in previous EOB work. Second, we need to compute the eﬀective Hamiltonian corresponding to the (canonically\ntransformed) realHamiltonian (2.1).\nWe start by performing the purely orbital canonical transformat ion which was found to be needed in Refs. [2, 4] to\ngo from the ADM coordinates (used in the PN-expanded dynamics) t o the coordinates used in the EOB dynamics.\nThis orbital canonical transformation is (implicitly) given by\nx′i=xi+∂Go(x,p′)\n∂p′\ni, p′\ni=pi−∂Go(x,p′)\n∂xi. (3.1)\nHere the orbital generating function Go(q,p′) has been derived to 2PN accuracy in [2], and to 3PN accuracy in [4]. In\nthe present paper, as we are only concerned with the additional sp in-orbit terms, treated to 1PN fractional accuracy,\nit is enough to work with the 1PN-accurate generating function Go(x,p′). In terms of the rescaled variables, the\nrescaled 1PN-accurate orbital generating function reads\n¯Go(r,p)≡Go(r,p)\nGMµ6\n=1\nc2(r·p)/parenleftBigg\n−1\n2νp2+/parenleftBig\n1+1\n2ν/parenrightBig1\nr/parenrightBigg\n. (3.2)\nThis transformation changes the phase-space variables from ( r,p,¯S,¯S∗) to (r′,p′,¯S,¯S∗). At the linear order in the\ntransformation (which will be enough for our purpose), the eﬀect of the transformation on any of the phase-space\nvariable, say y, isy′=y+{y,Go}, where{·,·}denotes the Poisson bracket. As Gois independent of time, it leaves the\nHamiltonian numericallyinvariant: H′(y′) =H(y). This means that it changesthe functional formofthe Hamiltonian\naccording to H′(y′) =H(y′−{y,Go}) =H(y′)−{H,Go}. Note the appearance of the opposite sign in front of the\nPoisson bracket, with respect to the eﬀect of the generating fun ction on the phase-space variables.\nAsGois of order 1 /c2, its explicit eﬀect on the two separate terms, Hso\nLOandHso\nNLO, in the PN expansion of the\nspin-orbit Hamiltonian is given by:\nH′so\nLO(r′,p′,¯S,¯S∗) =Hso\nLO(r′,p′,¯S,¯S∗), (3.3a)\nH′so\nNLO(r′,p′,¯S,¯S∗) =Hso\nNLO(r′,p′,¯S,¯S∗)\n−{Hso\nLO,¯Go}(r′,p′,¯S,¯S∗). (3.3b)\nItwillbeconvenientinthefollowingtofurthertransformthephase -spacevariablesbyperformingasecondary,purely\nspin-dependent canonical transformation, aﬀecting only the NLO spin-orbit terms. The associated new generating\nfunction, Gs(r,p,¯S,¯S∗) (assumed to be proportional to the spins and of order 1 /c4) will change the variables ( y′)≡\n(r′,p′,¯S,¯S∗) into (y′′)≡(r′′,p′′,¯S′′,¯S′′∗) according to the general rule6y′′=y′+{y′,Gs}. For the same reason as\nabove, the (ﬁrst-order) eﬀect of Gson the functional form of the Hamiltonian will involve a Poisson bracke t with the\nopposite sign: H′′(y′′) =H(y′′)−{H,Gs}.\nWe shall consider a generating function whose unrescaled form rea ds\nGs(R,P,S,S∗) =G\nµc41\nR3(R·P)(R×P)·/parenleftBig\na(ν)S+b(ν)S∗/parenrightBig\n, (3.4)\nwhile its rescaled form reads\n¯Gs(r,p,¯S,¯S∗)≡Gs(R,P,S,S∗)\nGMµ\n=1\nc4ν(n·p)\nr/parenleftBigg\na(ν)/parenleftbig¯S,n,p/parenrightbig\n+b(ν)/parenleftbig¯S∗,n,p/parenrightbig/parenrightBigg\n. (3.5)\nHerea(ν) andb(ν) are two arbitrary, ν-dependent dimensionless coeﬃcients.7Similarly to the result above, the\nexplicit eﬀect of this new canonical transformation on the two sepa rate terms, H′so\nLOandH′so\nNLO, in the PN expansion\nof the spin-orbit Hamiltonian reads:\nH′′so\nLO(r′′,p′′,¯S′′,¯S′′∗) =H′so\nLO(r′′,p′′,¯S′′,¯S′′∗), (3.6a)\nH′′so\nNLO(r′′,p′′,¯S′′,¯S′′∗) =H′so\nNLO(r′′,p′′,¯S′′,¯S′′∗)\n−{HoN,¯Gs}(r′′,p′′,¯S′′,¯S′′∗), (3.6b)\nwhereHoNis the Newtonian orbital Hamiltonian. In the following, we shall, for simp licity of notation, omit the\ndouble primes on the new phase-space variables (and on the corres ponding Hamiltonian).\nThe secondoperationweneedto doisto connectthe “real”Hamilton ianHtothe “eﬀective”one Heﬀ, which ismore\nclosely linked to the description of the EOB quasi-geodesic dynamics. The relation between the two Hamiltonians is\nquite simple [2, 4]:\nHeﬀ\nµc2≡H2−m2\n1c4−m2\n2c4\n2m1m2c4, (3.7)\n6Note that while Godid not aﬀect the spin variables, the spin-dependent genera ting function Gswill now aﬀect them.\n7The coeﬃcients a(ν) andb(ν) can be thought of as being two “gauge” parameters, related t o the arbitrariness in choosing a spin-\nsupplementary condition, and in deﬁning a local frame to mea sure the spin vectors.7\nwhere we recall that the real Hamiltonian Hcontains the rest-mass contribution Mc2= (m1+m2)c2. Let us also\nnote that Eq. (3.7) is equivalent to\nHeﬀ\nµc2= 1+HNR\nµc2+1\n2ν(HNR)2\nµ2c4, (3.8)\nwhereHNRdenotes the “non relativistic” part of the total Hamiltonian H, i.e.,HNR≡H−Mc2, or more explicitly\nHNR=/parenleftBig\nHoN+Ho1PN\nc2+Ho2PN\nc4+Ho3PN\nc6/parenrightBig\n+/parenleftBigHso\nLO\nc2+Hso\nNLO\nc4/parenrightBig\n. (3.9)\nBy expanding (in powers of 1 /c2and in powers of the spins) the exact eﬀective Hamiltonian (3.7), one easily ﬁnds\nthat the “spin-orbit part” of the eﬀective Hamiltonian Heﬀ(i.e. the part which is linear-in-spin) diﬀers from the\ncorresponding part Hsoin the “real” Hamiltonian by a factor ≃1+νˆHNR/c2≃1+νˆHoN/c2, so that we get, for the\nexplicit PN expansion of Hso\neﬀ,\nHso\neﬀ\nµ=1\nc2ˆHso\nLO+1\nc4/parenleftBig\nˆHso\nNLO+νˆHoNˆHso\nLO/parenrightBig\n. (3.10)\nCombining this result with the eﬀect of the two generating functions discussed above (and omitting, as we already\nsaid, the double primes on the new phase-space variables ( r′′,p′′,¯S′′,¯S′′∗)), we get the transformed spin-orbit part of\nthe eﬀective Hamiltonian in the form\nHso\neﬀ\nµ=ν\nc2r2(n×p)·/parenleftBig\ngeﬀ\nS¯S+geﬀ\nS∗¯S∗/parenrightBig\n, (3.11)\nwhich corresponds to the following unrescaled form (with L≡R×P):\nHso\neﬀ=G\nc2L\nR3·/parenleftBig\ngeﬀ\nSS+geﬀ\nS∗S∗/parenrightBig\n. (3.12)\nHere the two “eﬀective gyro-gravitomagnetic” ratios geﬀ\nSandgeﬀ\nS∗diﬀer from the “ADM” ones introduced above by\nthree eﬀects: (i) a factor ≃1+νˆHNR/c2≃1+νˆHoN/c2due to the transformation from HtoHeﬀ, (ii) the eﬀect of\nthe orbital generating function Gogoing from ADM to EOB coordinates, and (iii) the eﬀect of the spin-de pendent\ngenerating function Gs, which involves the gauge parameters a(ν) andb(ν). Their explicit expressions are then found\nto read\ngeﬀ\nS≡2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg/parenleftBig3\n8ν+a(ν)/parenrightBig\np2−/parenleftBig9\n2ν+3a(ν)/parenrightBig\n(n·p)2−1\nr/parenleftBig\nν+a(ν)/parenrightBig/parenrightbigg /parenrightbigg/parenrightbigg\n, (3.13a)\ngeﬀ\nS∗≡3\n2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg/parenleftBig\n−5\n8+1\n2ν+b(ν)/parenrightBig\np2−/parenleftBig15\n4ν+3b(ν)/parenrightBig\n(n·p)2−1\nr/parenleftBig1\n2+5\n4ν+b(ν)/parenrightBig/parenrightbigg /parenrightbigg/parenrightbigg\n. (3.13b)\nThe choice of the two “gauge” parameters a(ν) andb(ν) is arbitrary, and physical results should not depend on\nthem.8This would be the case if we were dealing with the exact Hamiltonian. How ever, as we work only with an\napproximation to the exact Hamiltonian, there will remain some (weak ) dependence of our results on the choice of\na(ν) andb(ν). We can use this dependence to try to simplify, and/or to render m ore accurate, the spin-orbit eﬀects\nimplied by the above expressions. In particular, we shall focus in this paper on a special simplifying choice of these\ngauge parameters: namely, the values\na(ν) =−3\n8ν, b(ν) =5\n8−1\n2ν, (3.14)\nwhich suppress the dependence of the eﬀective gyro-gravitomag netic ratios on p2. With this particular choice, the\nexplicit expressions of these ratios become\n8Note in particular that the gyro-gravitomagnetic ratios do not depend on a(ν) andb(ν) when considering circular orbits, i.e. when\np2= 1/rand (n·p) = 0.8\ngeﬀ\nS≡2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg\n−27\n8ν(n·p)2−5\n8ν1\nr/parenrightbigg /parenrightbigg/parenrightbigg\n, (3.15a)\ngeﬀ\nS∗≡3\n2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg\n−/parenleftBig15\n8+9\n4ν/parenrightBig\n(n·p)2−/parenleftBig9\n8+3\n4ν/parenrightBig1\nr/parenrightbigg /parenrightbigg/parenrightbigg\n. (3.15b)\nIV. SPIN-DEPENDENT EFFECTIVE-ONE-BODY HAMILTONIAN\nUp to now we only considered PN-expanded results. In this Section, we shall generalize the approach of [1] in\nincorporating, in a resummed way, the spin-dependent eﬀects with in the EOB approach. Let us ﬁrst recall that the\napproach of [1] consists in combining three diﬀerent ingredients:\n•a generalized Hamilton-Jacobi equation involving higher powers of th e momenta (as is necessary at the 3PN\naccuracy [4]);\n•aν-deformed Kerr-type metric gαβ\neﬀ, which depends on the choice of some basic “eﬀective spin vector” Si\neﬀ;\n•the possible consideration of an additional spin-orbit interaction te rm ∆Hso(r,p,S0,σ) in the eﬀective Hamil-\ntonian, whose aim is to complete the spin-dependent interaction inco rporated in the deﬁnition of the Hamilton-\nJacobi equation based on a certain choice of “eﬀective spin vector ”Si\neﬀ.\nAt the LO in spin-orbit and spin-spin interactions, Ref. [1] showed th at one had the choice between two possibilities:\n(i) use as eﬀective spin vector the combination S+3\n4S∗which correctly describes the LO spin-orbit eﬀects, but only\napproximately describes the LO spin-spin eﬀects;9or\n(ii) use as eﬀective spin vector the combination\nS0≡S+S∗=/parenleftBig\n1+m2\nm1/parenrightBig\nS1+/parenleftBig\n1+m1\nm2/parenrightBig\nS2, (4.1)\nwhich correctly describes the full LO spin-spin interaction (see (2.1 7) above), and complete the description of the LO\nspin-orbit eﬀects by adding a term ∆ Hso(r,p,S0,σ) involving a suitably deﬁned spin combination σ. (At LO, Ref.\n[1] deﬁned σLO=−1\n4S∗.)\nIntuitively speaking, the second possibility consists in considering th at the “eﬀective particle” is endowed not only\nwith a mass µ, but also with a “spin” proportionalto σ, so that it interacts with the “eﬀective background spacetime”\nboth via a geodesic-type interaction (described by the generalized Hamilton-Jacobi equation), and via an additional\nspin-dependent interaction proportional to its spin ∝σ.\nAt the present, NLO approximation, where it is crucial to accurate ly describe the spin-orbit interaction, as well as,\nby consistency, the LO spin-spin ones, we have chosen to follow the second possibility, which oﬀers more ﬂexibility,\nand which looks natural in view of the remarkably simple LO result (2.17 ) for the spin-spin interaction (see, however,\nthe suggestion at the end of the concluding Section 6).\nTherefore we shall successively introduce the ingredients needed to deﬁne\n•the Hamilton-Jacobi equation (describing the basic “geodesic-typ e” part of the eﬀective Hamiltonian);\n•the eﬀective, ν-deformed Kerr-type metric gαβ\neﬀ;\n•the “eﬀective spin vector” Si\neﬀentering the previous Kerr-type metric;\n•the additional spin-orbit interaction ∆ Hso(r,p,S0,σ) involving a new, speciﬁc NLO spin combination σ.\nThe modiﬁed Hamilton-Jacobi equation [4] is of the form\ngαβ\neﬀPαPβ+Q4(Pi) =−µ2c2, (4.2)\n9One can then correct for the missing terms by adding an explic it supplementary term in the Hamiltonian, quadratic in the s pins.9\nwhereQ4(Pi) is a quartic-in-momenta term (which only depends on the space mom entum components Pi). For\ncircular orbits Q4(Pi) will be zero (see [1, 4]), so that we will not need its explicit expression in the present paper.\nThe role of the Hamilton-Jacobi equation above is to allow one to comp ute the main part (modulo the additional\nspin-orbit interaction added later) of the eﬀective Hamiltonian Hmain\neﬀ=Eeﬀ≡ −P0cby solving (4.2) with respect to\nP0. The result can be written as\nHmain\neﬀ=Eeﬀ=βiPic+αc/radicalBig\nµ2c2+γijPiPj+Q4(Pi), (4.3)\nwhere we have introduced the auxiliary notation\nα≡(−g00\neﬀ)−1/2, βi≡g0i\neﬀ\ng00\neﬀ, γij≡gij\neﬀ−g0i\neﬀg0j\neﬀ\ng00\neﬀ. (4.4)\nThe next crucial ingredient consists in deﬁning the (spin-dependen t) eﬀective metric entering the Hamilton-Jacobi\nequation, and thereby the eﬀective Hamiltonian (4.3). We shall follow here Ref. [1] in employing an eﬀective co-metric\nof the form (here Pt≡cP0)\ngαβ\neﬀPαPβ=1\nR2+a2cos2θ/parenleftbigg /parenleftbigg/parenleftbigg\n∆R(R)P2\nR+P2\nθ\n+1\nsin2θ/parenleftBig\nPφ+asin2θPt\nc/parenrightBig2\n−1\n∆t(R)/parenleftBig\n(R2+a2)Pt\nc+aPφ/parenrightBig2/parenrightbigg /parenrightbigg/parenrightbigg\n, (4.5)\nwhere the functions ∆ tand ∆ Rare deﬁned as\n∆t(R)≡R2Pn\nm/bracketleftBig\nA(R)+a2\nR2/bracketrightBig\n, (4.6a)\n∆R(R)≡∆t(R)D−1(R), (4.6b)\nand where the Kerr-like parameter ais deﬁned as a≡Seﬀ/(Mc), where Seﬀdenotes the modulus of the “eﬀective\nspin vector” Si\neﬀentering the deﬁnition of the Kerr-like metric above. We shall come b ack below to the choice of this\nvectorSi\neﬀ(which is one of the ingredients in the deﬁnition of a spin-dependent E OB formalism). In Eq. (4.6a) Pn\nm\ndenotes the operation of taking the ( n,m)-Pad´ e approximant,10and the PN expansions of the metric coeﬃcients A\nandD−1equal (here ˆ u≡GM/(Rc2))\nA(ˆu) = 1−2ˆu+2νˆu3+/parenleftBig94\n3−41\n32π2/parenrightBig\nνˆu4, (4.7a)\nD−1(ˆu) = 1+6 νˆu2+2(26−3ν)νˆu3. (4.7b)\nFor pedagogical clarity, we have given above the expression of the eﬀective EOB metric in a Boyer-Lindquist-type\ncoordinate system aligned with the instantaneous direction of the ( time-dependent) eﬀective spin vector Si\neﬀ. This\nexpression will suﬃce in the present paper where we will only consider situations where the spin vectors are aligned\nwith the orbital angular momentum, so that they are ﬁxed in space. As emphasized in [1], when applying the EOB\nformalism to more general situations (non aligned spins) one must re write the eﬀective co-metric components in a\n“ﬁxed” Cartesian-like coordinate system. This is done by introducin g\nni≡xi/R, si≡Si\neﬀ\nSeﬀ,cosθ≡nisi,\nρ≡/radicalbig\nR2+a2cos2θ, (4.8)\n10Let us recall that the ( n,m)-Pad´ e approximant of a function c0+c1u+c2u2+···+cn+mun+mis equal to Nn(u)/Dm(u), where Nn(u)\nandDm(u) are polynomials in uof degrees nandm, respectively.10\nand rewriting the co-metric components as\ng00\neﬀ=−(R2+a2)2−a2∆t(R)sin2θ\nρ2∆t(R), (4.9a)\ng0i\neﬀ=−a(R2+a2−∆t(R))\nρ2∆t(R)(s×R)i, (4.9b)\ngij\neﬀ=1\nρ2/parenleftBig\n∆R(R)ninj+R2(δij−ninj)/parenrightBig\n−a2\nρ2∆t(R)(s×R)i(s×R)j. (4.9c)\nMaking use of Eqs. (4.9) one computes\nα=ρ/radicalBigg\n∆t(R)\n(R2+a2)2−a2∆t(R)sin2θ, (4.10a)\nβi=a(R2+a2−∆t(R))\n(R2+a2)2−a2∆t(R)sin2θ(s×R)i, (4.10b)\nγij=gij\neﬀ+βiβj\nα2. (4.10c)\nReplacing the latter expressions in the general form of the eﬀectiv e energy (4.3) yields the most general form of the\nmain part of the eﬀective Hamiltonian Hmain\neﬀ(x,P,Sa).\nThe deﬁnition of Hmain\neﬀ(x,P,Sa) crucially depends on the choice of eﬀective Kerr-type spin vector . In order to\nautomatically incorporate, in a correct manner, the LO spin-spin te rms, we shall use here\nMca≡Seﬀ≡S0=S+S∗=/parenleftBig\n1+m2\nm1/parenrightBig\nS1+/parenleftBig\n1+m1\nm2/parenrightBig\nS2. (4.11)\nNote that, besides its usefulness in treating spin-spin eﬀects, this deﬁnition has several nice features. For example, if\nwe introduce the Kerr parameters of the individual black holes, a1≡S1/(Mc),a2≡S2/(Mc), the Kerr parameter\na0≡S0/(Mc) (where we naturally take m0=M=m1+m2) associated to the spin combination (4.1) is simply\na0=a1+a2. (4.12)\nLet us also note that the corresponding dimensionless spin parameters (with, again, m0=M=m1+m2)\nˆai≡cSi\nGm2\ni, i= 0,1,2, (4.13)\nsatisfy\nˆa0=X1ˆa1+X2ˆa2, (4.14)\nwhereX1≡m1/MandX2≡m2/Mare the two dimensionless mass ratios (with X1+X2= 1 and X1X2=ν).\nThis last result shows that, in ˆa-space, the “point” ˆa0is on the straight-line segment joining the two “points” ˆa1and\nˆa2. The individual Kerr bounds tell us that each point ˆa1andˆa2is contained within the unit Euclidean sphere. By\nconvexity of the unit ball, we conclude that the “eﬀective” dimension les spin parameter ˆa0will also automatically\nsatisfy the Kerr bound |ˆa0| ≤1. This is a nice consistency feature of the deﬁnition of the associat ed Kerr-type metric.\nIt remains to deﬁne the additional “test-spin” vector σ, and the associatedadditional eﬀective spin-orbit interaction\nterm. Following the logic of [1] (and generalizing the LO results given in E qs. (2.56)–(2.58) there), these quantities\nare deﬁned by\nσ≡1\n2geﬀ\nSS+1\n2geﬀ\nS∗S∗−Seﬀ\n=1\n2/parenleftbig\ngeﬀ\nS−2/parenrightbig\nS+1\n2/parenleftbig\ngeﬀ\nS∗−2/parenrightbig\nS∗, (4.15)\nand11\n∆Hso(x,P,S0,σ)≡R2+a2\n0−∆t(R)\n(R2+a2\n0)2−a2\n0∆t(R)sin2θ0(P,σ,R)\nM, (4.16)\nwherea0≡S0/(Mc) andcos θ0≡niSi\n0/|S0|. Thejustiﬁcation forthesedeﬁnitions isthat the“main”Hamilton-J acobi\npart of the eﬀective Hamiltonian contains, as spin-orbit (i.e. linear-in -spin) part, the following term\nHmaineﬀ\nso=cPi/parenleftbig\nβi/parenrightbig\nlinear-in-spin\n=cPi/parenleftbiggR2+a2\n0−∆t(R)\n(R2+a2\n0)2−a2\n0∆t(R)sin2θ0(a0×R)i/parenrightbigg\nlinear-in-spin\n=2GM\ncR3Pi(a0×R)i+(NNLO corrections)\n= 2G\nc2L\nR3·S0+(NNLO corrections) , (4.17)\nwhere the factor 2 GMcomes from the second term in the PN expansion of ∆ t(R) =R2−2GMR/c2+\n2ν(GM)3/(Rc6)+(quadratic-in-spin terms). Note that the absence of c−4correction in the eﬀective metric function\nA(R) means that the leading term ∝2GMin the spin-orbit part of Hmainis validbothto LO and to NLO, i.e., up\nto “next to next to leading order” (NNLO).\nWhen comparing this result to the NLO result (3.12), we see that the “main” part of the eﬀective Hamiltonian\ncontains a spin-orbit piece which is equivalent to having eﬀective gyro -gravitomagnetic ratios equal to gmaineﬀ\nS= 2\nandgmaineﬀ\nS∗= 2, instead of the correct values derived above. One then easily ch ecks that the deﬁnition above of σ\nand of the associated supplementary spin-orbit interaction ∆ Hso(x,P,S0,σ) has the eﬀect of including the full result\nfor the NLO spin-orbit interaction. It is also to be noted that the ad ditional spin-orbit interaction ∆ Hsogoes to zero\nproportionally to νin the test mass limit m2→0 because, on the one hand, geﬀ\nS−2 is proportional to ν(ifa(ν) is),\nand, on the other hand, though geﬀ\nS∗−2does not tend to zero with ν, the second spin combination S∗doestend to\nzero proportionally to ν[see Eqs. (5.2) below].\nSummarizing: we propose to deﬁne a total eﬀective spin-dependen t Hamiltonian of the form\nHeﬀ(x,P,S1,S2)≡Hmain\neﬀ(x,P,S0)+∆Hso(x,P,S0,σ), (4.18)\nwhereHmain\neﬀ(x,P,S0) is given by the right-hand side of Eq. (4.3) computed for the eﬀect ive spin variable equal to S0\n[deﬁned in Eq. (4.1)] and where ∆ Hso(x,P,S0,σ) is the additional spin-orbit interaction term deﬁned above [with\na0≡S0/(Mc)].\nFinally, the real EOB-improved Hamiltonian (by contrast to the “eﬀective” one) is deﬁned by solving Eq. (3.7) with\nrespect to Hreal=HNR+Mc2:\nHreal=Mc2/radicalBigg\n1+2ν/parenleftBigHeﬀ\nµc2−1/parenrightBig\n, (4.19)\nwhereHeﬀis given in Eq. (4.18).\nV. DYNAMICS OF CIRCULAR ORBITS\nIn this Section we shall apply the construction of the NLO spin-depe ndent EOB Hamiltonian to the study of the\ndynamics of circular orbits of binary black hole systems.\nBesides the dimensionless spin parameters ˆa1andˆa2already introduced above, it is convenient to introduce the\ndimensionless spin variables corresponding to the basic spin combinat ionsSandS∗, namely\nˆa≡cS\nGM2,ˆa∗≡cS∗\nGM2. (5.1)\nLet us note in passing the various links between the dimensionless spin parameters that one can deﬁne [including\nˆa0≡cS0/(GM2) already introduced above],\nˆa=X2\n1ˆa1+X2\n2ˆa2,ˆa∗=νˆa1+νˆa2, (5.2a)12\nˆa0=ˆa+ˆa∗=X1ˆa1+X2ˆa2. (5.2b)\nHere as above we use the mass ratios X1≡m1/M,X2≡m2/Msuch that X1+X2= 1 and X1X2=ν. Let\nus note that for equal-mass binaries ( m1=m2,X1=X2=1\n2), with arbitrary (possibly unequal) spins, one has\nˆa=ˆa∗=1\n4(ˆa1+ˆa2) =1\n2ˆa0. Note also that, in the test-mass limit, say m1≫m2so thatX1→1 andX2→0, one\nhas\nˆa=ˆa0=ˆa1,ˆa∗= 0. (5.3)\nIn the general case where the spin vectors are not aligned with the (rescaled) orbital angular momentum vector11\nℓ,\nℓ=rn×p, (5.4)\nthere exist no circular orbits. However, there exist (at least to a g ood approximation) some “spherical orbits”, i.e.\norbits that keep a constant value of the modulus of the radius vect orr, though they do not stay within one ﬁxed\nplane. As discussed in [1] one can analytically study these spherical o rbits within the EOB approach, and discuss, in\nparticular, the characteristics of the last stable spherical orbit .\nFor simplicity, we shall restrict ourselves here to the situation wher e both individual spins are parallel (or antiparal-\nlel) to the orbital angular momentum vector ℓ. In that case, we can consistently set everywhere the radial mom entum\nto zero,pr=n·p= 0, and express the (real) EOB Hamiltonian as a function of r,ℓ=pϕ(usingp2=ℓ2/r2, where\nℓ≡ |ℓ|), and of the two scalars ˆ a,ˆa∗measuring the projections of our basic spin combinations on the dire ction of the\norbital angular momentum ℓ. They are such that\nˆa·ℓ= ˆaℓ,ˆa∗·ℓ= ˆa∗ℓ. (5.5)\nThe scalars ˆ aand ˆa∗can be either positive or negative, depending on whether, say, ˆais parallel or antiparallel to ℓ.\nThe sequence of circular (equatorial) orbits is then determined by t he constraint\n∂Hreal(r,ℓ,ˆa,ˆa∗)\n∂r= 0. (5.6)\nThen, the angular velocity along each circular orbit is given by\nΩ≡1\nGMµ∂Hreal(r,ℓ,ˆa,ˆa∗)\n∂ℓ. (5.7)\nAs mentioned above, we have chosen the special values a(ν) =−3\n8ν,b(ν) =5\n8−1\n2νof the two gauge parameters, to\nsimplify the expression of the Hamiltonian.\nIn Figs. 1–4 we explore several aspects of the dynamics of circular orbits, using as basic diagnostic the relation\nbetween the energy and the angular velocity along the sequence of circular orbits (“binding energy curve”). More\nprecisely, we plot the dimensionless “non relativistic” energy\ne≡Hreal\nMc2−1, (5.8)\nas a function of the dimensionless angular velocity:\nˆΩ≡GM\nc3Ω. (5.9)\nFor simplicity, we shall restrict most of our studies to symmetric binary systems, i.e. systems with m1=m2and\na1=a2. For such systems the dimensionless eﬀective spin parameter is ˆ a0= ˆa1= ˆa2. The information contained in\nthese ﬁgures deals with the following aspects of the description of t he dynamics:\n•As a warm up, and a reminder, Fig. 1 considers the case of non-spinning binaries (i.e. ˆa0= 0). This ﬁgure\ncontrasts the behaviour of the successive PN versions of the EOB dynamics, with that of the successive PN\nversions of the non-resummed, “Taylor-expanded” Hamiltonian. T he numbers 1,2,3 refer to 1PN, 2PN, and\n11In the following, we switch again to the use of scaled variabl es:r≡R/(GM),ℓ≡L/(GMµ), andp≡P/µ.13\n0.00 0.05 0.10 0.15 0.20/Minus0.04/Minus0.03/Minus0.02/Minus0.010.000.01\n/CapOmega/Hate\na/Hat\n1/Equala/Hat\n2/Equal0\nT/LParen13,/Star/RParen1T/LParen12,/Star/RParen1T/LParen11,/Star/RParen1E/LParen13,/Star/RParen1E/LParen12,/Star/RParen1E/LParen11,/Star/RParen1\nE/LParen14,/Star/RParen1,a5/Equal/Plus60E/LParen14,/Star/RParen1,a5/Equal/Plus25T/LParen11,/Star/RParen1\nT/LParen12,/Star/RParen1T/LParen13,/Star/RParen1\nFIG.1: Bindingenergycurvesfor circular orbitsofsymmetr icnon-spinning binaries ( m1=m2andˆa1=ˆa2=0): dimensionless\n“non relativistic” energy eversus dimensionless angular frequency ˆΩ. The notation E( n,∗) means computation of the energy\nusing the EOB-improved real Hamiltonian (4.19) with the nPN-accurate metric function ∆ t(R); the function ∆ t(R) was\ncomputed by means of Eq. (4.6a) using the (1 ,n) Pad´ e approximant at the nPN order. Here n= 1,2,3,4, where n= 4 refers\nto the “4PN” case where a term + a5νˆu5is added to the function A(ˆu). For the curves labelled by T( n,∗) the computation was\ndone with the direct PN-expanded (ADM-coordinates) orbita l Hamiltonian (2.2) with the terms up to the nPN order included.\n3PN, while the letter “E” refers to “EOB” and the letter “T” refers to “Taylor”. For instance, E(3 ,∗) refers\nto thee(ˆΩ) binding energy curve computed with the 3PN-accurate EOB Hamilt onian. [The star in E(3 ,∗)\nreplaces the label we shall use below to distinguish LO versus NLO tre atment of spin-orbit eﬀects. In the\npresent non-spinning case we are insensitive to this distinction.] To b e precise, the notation E( n,∗) refers to\na computation of the circular orbits using the ˆ a0→0 limit12of the EOB-improved real Hamiltonian (4.19)\nwith the nPN-accurate metric function ∆ t(R); where ∆ t(R) was computed by means of Eq. (4.6b) using the\nfollowing Pad´ e approximants: (1,1) at the 1PN order, (1,2) at the 2 PN order, and (1,3) at the 3PN order. As\nfor the Taylor-based approximants to the binding energy curve, T (n,∗), they were computed by using as basic\nHamiltonian (to deﬁne the dynamics) the nPN-accurate Taylor-expanded Hamiltonian, in ADM coordinates,\n(2.2), without doing any later PN re-expansion.13\nIt is interesting to note that the successive PN-approximated EOB binding energy curves are stacked in a\nmonotonically decreasing fashion, when increasing the PN accuracy, and all admit a minimum at s ome value of\nthe orbital frequency. This minimum corresponds to the last stable circular orbit (see below). The monotonic\nstacking of the EOB energy curves therefore implies that a higher P N accuracy predicts circular orbits which\nare more bound, and can reach higher orbital frequencies. Let us note in this respect that recent comparisons\nbetween EOB and numerical relativity data have found the need to a dd apositive4PN additional term + a5νˆu5\ninthe basic EOBradialpotential A(ˆu) ofEq.(4.7a) above, with a5somewherebetween +10and +80[7, 8, 9, 10].\nThough we do not know yet what is the “real” value of the 4PN coeﬃcie nta5we have included in Fig. 1 two\nillustrative14values of this “4PN” orbital parameter, namely a5= +25 and a5= +60. Note that the eﬀect of\nsuch positive values of a5is to push the last few circular orbits towards more bound, higher or bital frequency\norbits. This eﬀect will compound itself with the eﬀects of spin explore d below, and should be kept in mind when\nlooking at our other plots.\nBy contrast with the “tame” and monotonic behaviour of successiv e EOB approximants, we see on Fig. 1 that\n12Note that the ˆ a0→0 limit of the Pad´ e resummation of some ˆ a0-dependent metric coeﬃcient is not necessarily the same as t he Pad´ e\napproximant one might normally consider in the non-spinnin g case.\n13As is well-known there are always many non-equivalent ways o f deﬁning any “ nPN” result, depending of where, and how, in the\ncalculation one is replacing a function by a PN-expanded pol ynomial. For instance, one could PN re-expand the function g iving the\nenergyein terms of the orbital frequency ˆΩ, or the function giving ein terms of the orbital angular momentum L(see Ref. [4] for\nthe computation of several such functions in the non-spinni ng case). However, we are ultimately interested (for gravit ational-wave\npurposes) in deﬁning a complete dynamics for coalescing spi nning binaries. Therefore, we focus here on the results pred icted by\nHamiltonian functions H(x,p,···).\n14These two values of the 4PN parameter a5were found in Refs. [7, 9] to be representative of the values o fa5that improve the agreement\nbetween EOB waveforms and numerical relativity ones.14\n0.00 0.05 0.10 0.15 0.20 0.25 0.30/Minus0.05/Minus0.04/Minus0.03/Minus0.02/Minus0.010.000.01\n/CapOmega/HateE/LParen13,1/RParen1\na/Hat\n0/Equal/Plus1.0a/Hat\n0/Equal/Plus0.5a/Hat\n0/Equal0a/Hat\n0/Equal/Minus0.5a/Hat\n0/Equal/Minus1.0\nFIG. 2: Binding energy curves for circular orbits of symmetr icparallely spinning binaries ( m1=m2andˆa1=ˆa2∝r×p):\ndimensionless energy eversus dimensionless angular frequency ˆΩ along circular orbits for various values of the dimensionl ess\neﬀective spin parameter ˆ a0≡cS0/(GM2) = ˆa1= ˆa2within the eﬀective-one-body approach. The label E(3 ,1) means that\nwe use the EOB Hamiltonian with 3PN-accurate orbital eﬀects and NLO spin-orbit coupling, i.e. Eq. (4.15) was used with th e\nNLO gyro-gravitomagnetic ratios geﬀ\nSandgeﬀ\nS∗, Eqs. (3.13).\nthe successive Taylor-Hamiltonian approximants T( n,∗) have a more erratic behaviour. Note in particular, that\nthe 3PN-accurate energy curve does not admit any minimum as the o rbital frequency increases (in other words,\nthere is no “last” stable circular orbit). In view of this bad behaviour of the 3PN-accurate orbital Taylor-\nHamiltonian, we shall not consider anymore in the following ﬁgures the predictions coming from such Taylor\nHamiltonians.15\n•In Fig. 2 we study the eﬀect of changing the amount of spin on the bla ck holes of our binary system. We use\nhere our new, NLO spin-orbit EOB Hamiltonian, as indicated by the not ation E(3 ,1), where the ﬁrst label, 3,\nrefers to the 3PN accuracy, and the second label, 1, to the 1PN fr actional accuracy of the spin-orbit terms (i.e.,\nthe NLO accuracy). Note that the EOB binding energy curves are s tacked in a monotonically decreasing way\nas the dimensionless eﬀective spin ˆ a0increases from ˆ a0=−1 (maximal spins antiparallel to the orbital angular\nmomentum) to ˆ a0= +1 (maximal spins parallel to the orbital angular momentum). Note also that this curve\nconﬁrms the ﬁnding of [1] that parallel spins lead to the possibility of c loser and more bound circular orbits.\n•Fig.3 contraststhe eﬀect ofusing the NLO spin-orbitinteractionin stead ofthe LOone in the EOBHamiltonian.\nWeusethefull3PNaccuracy,andincludetheLOspin-spininteractio n. E(3,0)denotesaresultobtainedwiththe\n3PN-accurate EOB Hamiltonian using the LO (or 0PN-accurate) spin -orbit terms, while E(3 ,1) uses the 3PN-\naccurate EOB Hamiltonian with NLO (1PN-accurate) spin-orbit term s. Each panel in the Figure corresponds\nto a speciﬁc value of the dimensionless eﬀective spin ˆ a0. To guide the eye we use in all our ﬁgures a solid line\nto denote our “best” description, i.e. the 3PN-NLO EOB E(3 ,1). Note that the addition of the NLO eﬀects\nin the spin-orbit interaction has the clear eﬀect of moderating the inﬂuence of the spins (especially for positive\nspins). While the binding energy curves using the LO spin-orbit eﬀect s tend to abruptly dive down towards\nvery negative energies when the spins are large and positive,16the corresponding NLO curves have a much more\nmoderate behaviour.\nAmong the binding energy curves shown above, all the EOB ones (at least when the eﬀective spin is not too\nlarge and positive), and some of the Taylor ones, admit a minimum for a certain value of the orbital frequency ˆΩ.\nThis minimum corresponds to an inﬂection point in the corresponding ( EOB or Taylor) Hamiltonian considered as a\n15Indeed, in the physically most important case of parallel (r ather than anti-parallel) spins, the spin-orbit coupling w ill be repulsive (like\nthe eﬀect of a positive a5), and will tend to reinforce the “bad” behaviour of the 3PN or bital Taylor Hamiltonian (i.e. the absence of\nany last stable orbit).\n16As discussed in Section 3C of Ref. [1], this is due to the then repulsive character of the spin-orbit (and spin-spin) interaction.15\n0.00 0.02 0.04 0.06 0.08/Minus0.010/Minus0.0050.0000.005\n/CapOmega/Hate\na/Hat\n0/Equal/Minus1.0E/LParen13,1/RParen1E/LParen13,0/RParen1\n0.00 0.02 0.04 0.06 0.08 0.10 0.12/Minus0.015/Minus0.010/Minus0.0050.0000.005\n/CapOmega/Hate\na/Hat\n0/Equal/Minus0.5\n0.00 0.05 0.10 0.15 0.20/Minus0.020/Minus0.015/Minus0.010/Minus0.0050.0000.0050.010\n/CapOmega/Hate\na/Hat\n0/Equal0E/LParen13,/Star/RParen1\n0.00 0.05 0.10 0.15 0.20/Minus0.025/Minus0.020/Minus0.015/Minus0.010/Minus0.0050.0000.005\n/CapOmega/Hate\na/Hat\n0/Equal/Plus0.25\n0.0 0.1 0.2 0.3 0.4/Minus0.08/Minus0.06/Minus0.04/Minus0.020.00\n/CapOmega/Hate\na/Hat\n0/Equal/Plus0.5\n0.00 0.05 0.10 0.15 0.20 0.25 0.30/Minus0.05/Minus0.04/Minus0.03/Minus0.02/Minus0.010.000.01\n/CapOmega/Hate\na/Hat\n0/Equal/Plus1.0\nFIG. 3: Energy eversus angular frequency ˆΩ along circular orbits for various values of the parameter ˆ a0, as predicted by the\nEOB Hamiltonian. We have assumed m1=m2,a1=a2, andθ0=π/2. As before E( n,s) refers to an EOB Hamiltonian, with\nnPN accuracy in the orbital terms, and an accuracy in the spin- orbit coupling equal to the LO one if s= 0, and the NLO one\nifs= 1. In all cases, we include the full LO spin-spin coupling.\nfunction of r. In other words, the minimum is the solution of the two equations17\n∂Hreal\n∂r= 0,∂2Hreal\n∂r2= 0. (5.10)\n17Note in passing that, in the EOB case, the two Eqs. (5.10) are e quivalent to the two similar equations involving the eﬀective Hamiltonian:\n∂Heﬀ/∂r= 0,∂2Heﬀ/∂r2= 0.16\nTABLE I: LSO parameters for symmetric binary systems (with m1=m2and ˆa1= ˆa2= ˆa0) for the 3PN-NLO EOB\nHamiltonian E(3 ,1).\nˆa0 e ˆΩ\n−1.00 −0.01039 0.04473\n−0.75 −0.01143 0.05139\n−0.50 −0.01270 0.05989\n−0.25 −0.01437 0.07143\n0.00 −0.01670 0.08822\n0.25 −0.02026 0.11521\n0.50 −0.02660 0.16444\n0.75 −0.03701 0.23249\n1.00 −0.03826 0.22210\nThesolutionsofthesetwosimultaneousequationscorrespondtow hatweshallcallheretheLastStable(circular)Orbit\n(LSO).18Several methods have been considered in the literature [4, 23] for using PN-expanded results to estimate\nthe characteristics of the LSO. One of these methods consists in c onsidering the minima in the Taylor expansion of\nthe function e(ˆΩ). These minima (called “Innermost Circular Orbit” (ICO) in Refs. [15 , 23], where they were used to\nestimate the LSO of spinning binaries) diﬀer from the minima in the Taylo r energy curves considered in Fig. 1 above,\nwhich were based on using a Taylor-expanded Hamiltonian. The advan tage of consistently working (as we do here)\nwithin a Hamiltonian formalism is that we are guaranteed that the minima in the corresponding energy curves, when\nthey exist, do correspond to a Last Stable orbit (and an associate d inﬂection point) for some well-deﬁned underlying\ndynamics. By contrast the dynamical meaning (if any) of a minimum of the Taylor-expanded function eTaylor(ˆΩ) is\nunclear. Anyway, as we saw above that the 3PN-accurate Taylor- expanded orbital Hamiltonian does not admit any\nLast Stable Orbit, we have not plotted in Fig. 4 the Taylor-based pre dictions for spinning binaries because they do\nnot seem to lead to reasonable results.\nConcerning the dynamical meaning of the LSO, let us recall that it ha d been analytically predicted in [3] (and con-\nﬁrmed in recent numerical simulations [11]) that the transition betw een inspiral and plunge is smooth and progressive,\nso that the passage through the LSO is blurred. In spite of the inherent “fuzziness” in the deﬁnition of the LSO, it\nis still interesting to delineate its dynamical characteristics becaus e they strongly inﬂuence some of the gross features\nof the GW signal emitted by coalescing binaries (such as the total em itted energy, and the frequency of maximal\nemission).\nLet us comment on the results of our study of the characteristics of LSO’s:\n•In Fig. 4 we plot the LSO binding energy, predicted by the EOB approa ch, as a function of the dimensionless\neﬀective spin parameter ˆ a0. We contrast LO spin-orbit versus NLO spin-orbit (1 versus 0). We use 3PN\naccuracy (for the orbital eﬀects) in all cases, and always include t he LO spin-spin interaction. The upper panel\nshows that the use of LO spin-orbit interactions leads to dramatica lly negative LSO binding energies when\nthe spins become moderately large. [The middle panel is a close-up of t he upper one, and focuses on spins\nˆa0≤+0.2.] We ﬁnd that the 3PN-LO EOB Hamiltonian E(3 ,0) admits an LSO only up to spins as large as:\nˆa0≤+0.9. However, as ﬁrst found in [1], spin eﬀects become dramatically (a nd suspiciously) large already\nwhen ˆa0≥+0.5. By contrast, as we found above, the inclusion of NLO spin-orbit in teractions has the eﬀect\nofmoderating the dynamical inﬂuence of high (positive) spins. The bottom panel f ocusses on our “best bet”\n3PN-NLO Hamiltonian E(3 ,1).\nAs mentioned above, Ref. [15] has considered, instead of the Taylo r-Hamiltonian LSO, the minimum of the\nTaylor-expandedfunction eTaylor(ˆΩ)(or“ICO”).Forthetwocasesˆ a0=−1,0(correspondingtotheir κi=−1,0),\nthey found, in the 3PN-NLO case, energy minima equal to e≡EICO/m=−0.0116,−0.0193 for corresponding\norbital frequencies ˆΩ≡mωICO= 0.059,0.129. These numerical values should be compared with the numerical\nvalues we quote in Table I below. On the other hand, for the large and parallel spin case ˆ a0= +1 Ref. [15] found\nthat the Taylor-expanded function eTaylor(ˆΩ) has no minimum. Finally, note that the qualitative shape of the\ncurve giving the (EOB) LSO energy as a function of the spin paramet ers ˆa0is similar both to the corresponding\n18As we recalled above, spinning binaries admit, in general, o nlyspherical orbits, rather than circular ones. Reference [1] studied the\nbinding energies of the Last Stable Spherical Orbits (LSSO) . Here, however, we restrict ourselves to the parallel spin, where it makes\nsense to study circular, equatorial orbits.17\n/Minus1.0/Minus0.5 0.0 0.5 1.0/Minus0.30/Minus0.25/Minus0.20/Minus0.15/Minus0.10/Minus0.050.00\na/Hat\n0eLSO\nE/LParen13,1/RParen1E/LParen13,0/RParen1\n/Minus1.0/Minus0.8/Minus0.6/Minus0.4/Minus0.2 0.0 0.2/Minus0.040/Minus0.035/Minus0.030/Minus0.025/Minus0.020/Minus0.015/Minus0.010/Minus0.005\na/Hat\n0eLSO\nE/LParen13,1/RParen1E/LParen13,0/RParen1\n/Minus1.0/Minus0.5 0.0 0.5 1.0/Minus0.040/Minus0.035/Minus0.030/Minus0.025/Minus0.020/Minus0.015/Minus0.010\na/Hat\n0eLSO\nE/LParen13,1/RParen1E/LParen13,0/RParen1\nFIG. 4: Binding energy of the Last Stable (circular) Orbit (L SO) predicted by the EOB approach. We study the eﬀect of\nincluding NLO spin-orbit terms by contrasting the LO and NLO predictions. We plot the dimensionless energy eLSOof the\nLSO versus ˆ a0. We have assumed m1=m2,a1=a2, andθ0=π/2. For E(3 ,0) a LSO exists up to ˆ a0≤+0.9.\ncurve for a spinless test-particle in a Kerr background (see, e.g., F ig. 7 below), and to the curve giving the LSO\nenergy of a spinning test particle in a Kerr background, as a function of the test spin (see Fig. 4 in Ref . [24]).\nTo complement the information displayed in Figs. 1–4, we give in Table I t he numerical values of the main LSO\ncharacteristics (binding energy and orbital frequency) for our “ best bet” Hamiltonian, namely the 3PN-NLO EOB\none E(3,1).\nIn Figs. 5 and 6 we study the eﬀective-spin-dependence of anothe r LSO-related physical quantity of relevance for\nthe dynamics of coalescing binaries: the total (orbital plus spin) an gular momentum of the binary when it reaches18\n/Minus1.0/Minus0.5 0.0 0.5 1.00.40.60.81.0\na/Hat\n0a/Hat\nJLSOE/LParen13,1/RParen1E/LParen13,0/RParen1E/LParen14,1/RParen1,a5/Equal/Plus60E/LParen14,1/RParen1,a5/Equal/Plus25Rezzollaetal./LBracket128/RBracket1\nFIG. 5: The dimensionless total angular momentum Kerr param eter ˆaLSO\nJ, Eq. (5.13), at the LSO, versus ˆ a0. We have assumed\nm1=m2,a1=a2, andθ0=π/2. The parameter ˆ aLSO\nJis computed from Eq. (5.13) with ˆjLSO=ˆℓLSO+ˆa1+ˆa2=ˆℓLSO+2ˆa0.\nWe compare the various EOB predictions obtained either by im proving the accuracy of spin-orbit terms [E(3 ,1) versus E(3 ,0)],\nor by improving the accuracy of orbital terms [E(4 ,1) versus E(3 ,1)]. We use two representative values of the 4PN parameter\na5= +25 and a5= +60. For comparison, we also include a ﬁt to recent numerica l estimates of the ﬁnalKerr parameter of the\nblack hole resulting from the coalescence of the two constit uent black holes.\nthe LSO [i.e., at the end of the (approximately) adiabatic inspiral, just before the plunge],\nJ≡L+S1+S2. (5.11)\nIn terms of rescaled dimensionless variables, this becomes\nˆj≡c\nGMµJ=ˆℓ+m1\nm2ˆa1+m2\nm1ˆa2, (5.12)\nwhereˆℓ≡cℓ. Actually, the most relevant quantity is the dimensionless Kerr para meter associated to the total LSO\nmass-energy and the total LSO angular momentum, i.e., the value at the LSO of the ratio\nˆaJ≡cJ\nG/parenleftbig\nHreal/c2/parenrightbig2=νˆj\n/parenleftbig\nHreal/(Mc2)/parenrightbig2, (5.13)\nwhereˆjis the modulus of ˆj.\n•In Fig. 5 we contrast the dependence of ˆ aLSO\nJon the dimensionless eﬀective spin parameter ˆ a0for several EOB\nmodels: the two 3PN-accurate ones [E(3 ,0) using LO-accurate spin-orbit, and E(3 ,1) using NLO-accurate spin-\norbit], and two illustrative [7, 9] “4PN-accurate” NLO-spin-orbit mo dels E(4,1) (using either a5= +25 or\na5= +60, as in Fig. 1). [Here, we are still considering fully symmetric syst ems with m1=m2anda1=a2,\nso that ˆa0= ˆa1= ˆa2.] Again we see the moderating inﬂuence of NLO corrections. The EOB -LO curve E(3 ,0)\nexhibits a sudden drop down (pointed out in [1]) before rising up again ( and disappearing at ˆ a0= +0.9 when\nthe LSO ceases to exist). By contrast, the NLO curve E(3 ,1) exhibits a much more regular dependence on ˆ a0,\nwhich is roughly linear over the entire range of values −1≤ˆa0≤1. The two illustrative E(4 ,1) curves exhibit\na “mixed” behaviour where a “drop” similar to the one featuring in the LO curve is still present, though it is\nmoderated by NLO spin-orbit eﬀects. This sensitivity to the inclusion of a 4PN contribution in A(ˆu) is due to a\ndelicate interplay between the modiﬁed shape of the basic spin-indep endent “radial potential” A(ˆu,a5) and the\nuse of a (1,4) Pad´ e resummation of the “eﬀective spin-dependent radial potential” ∆ t(R), Eq. (4.6a). Indeed,\nthe additional contributions proportional to a5anda2areboth repulsive , and tend to compound their eﬀect,\nwhich is to push the LSO toward closer, more bound orbits [1].\nWe have also indicated in Fig. 5 the ﬁnal(i.e., after coalescence) dimensionless Kerr parameter of (symmet ric)\nspinning binaries, as obtained in recent numerical simulations [26, 27 , 28, 29]. For simplicity, we have shown the\nsimple analytic ﬁt proposed in [28]. The fact that the 3PN-NLO-accu rate EOB LSO Kerr parameter [E(3 ,1)]\nis systematically abovethe ﬁnal Kerr parameter is in good agreement with the fact that, a fter reaching the19\n/Minus1.0/Minus0.5 0.0 0.5 1.00.60.70.80.91.01.1\na/Hat\n2a/Hat\nJLSOE/LParen13,1/RParen1\na/Hat\n1/Equal1.00a/Hat\n1/Equal0.75a/Hat\n1/Equal0.50a/Hat\n1/Equal0.25a/Hat\n1/Equal0.00\nFIG. 6: The dimensionless total angular momentum Kerr param eter ˆaLSO\nJat the E(3,1) LSO versus ˆ a2for various values of the\nparameter ˆ a1. Here we consider spin-dissymmetric systems with a1∝negationslash=a2(but still m1=m2andθ0=π/2). The parameter\nˆaLSO\nJis computed from Eq. (5.13) with ˆjLSO=ˆℓLSO+ˆa1+ˆa2.\n/Minus1.0/Minus0.5 0.0 0.5 1.0/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0\na/Hat\n0e/Hat\nLSO\nE/LParen13,1/RParen1exactKerr\nFIG. 7: Comparison of the test-mass limit m2≃µ→0 (with ﬁxed ˆ a2; so that S2/m2→0) for two Hamiltonians. We consider\nthe speciﬁc “non relativistic” binding energy ˆ eLSO≡eLSO/ν= (ELSO−Mc2)/(µc2) at the LSO versus ˆ a0. The solid curve is\nthe result of taking the test-mass limit of the EOB Hamiltoni an, while the short-dashed curve is the result for a test part icle\nmoving in the Kerr spacetime.\nLSO, the system will still loose a signiﬁcant amount of angular moment um19during the plunge and the merger-\nplus-ringdown. In the case of non-spinning binaries, it has been shown that, by using the EOB formalism up\nto the end of the process [i.e., by taking into account the losses of JandEduring plunge, as well as during\nmerger-plus-ringdown], there was a good agreement (better tha n∼2%) between EOB and numerical relativity\nfor the ﬁnal spin parameter [12]. We hope that the same type of agr eement will hold also in the case of spinning\nbinaries considered here.\n•In Fig. 6 we plot the LSO dimensionless Kerr parameter of Eq. (5.13) f orspin-dissymmetric systems, namely\na1∝ne}ationslash=a2(but with m1=m2), computed with the 3PN-NLO EOB Hamiltonian model E(3 ,1). This plot\nillustrates that the LSO spin parameter is a smooth (and essentially lin ear) function of the two individual spins.\n•Finally, we compare in Fig. 7 the spinless test particle limit [i.e., m2→0, together with a2=S2/(m2c)→0, as\nappropriate to black holes for which ˆ a≤1] for two Hamiltonians: the 3PN-NLO EOB one E(3 ,1), and the exact\n19We use here the fact (found in numerical calculations, and im plied by the analytical EOB approach), that, fractionally s peaking, the\nangular momentum loss after the LSO is signiﬁcantly higher t han the corresponding energy loss.20\none, as known from the geodesic action of a spinless test particle in t he Kerr metric. For non-spinning systems\nthe EOB Hamiltonian is constructed so as to reduce to the exact Sch warzschild-derived one in the test-particle\nlimit. However, for spinning systems, we have chosen in Eq. (4.6a) to deﬁne the crucial metric coeﬃcient ∆ t(R)\nby Pad´ e-resummingthe sum of A(R;ν)+a2/R2. This Pad´ e-resummationis indeed useful for generally ensuring,\nfor comparable mass systems, that ∆ t(R) have a simple zero at some “eﬀective horizon” rH. However, in the\ntest-mass limit ν→0, while the Taylor-approximant to A(R;ν) +a2/R2would coincide with the exact Kerr\nanswer, the Pad´ e-resummed version of A(R;ν) +a2/R2diﬀers from it. We see, however, on Fig. 7 that the\nresulting diﬀerence has a very small eﬀect on the LSO energy per un it (µ) mass, except when the dimensionless\neﬀective spin ˆ a0is very close to +1. On the other hand, as we saw above when discuss ing Fig. 5, the issue of\nthe Pad´ e resummation of ∆ t(R) becomes more subtle when one considers the comparable-mass ca se, together\nwith the inclusion of a repulsive 4PN parameter a5.\nVI. CONCLUSIONS\nThe main conclusions of this work are:\n•We have prepared the ground for an accurate Eﬀective One Body ( EOB) description of the dynamics of binary\nsystems made of spinning black holes by incorporating the recent computation of the next-t o-leading order\n(NLO) spin-orbit interaction Hamiltonian [13] (see also Refs. [14, 15 ]) into a previously developed extension of\nthe EOB approach to spinning bodies [1].\n•We found that the inclusion of NLO spin-coupling terms has the quite s igniﬁcant result of moderating the eﬀect\nof the LO spin-coupling, which would, by itself (as found in Ref. [1]), p redict that the Last Stable (circular)\nOrbit (LSO) of parallely-fast-spinning black holes can reach very lar ge binding energies of the order of 30%\nof the total rest-mass energy Mc2. By contrast, the inclusion of NLO spin-orbit terms predicts that t he LSO\nof parallely-fast-spinning systems, though signiﬁcantly more boun d than that of non-spinning holes, can only\nreach binding energies of the order of 4% of the total rest-mass e nergyMc2(see Fig. 4 above). This reduction\nin the inﬂuence of the spin-orbit coupling is due to the fact that the ( eﬀective) “gyro-gravitomagnetic ratios”\narereducedby NLO eﬀects from their LO values gLO\nS= 2,gLO\nS∗=3\n2to the values (here considered along circular\norbits)\ngcirceﬀ\nS= 2−5\n8νx,\ngcirceﬀ\nS∗=3\n2−/parenleftBig9\n8+3\n4ν/parenrightBig\nx, (6.1)\nwherex≃GM/(Rc2)≃(GMΩ/c3)2/3. This reduction then reduces the repulsive eﬀect of the spin-orbit\ncoupling which is responsible for allowing the binary system to orbit on v ery close, and very bound, orbits (see\ndiscussion in Section 3C of Ref. [1]).\n•We studied the dependence of the dimensionless Kerr parameter of the binary system, ˆ aJ≡cJ/(G(Hreal/c2)2),\ncomputed at the LSO, on the spins of the constituent black holes. A gain the moderating eﬀect of including\nNLO spin-orbit terms is very signiﬁcant (compare the solid and the da shed20lines in Fig. 5). Thanks to this\nmoderating eﬀect the LSO Kerr parameter ˆ aLSO\nJis found to have a monotonic, and roughly linear, dependence\non the spin parameters of the individual black holes (see solid line in Fig. 5 and the various curves in Fig. 6).\nWe also studied the eﬀect of including the type of 4PN parameter a5found useful in recent work [7, 8, 9, 10]\nfor improving the agreement between EOB waveforms and numerica l ones.\n•We leave to future work the analog of what was initiated for spinning s ystems in Ref. [5], and recently completed\nfor the case of non-spinning black holes in Ref. [12], i.e., a full dynamica l study, within the EOB approach, of the\nKerr parameter of the ﬁnalblack hole resulting from the merger of spinning black holes which take s into account\nthe angular momentum losses that occur after the LSO, during the plunge, the merger, and the ringdown. Let\nus also note that Ref. [30] has recently proposed an approximate a nalytical approach (which is similar in spirit\n20Compare also with Fig. 2 of Ref. [1] where the relevant LO resu lt is the curve labelled “DJS” which reaches a maximum around\nˆa≡7\n8ˆa0≃0.31, in agreement with the (local) maximum in the dashed line o f our Fig. 5 reached around ˆ a0≃0.36.21\nto the approximation used in Refs. [1, 3, 5] and above, namely that of considering the Kerr parameter of an\neﬀective test particle at, or after, the LSO) towards estimating t he ﬁnal spin of a binary black hole coalescence.\nThe resulting prediction is, however, only in coarse agreement ∼10% with numerical results. Note in this\nrespect that, as displayed in Fig. 5, the “zeroth order” EOB result [corresponding to using the Kerr parameter\nfor E(3,1) at the LSO, without taking into account the later losses of angula r momentum] is already in ∼20%\nagreement with the ﬁt to the numerical data [28]. The fact (displaye d on Fig. 5) that the E(3,1) EOB LSO Kerr\nparameter is systematically abovethe ﬁnal (after coalescence) Kerr parameter determined by rec ent numerical\nsimulations [26, 27, 28, 29] is in qualitative agreement with the fact th at the system will loose a signiﬁcant\namount of angular momentum during the plunge and the merger-plus -ringdown. Note, however, the sensitivity\nof ˆaLSO\nJto a “4PN deformation” of the EOB Hamiltonian by the parameter a5. As said above, this sensitivity\nis due to the fact that the radial function ∆ t(R)/R2combines the additional repulsive eﬀects of both a positive\n4PN contribution + a5ν(GM/(c2R))5and a positive spin-dependent contribution + a2/R2. We leave to future\nwork an exploration of this issue, which might need the use of a diﬀere nt Pad´ e resummation than the (1,4) one\nused in (4.6a).\nItremainstobe seenwhetherthe EOB/NumericalRelativitycompar isonfortheﬁnalKerrparameterofspinning\nsystems will be as good as it was found to be for the non-spinning cas e [12], i.e., at the 2% level. If this is the\ncase, it will establish the physical relevance of the improved EOB Ham iltonian constructed in the present paper.\n•Let us ﬁnally note that there is some ﬂexibility in the improved spin-dependent EOB Hamiltonian proposed\nabove (besides the ﬂexibility in the choice of the Pad´ e resummation m entioned above). On the one hand, the\nchoice (3.14) for the gauge parameters a(ν) andb(ν) might be replaced by other choices. On the other hand,\nthe choice (4.11) for the eﬀective spin vector might also be replaced by other ones. In particular, it might be\ninteresting to consider the alternative deﬁnition\nMcanew≡Seﬀ new≡1\n2geﬀ new\nSS0\n=1\n2geﬀ new\nS/parenleftbig\nS+S∗/parenrightbig\n. (6.2)\nThis deﬁnition coincides with the one used above at LO in spin-orbit eﬀe cts (because geﬀ new\nS= 2+O(ν/c2)),\nand allows one to use a simpliﬁed supplementary spin-orbit contributio n, built with\nσnew≡1\n2/parenleftbig\ngeﬀ\nS∗−geﬀ\nS/parenrightbig\nS∗, (6.3)\ninsteadof (4.15). Itmightbeinterestingtoexplorewhichofthesep ossibledeﬁnitionsexhibitsthebestagreement\nwith current numerical results.\nAcknowledgments\nThis work was supported in part by the KBN Grant no 1 P03B 029 27 (t o P.J.) and by the Deutsche Forschungs-\ngemeinschaft (DFG) through SFB/TR7 “Gravitational Wave Astro nomy”.\n[1] T. Damour, “Coalescence of two spinning black holes: An e ﬀective one-body approach,” Phys. Rev. D 64, 124013 (2001)\n[arXiv:gr-qc/0103018].\n[2] A. Buonanno and T. 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Groth,5Xavier Waintal,5and Stephan Roche1, 6\n1Catalan Institute of Nanoscience and Nanotechnology (ICN2),\nCSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n2Department of Physics, Universitat Aut\u0012 onoma de Barcelona,\nCampus UAB, Bellaterra, 08193 Barcelona, Spain\n3Universitat Aut\u0012 onoma de Barcelona, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n4School of Physics, Trinity College Dublin, Dublin 2, Ireland\n5Univ. Grenoble Alpes, CEA, IRIG-PHELIQS, 38000 Grenoble, France\n6ICREA{Instituci\u0013 o Catalana de Recerca i Estudis Avan\u0018 cats, 08010 Barcelona, Spain\nImproved fabrication techniques have enabled the possibility of ballistic transport and unprece-\ndented spin manipulation in ultraclean graphene devices. Spin transport in graphene is typically\nprobed in a nonlocal spin valve and is analyzed using spin di\u000busion theory, but this theory is not\nnecessarily applicable when charge transport becomes ballistic or when the spin di\u000busion length is\nexceptionally long. Here, we study these regimes by performing quantum simulations of graphene\nnonlocal spin valves. We \fnd that conventional spin di\u000busion theory fails to capture the crossover\nto the ballistic regime as well as the limit of long spin di\u000busion length. We show that the latter can\nbe described by an extension of the current theoretical framework. Finally, by covering the whole\nrange of spin dynamics, our study opens a new perspective to predict and scrutinize spin transport\nin graphene and other two-dimensional material-based ultraclean devices.\nSince the seminal work of Tombros and cowork-\ners [1], who \frst measured long spin di\u000busion length\nin graphene nonlocal spin devices, a considerable\nnumber of studies have explored how to improve\nthe material quality and the e\u000eciency of spin injec-\ntion and detection so as to reach the upper limit of\nspin transport [2{11]. After \ffteen years of progress,\nthe fabrication of ultraclean (ballistic) graphene de-\nvices is now a reality with mean free paths reaching\nhundreds of nanometers and as long as tens of \u0016m\nat lower temperatures [12, 13]. Theoretical anal-\nysis of experimental data is usually based on the\nspin di\u000busion equations [14{20], but their validity in\nnew regimes of spin transport, especially the ballistic\nregime, is under question [21, 22]. A solution is to\nuse quantum transport simulations to describe the\nspin dynamics in a realistic device geometry [23, 24].\nHowever, theoretical e\u000bort in this direction is cur-\nrently lacking, which not only limits the understand-\ning of spin transport in ultraclean devices but also\nrestrains further improvement for spintronic appli-\ncations based on two-dimensional materials and van\nder Waals heterostructures [25].\nIn this Letter we use quantum transport simu-\nlations to explore the physics of spin dynamics in\ngraphene nonlocal spin valves (NSVs). By calculat-\ning exactly the nonlocal resistance Rnl, we are able\nto capture regimes that conventional analysis fails\nto describe. In the di\u000busive regime we show thatthe typically overlooked drain and reference elec-\ntrodes (see Fig. 1 below) play a fundamental role\nin limitingRnlwhen spin relaxation is weak. When\napproaching the quasiballistic regime of spin trans-\nport, simulated Hanle precession curves reveal the\nfailure of the di\u000busive equations. By extending the\ntheory of spin di\u000busion following Refs. [26, 27], we\nobtain more general formulas for properly obtain-\ning the spin di\u000busion length \u0015sin the former case,\nand for understanding the evolution of Hanle curves\nin the crossover from di\u000busive to ballistic transport.\nOur \fndings demonstrate that brute force quantum\nsimulation of nonlocal transport is fundamental to\nproperly analyze spin dynamics in unconventional\nregimes and for allowing a direct comparison with\nexperimental data.\nLateral NSVs are widely used to probe spin trans-\nport in disordered materials because they decouple\nelectrical currents from spin currents, allowing for\nbetter device sensitivity [14, 26, 28{31]. In such a de-\nvice con\fguration, shown in Fig. 1, a ferromagnetic\n(FM) electrode (labeled \\2\") drives a spin-polarized\ncurrentI0to a drain electrode (\\1\"), a spin accu-\nmulation develops below the FM, and this spin dif-\nfuses to the right along the channel. This spin is\ndetected by another FM contact (\\3\") as a nonlo-\ncal voltage Vnl, which is normalized to a nonlocal\nresistanceRnl\u0011Vnl=I0. Owing to inherent spin\nrelaxation, Rnlusually decays exponentionally witharXiv:1910.06194v3  [cond-mat.mes-hall]  22 May 2020channel length d,Rnl/exp(\u0000d=\u0015s). Extracting \u0015s\nfrom the length dependence of Rnlis quite di\u000ecult\nexperimentally, but this can be avoided by applying\na perpendicular magnetic \feld which provokes pre-\ncession and additional dephasing of the spins. Ac-\ncording to traditional spin di\u000busion theory, Rnlthen\ntakes the form [14, 26, 30{33]\nRnl=PiPd\n2w\u001bRe\u001ae\u0000d\u000b\n\u000b\u001b\n; (1)\nwhere\u000b=q\n1\n\u00152s+i!\nD,Pi(Pd) is the polarization of\nthe injector (detector) FM contact, \u001bis the electri-\ncal conductivity, wis the channel width, Dis the\ndi\u000busion coe\u000ecient, and !=g\u0016BB=~is the Larmor\nspin precession frequency induced by the magnetic\n\feldBwithgthe g-factor, \u0016Bthe Bohr magneton,\nand~the Planck constant. By \ftting this expres-\nsion to a measurement of Rnlvs.B, one can extract\nthe spin di\u000busion length \u0015s.\nThis approach, known as a Hanle measurement,\nhas been a cornerstone of the exploration of spin\ndynamics in a large variety of materials including\nmetals [26, 34], semiconductors [14, 35{38], and\ngraphene [7, 21]. However, Eq. (1) is based on sev-\neral assumptions which may be violated in ultraclean\ndevices. Speci\fcally, Eq. (1) assumes that transport\nis fully di\u000busive, that relaxation is fast enough so\nthat no spin signal reaches the reference electrode\n(lead 4), and neglects di\u000busion along the left-hand\npart of the NSV (between leads 2 and 1). It is\ntherefore important to revisit and extend the cur-\nrent theoretical framework to cope with new spin\ntransport regimes which are emerging in today's ul-\ntraclean nonlocal spin devices.\nTo study the behavior of graphene NSVs, we em-\nploy the Landauer-B uttiker formalism, as imple-\nmented in Kwant [39], to the device setup in Fig. 1.\nThe graphene layer is described in a single- \u0019-orbital\ntight-binding basis, with a Hamiltonian given by\n^H=tX\nhi;jicy\ni\u0011cj\u0011+X\ni\u000eUicy\ni\u0011ci\u0011+X\ni\u0011\u00110cy\ni\u0011[s\u0001Ji]\u0011\u00110ci\u00110\n+\u0016BX\ni\u0011\u00110cy\ni\u0011[s\u0001Bi]\u0011\u00110ci\u00110; (2)\nwherecy\ni\u0011(ci\u0011) is the creation (annihilation) oper-\nator with spin \u0011on sitei. The \frst term ( t=\n\u00002:6 eV) denotes nearest-neighbor hopping in the\ngraphene honeycomb lattice. The second term is\nAnderson disorder de\fned by a random potential\nFIG. 1. Sketch of the lateral nonlocal spin valve. Red\n(black) regions denote the contacts (sample). The injec-\ntor and detector contacts, labeled 2 and 3 respectively,\nare ferromagnetic with their magnetization indicated by\narrows. Contacts 1 and 4 represent the drain and refer-\nence electrodes, respectively.\nuniformly distributed at each site i, with\u000eUi2\n[\u0000U=2;U=2]. The third term is magnetic disor-\nder mainly a\u000becting the spin dynamics. It is de-\n\fned as a magnetic exchange coupling with strength\nJand random orientation at each site i,Ji=\nJ[sin(\u0012i) cos(\u001ei);sin(\u0012i) sin(\u001ei);cos(\u0012i)], with\u0012and\n\u001espherical angles and sthe spin Pauli matrices. The\nlast term is the Zeeman exchange induced by an ex-\nternal magnetic \feld B(note that orbital e\u000bects of\nthe magnetic \feld are neglected). In general, Uis\ntaken to be much larger than J, such that Udic-\ntates the charge transport regime, whereas the spin\nrelaxation is driven by J. The modeling of the leads\nis described in Supplemental Material [40].\nThe calculation of Rnlis performed by evaluat-\ning all transmission probabilities between di\u000berent\nleads. We then construct the conductance matrix G\n[41] and solve the linear system I=GV, whereI\nandVare vectors including the current and voltage\nconditions at each electrode. We \fx a current I0\nfrom lead 2 (injector) to lead 1 (drain) while enforc-\ning that no current \rows in leads 3 (detector) and 4\n(reference). This ensures zero charge current in the\nchannel since any current going to the right from\nthe injector will be compensated with an oppositely\nspin-polarized current injected by the reference lead.\nWe also ground the drain ( V1= 0) and solve the\nsystem to obtain the other voltages. The nonlocal\nresistance is then calculated as Rnl= (V3\u0000V4)=I0.\nWe \frst investigate spin dynamics in the di\u000busive\nregime of charge transport, which is identi\fed from\nthe scaling of the two-terminal conductance G2T\nwith the channel length x=L+lby removing leads\n22 and 3. We evaluate the mean free path leby \ftting\nthe numerical result to G2T= 2e2=h\u0002Mle=x, with\nMthe number of propagating modes per spin. Also,\nforx\u001dlewe calculate the localization length llocus-\ninghln(G2T)i/\u0000x=lloc. By choosing w= 20:1 nm\n(164-aGNR), Fermi energy EF= 0:4 eV,M= 9 and\nU= 1:04 eV, we obtain le= 117 nm and lloc= 880\nnm [40]. We take L= 250 nm and l= 1000 nm\nso that most of the transport occurs between leand\nlloc, and we compute Rnlvs. the channel length dat\nB= 0 for di\u000berent magnetic disorder strengths J.\nThe results are plotted in Fig. 2.\nFor large values of J,Rnldecays exponentially\nwith channel length, as predicted by Eq. (1). How-\never, as spin relaxation slows with decreasing J, thedecay ofRnlbecomes linear instead of exponential.\nEven forJ= 0, corresponding to \u0015s!1 , there is a\nloss of spin signal with channel length [40], a result\nnot captured by Eq. (1). Conventional spin di\u000bu-\nsion theory assumes the spin accumulation vanishes\natx!+1, or at least at x=l[14, 31]. However,\nthis condition is violated for the lowest values of Jin\nour simulations, and may also be the case in recent\nexperiments for which \u0015sreaches tens of \u0016m [42].\nTo describe the proper length dependence of Rnl,\nwe solve the spin di\u000busion equations taking the full\ndevice geometry into account [40]; not only are spins\ninjected from lead 2, but leads 1 and 4 are explicitly\nincluded (lead 3 does not perturb the system). From\nthis,Rnlbecomes\nRnl=PiPd\n2w\u001bRe\u001a[\fcosh(L\u000b) + 4 sinh(L\u000b)]\u0001[\fcosh(\u000b(d\u0000l))\u00004 sinh(\u000b(d\u0000l))]\n\u000b[4\fcosh((L+l)\u000b) + (8 +\f2=2) sinh((L+l)\u000b)]\u001b\n; (3)\nwhere\f=Rcw\u001b\u000b andRcis the contact resistance\nbetween leads 1 and 4 and the graphene. In the\ncase of perfectly transparent contacts, the interface\nresistance is not zero but dictated by the Sharvin\nresistanceRS=h=(2e2M) [43, 44]. If one takes the\nlimits\u0015s\u001cL;l, Eq. (1) is recovered. Importantly,\nEq. (3) becomes linear when \u0015s!1 ,\nRnl=PiPd\n2w\u001b(4RL+Rc)(\u00004d+ 4l+Rcw\u001b)\n8RL+ 8Rl+ 4Rc;(4)\nwhereRL=L=w\u001b andRl=l=w\u001b are the sheet re-\nsistance of the left and right device regions, respec-\ntively. The black dot-dashed lines in Fig. 2 show the\n\fts of the numerical results to Eq. (3), indicating\nthat this expression is able to capture the scaling of\nRnlfor any value of J.\nEquation (4) shows that when \u0015s\u0015L;lthe non-\nlocal spin signal still decays with length. This decay\nis no longer related solely to spin relaxation but also\nto charge di\u000busion and the presence of the leads.\nRecall that Rnldepends on the conductance matrix\nG, which consists of the transmission between all\nleads and the imposed current/voltage conditions.\nIn the limit of long \u0015s, the drain and reference elec-\ntrodes act as spin sinks, \fxing the value of Rnlin\norder to meet the conditions I=I0andI= 0 at\nleads 1 and 4, respectively. We note that this spin\nsinking e\u000bect occurs despite the absence of spin re-\nlaxation in the leads. Rather it is the result of these\nleads absorbing and reinjecting spin current under\nthe imposed boundary conditions. This contrasts\nwith the contact-induced spin dephasing discussed\nFIG. 2.Rnlas a function of injector-detector distance for\ndi\u000berent strengths of magnetic disorder, with le= 117\nnm. Error bars result from the averaging of several disor-\nder con\fgurations ( >130). All curves have similar error\nbars. Black dot-dashed lines are the \fts using Eq. (3).\nInset: comparison of \u0015sextracted from Eq. (1) (gray\nsquares) and Eq. (3) (black circles). The red line indi-\ncates 1=Jscaling of\u0015s.\nin some experiments [15, 19, 45]. Equation (4) shows\nthat at the reference electrode ( d=l)Rnlis propor-\ntional toRcto leading order. Thus, in the limit of\nweak spin relaxation a small Rcwill suppress the\nnonlocal spin signal. Another consequence of long\n\u0015sis that the transmission between the drain and\nreference electrodes becomes crucial. The condition\nof zero charge current in the channel forces the in-\njection of spin-down current from lead 4 to 1 so that\nlead 2 can inject up-spins that di\u000buse towards lead 3.\n3FIG. 3. Hanle spin precession curves for di\u000berent\nstrengths of magnetic disorder, with le= 117 nm and\nd= 500 nm. Error bars result from averaging several\ndisorder con\fgurations ( >90). All curves have similar\nerror bars. Black dot-dashed line is the \ft using Eq. (3).\nInset: comparison of \u0015sextracted from Eq. (1) (gray\nsquares) and Eq. (3) (black circles).\nIf lead 4 (1) is unable to inject (absorb) down-spins\nto (from) the system, up-spins will not be able to dif-\nfuse along the channel and Rnlwill be suppressed.\nIn Fig. S3 [40], such e\u000bect is evidenced further by\nchanging lead 1 from nonmagnetic to FM, which re-\nducesRnlby more than three orders of magnitude.\nThis suggests not employing FM materials for leads\n1 and 4 in experiments. Another important conse-\nquence is that since Eq. (1) does not account for this\nextra decay induced by leads 1 and 4, this reduction\nis absorbed in the value of \u0015s, which will be there-\nfore underestimated by Eq. (1). This is shown in\nFig. 2 (inset), where \u0015sis plotted vs. spin relaxation\nstrength. The gray squares are extracted from \fts\nto Eq. (1), while the black circles are from Eq. (3).\nThe spin di\u000busion length is the same when \u0015s<L;l\n(largeJ), but for small JEq. (1) signi\fcantly under-\nestimates the value of \u0015s. According to the theory\nof spin relaxation arising from exchange \ructuations,\n\u0015sshould scale as 1 =J[14, 40], which is captured by\nthe \fts to Eq. (3).\nWe now extend the analysis to Hanle precession\nand plotRnlvs.Bin Fig. 3 using a channel length\nofd= 500 nm. We note that large magnetic \felds\nare required for computational convenience (our de-\nvice is smaller than in experiments) but no spuri-\nous e\u000bect is introduced since the Zeeman splitting\nremains much smaller than the subband energy sep-\naration and orbital e\u000bects are excluded. The sim-\nulation data are \ftted with Eqs. (1) and (3) usingD=vFleand the resulting \u0015sare compared in Fig. 3\n(inset). Similar to Fig. 2, in the limit of weak spin\nrelaxation\u0015sis underestimated when using the con-\nventional equation.\nTo estimate how strong the underestimation of \u0015s\nmay be in state-of-the-art devices, we calculate a\nHanle curve with Eq. (3) using realistic parameters\n(L= 5\u0016m,l= 20 \u0016m,d= 15 \u0016m,\u0015s= 10 \u0016m,\nD= 0:05 m2/s) and \ft it with Eq. (1). We obtain\n\u0015s= 7:26\u0016m, about 25% less than the real value.\nMore importantly, the spin lifetime ( \u001cs=\u00152\ns=D) is\nunderestimated by nearly 100%; the real value is 2\nns while the \ft gives 1 :05 ns. These results thus call\nfor a revised analysis of Hanle spin precession mea-\nsurements taking into account the device geometry\nand our more general formula (Eq. (3)).\nFinally, we examine the quasiballistic limit. When\nle\u0018d, only a few scattering events occur dur-\ning transport through the channel. This situation\nhas been discussed for spin relaxation in ultraclean\ngraphene [46], but little is known about its impact\non Hanle measurements. We keep all simulation pa-\nrameters the same as before, including the channel\nlengthd= 500 nm, and reduce the Anderson disor-\nder toU= 0:52 eV, giving a mean free path le\u0018500\nnm. The solid lines in Fig. 4 show the Hanle curves\nfor this quasiballistic regime, and the dashed lines\nshow \fts using Eq. (3). The behavior of Rnlis\nnow substantially di\u000berent, especially with respect\nto the dependence on B. The unit B0corresponds\nto the magnetic \feld needed for spin to precess 2 \u0019\nradians upon reaching the detector ( B0=2\u0019vav\nF\n\rd,\nwith\r=g\u0016B=~the gyromagnetic ratio and vav\nF\nthe averaged Fermi velocity from all modes at the\nFermi level). The \frst rotation of the spins occurs\natB0= 1, but is followed by a dispersion of frequen-\ncies for larger B. This can be understood if we ex-\namine the origin of such oscillations. By performing\na simulation in a purely ballistic regime, U=J= 0,\nwe observe in Fig. 4 (inset) that the main oscillation\nhas the same period, but is superimposed with other\nfrequencies. This arises because the nonlocal signal\nis the sum of each propagating mode moving at a\ndi\u000berent velocity. To verify this, we follow Ref. 26\nand sum the contributions of spin over all transport\ntimes to get Rnl/PM\nicos (\rdB=v F;i) [40]. By tak-\ning only the Fermi velocities of each mode of the\nsystem, the simulations are very well reproduced.\nFrom these results we can conclude that in the\nquasiballistic regime the scattering is weak enough\nfor the precession to follow that of ballistic trans-\n4FIG. 4. Hanle spin precession curves in the quasiballis-\ntic regime, with le= 487 nm and d= 500 nm. Solid\nlines correspond to simulations (averaged from 12 dis-\norder con\fgurations), while dashed lines are \fts using\nEq. (3). Inset: Case with U=J= 0, solid (dashed) line\nshows the simulation ( Rnl/PM\nicos (\rdB=v F;i)).\nport, but is also strong enough to average the beat-\ning pattern to one main frequency. This explains\nwhy neither Eq. (3) nor the sum of cosines is able\nto \ft the quasiballistic Hanle curves. This di\u000eculty\nin capturing the crossover from di\u000busive to ballistic\ntransport in a single expression highlights the im-\nportance of brute force quantum simulations to un-\nderstand such a regime. Furthermore, in the supple-\nmental material [40] we simulate Hanle curves with\ndi\u000berentleand \fnd that when le>d=3, the spin dy-\nnamics enters the quasiballistic regime. Finally, in\nthe limit of a 2D graphene \rake, with most electrons\nmoving at the same Fermi velocity, one would expect\nthe signal to be determined by a single frequency. To\nobserve this e\u000bect at low magnetic \felds ( B\u00140:5\nT), the channel length needs to be d\u00152\u0019vF\n\rB\u001950\n\u0016m. We highlight the fact that this analysis can\nbe applied to other materials as well; depending on\nwhether there are electrons moving at the same or\ndi\u000berent Fermi velocities, one can expect single or\nmultiple precession frequencies, respectively.\nIn conclusion, we have performed fully quantum\nsimulations that provides a more global picture of\nnonlocal spin transport when the material quality\ndrives the system towards the quasiballistic regime,\nas well as an extended theoretical frame to ana-\nlyze systems with long spin di\u000busion lengths. In\nthis limit, the drain and reference electrodes become\nthe limiting factors, and one should aim for these\nto be nonmagnetic and optimize their contact resis-\ntance to reach the upper limit for spin informationtransfer. 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Under these conditions (such as originate from nuclear contact hyperﬁne\nﬁelds that do not affect orbital motion) a linear, spin-dependent dispersion for free electrons emerges from the\nLandau Hamiltonian. The result is that spins of opposite orientation ﬂow in opposite directions giving rise to a\npure spin current. A classical model of the spin and charge dynamics reveals intuitive aspects of the full quantum\nmechanical solution. We propose optical orientation or electrical polarization experiments to demonstrate this\noutcome.\nI. INTRODUCTION\nThe coupling of spin and orbital currents is integral to\nspintronics1,2. The (inverse) spin Hall effect is a hallmark\nexample where (spin) charge current is converted to (charge)\nspin current.3–6Other effects include: spin galvanic or “Edel-\nstein” effects (and their reciprocal) which convert charge cur-\nrent into spin polarization.7–9Each of these rely on the in-\ntrinsic coupling of spin and charge via the spin-orbit effect.\nDespite this there are a few examples of spin-charge cur-\nrent coupling not through spin-orbit effects, such as the spin\nGunn effect10,11or spin bottleneck effects in localized12or\nextended13,14materials, which rely on the Pauli exclusion\nprinciple and dynamical spin correlations.\nIn this article an alternate method of spin-charge coupling\nis described that relies on electron-nuclear spin coupling and\ndoes not require spin-orbit coupling, the Pauli exclusion prin-\nciple, or electronic spin-spin correlations.\nThe origin of the effect is dynamic nuclear polarization:\nlarge nuclear spin polarizations that can exert a sizable nu-\nclear ﬁeld on the electronic system15,16. These nuclear spin\npolarizations are generated by electron non-equilibrium spin\ntransfers to moment-carrying nuclei through the hyperﬁne in-\nteraction which accumulate due to the slow spin relaxation\ntime of nuclei. The resultant nuclear ﬁeld, although it is a\nmagnetic ﬁeld, is highly concentrated near the nuclei (Fermi\ncontact potential). Due to its localized character and the lack\nof an extended vector potential acting on the orbital motion,\nthis nuclear ﬁeld acts only on spin and not orbital motion, and\nis sometimes referred to as an effective ﬁeld. Here we desig-\nnate it as a “Zeeman-only ﬁeld”.\nThis absence of coupling between nuclear spin and elec-\ntronic orbital momentum entails the ﬁeld from polarized nu-\nclei does not contribute to the ordinary Hall effect but may\nsupport a larger anomalous Hall effect due to the increase in\nspin splitting17,18. How a Zeeman-only ﬁeld allows spin com-\nponents to be spatially separated is the subject of this article.\nHere we show that nuclei with patterned spin polarization,\ndue to their lack of orbital coupling and spatial inhomogene-\nity, can evince remarkable spin-dependent charge dynamics\nleading to pure spin currents or charge currents. Recent work\nhas demonstrated the importance of inhomogeneous nuclear\nxStern-Gerlach force, , for  FSG↓zy\nFSGFLvLorentz force,  for FL↑z\ndirection of nuclear polarization gradient, ∇Bn,zFLnegative test charge\nBn\n-2-1012-4-2024\nx(πgμBb/mωc2)y(πgBb/mωc2)y(πgμBb/mω2c)\nx(πgμBb/mω2c)-2-1012-4-2024\nx(πgμBb/mωc2)y(πgBb/mωc2)B0Lorentz force,  for FL↓zStern-Gerlach force, , for  FSG↑z\nFSGFLv\nv(a)\n(b)FIG. 1: Diagram of transverse geometry in n-GaAs where gradient\nforce is perpendicular to applied ﬁeld. Trajectories of Eqs. (17, 18)\nare shown. Regardless of initial conditions, spins travel in opposite\ndirections at the same speed. Blue trajectory is up spin and black\ntrajectory is down spin. Inset: orange arrows represent Lorentz force,\nred arrows represent Stern-Gerlach force, and green arrows represent\ncharge velocity.\nﬁelds on coupled electron-nuclear spin dynamics19,20but have\nnot explored the resulting spin-dependent motion. The spin-\nmotive force may emerge from either the Stern-Gerlach force\nof from the combination of Stern-Gerlach and Lorentz forces.\nThe effect of the net force is to separate up and down spins\nalong a direction longitudinal or transverse to the effective\nﬁeld gradient (Fig. 1(b) shows spins separating transverse\nto the gradient). In a longitudinal ﬁeld conﬁguration, a lin-arXiv:2208.03414v1  [cond-mat.mes-hall]  6 Aug 20222\near contribution to the dispersion relation appears for which\nwe calculate a spin current using a Landau-like Hamiltonian.\nWe also examine the longitudinal and transverse geometries\nwithin a semiclassical Drude-like model. Finally, we propose\nexperiments to see this effect by inducing dynamic nuclear\npolarization gradients via patterned optical or electrical orien-\ntation.\nII. NUCLEAR FIELD\nWe treat an electron ensemble in two or three dimensions\nwith charge q=\u0000e, effective electron mass m, and effective\nLand ´eg-factor g\u0003. We assume that the external ﬁeld, BBB0, is\nhomogeneous and any inhomogeneities lie in a Zeeman-only\nﬁeld, BBBZ(rrr). We choose a Zeeman ﬁeld based on the mag-\nnetic interaction between electrons and nuclei: BBBZ(rrr) =BBBn(rrr)\nwhere\nBBBn(rrr) =bnhIII(rrr)i=bnhI(rrr)iˆB0; (1)\nwith\nhI(rrr)i=B2\n0\nB2\n0+xB2\n`PPP(rrr)\u0001ˆB0: (2)\nHere bnis the Overhauser coefﬁcient and PPP(rrr) = PPP0+\nbbbf(rrr) =P0ˆB0+bf(rrr)ˆB0is the out-of-equilibrium electron\nspin polarization (we assume that thermal electron spin po-\nlarization is negligible) that is responsible for the dynamic\nnuclear polarization. The quantity bcontrols the magnitude\nof the non-uniformity whereas f(rrr)is the function specify-\ning the spatial structure of the Zeeman-only ﬁeld;p\nxB`is\nthe strength of the random local ﬁeld21. A coordinate sys-\ntem is chosen to maintain the external ﬁeld in the z-direction.\nBBBn(rrr)lies collinear with BBB0(we ignore Knight ﬁelds) but its\ngradient may not; we examine two different linear functions\nforf(rrr): longitudinal [ f(rrr) =z] and transverse [ f(rrr) =x].\nThese linear functional forms describe slowly varying expo-\nnential functions (\u0000e\u0000ri\u0018ri\u00001) of their respective Cartesian\ncoordinate, i2fx;y;zg, so that we can posit the existence of\nan constant spin-dependent force as shown later in this article.\nCombining these assumptions, the nuclear ﬁeld is\nBBBn(rrr) =cˆB0+briˆB0 (3)\nwhere\nb=bnB2\n0\nB2\n0+xB2\n`b; c=bnB2\n0\nB2\n0+xB2\n`P0: (4)III. QUANTUM MECHANICAL FORMULATION\nThe starting point is the Landau level description of free\nelectrons in a magnetic ﬁeld. The Hamiltonian for Landau\nlevels is\nH=p2\nx\n2m+1\n2mw2\nc(x\u0000x0)2(5)\nwhere the Landau gauge, AAA= (0;B0x;0)is assumed and wc=\neB0=m. The harmonic potential is centered at x0=\u0000~ky=eB0.\nThe lack of propagation in the bulk is shown by a quick com-\nputation of the velocity vy=~ky=m\u0000qAy(x0)=m=~ky=m\u0000\nqB0x0=m=0. The wave functions are\nY(rrr;x0) =ei(kyy+kzz)\np\n2nn!\u0010mwc\np~\u00111=4\ne\u0000mwc(x\u0000x0)2\n2~Hn\u0012r\nmwc\n~(x\u0000x0)\u0013\n(6)\nwhere Hnare Hermite polynomials of degree n. To the\nbest of our knowledge, prior work has not treated effective\nor Zeeman-only magnetic ﬁeld gradients within the Landau\nHamiltonian. Inhomogeneous real magnetic ﬁelds, that op-\nerate on spin and orbital degrees of freedom, lead to more\nchallenging Hamiltonians that exclude analytic solutions. For\ninstance, a linear magnetic ﬁeld gradient produces an anhar-\nmonic oscillator potential ( \u0018(x2\u0000x2\n0)2) for which there is\nno analytic solution22. By assuming a constant real mag-\nnetic ﬁeld in zand a linearly inhomogeneous Zeeman-only\nﬁeld (directed in zbut changes along x— we call this the\ntransverse geometry), difﬁculties are avoided since the Lan-\ndau level Hamiltonian is unchanged from Eq. (5) except for\nthe addition of a Zeeman term that contains the ﬁeld inhomo-\ngeneity:\nH=p2\nx+p2\ny+p2\nz\n2m+1\n2mw2\nc(x\u0000x0)2+ge\n2m(B0+c+bx)Sz:\n(7)\nSince the equation depends on neither ynorz, we express\nthose dependencies of the wave function as planes waves\nwhich leaves us, for each spin orientation s=\u00061, with, after\n“completing the square” and dropping terms of order b2, with\n\u0000~2\n2m¶2\n¶x2f(x) +1\n2mw2\nc\u0002\nx\u0000x0\n0\u00032f(x) =h\ne\u0000~2k2\nz\n2m\u0000gµb\n2(B0+c)s\u00001\n2gµBbx0si\nf(x); (8)\nwhere x0\n0=x0\u0000gµBb\n2mw2csis the new center of the harmonic\npotential and f(x)is the xpart of the separable wave func-tion. The eigenvalues of this modiﬁed Landau problem are3\nE=~wc(n+1\n2)which gives a total energy of\ne=~wc(n+1\n2) +~2k2\nz\n2m+gµb\n2(B0+c)s\u00001\n2gµBb~ky\neB0s;(9)\nwhich possesses a linear-in- kydispersion. The wave function\ndiffers only slightly from the Landau level case: Y(rrr;x0\n0).\nThe group velocity is deﬁned as vvvg=¶e=~¶kkk.vg;xis triv-\nially zero and vg;zis~kz=mbut on average also zero sinceR¥\n\u0000¥kzdkz=0. There is no type of current in xorz. How-\never the linear term remains for the ygroup velocity: vg;y=\n¶e=~¶ky=\u0000gµBb\n2eB0s=\u0000rSGwcswhere rSG=gµBb=2mw2\ncand\nwc=eB0=m. As expected, different spin orientations move inopposite directions. Summing over the two spins yields zero\ncharge current. If only a single spin orientation were present,\nthen a charge current would accompany the spin polarized cur-\nrent.\nA harmonic conﬁning potential can be added to mimic\nedges ( Vcon fine =mw2\n0x2=2), and remarkably the problem can\nstill be solved exactly; x0becomes\nx0!w2\nc\nw2c+w2\n0x0\n0=w2\nc\nw2c+w2\n0\u0000\nx0\u0000gµbb\n2mw2cs\u0001\n(10)\nand the eigenvalues are\ne= (n+1\n2)~(w2\nc+w2\n0)1=2\u00001\n2gµBb~kywc\nm(w2c+w2\n0)s+~2k2\ny\n2mw2\n0\nw2c+w2\n0+~2k2\nz\n2m+gµb\n2(B0+c)s (11)\nwhere the main difference between the unconﬁned example is\nthe presence of a kinetic energy with a modiﬁed effective mass\n(3rd term). The dispersion relation contains spin-dependent\nlinear and spin-independent quadratic elements.The eigen-\nstates are not signiﬁcantly altered beyond a redeﬁnition of\nx0and a new normalization factor23. Conﬁnement does not\nchange the results in any signiﬁcant way — a pure spin cur-\nrent is still generated traveling in the \u0007ydirection:\nvg;y=\u00001\n2gµBbwc\nm(w2c+w2\n0)s; (12)\nwhere the effect of the harmonic potential is to reduce the ve-\nlocity.\nThe presence of the soft potential allows us to avoid the\nunphysical fact that the velocity diverges as wc!0 in the\nunconﬁned model. With the soft potential in place, the trans-\nverse velocity also vanishes if the applied ﬁeld vanishes in\naccordance with expectations.\nIV . SEMI-CLASSICAL FORMULATION\nThe spin separation is naturally seen within a simple clas-\nsical framework that includes discrete spins. Only real ﬁelds\nexert a Lorentz force, FFFL=\u0000evvv\u0002BBB0, while the gradient of\neither ﬁeld ( BBB0orBBBn) may exert a Stern-Gerlach force; since\nBBB0is uniform, the spin-dependent forces are\nFFFSG=\u0000Ñ(\u0000µµµ\u0001BBB) =\u0000gµB\n~Ñ(SSS\u0001BBB) =\u0000g\n2µBsbˆz;(longitudinal)\n(13)\nFFFSG=\u0000Ñ(\u0000µµµ\u0001BBB) =\u0000gµB\n~Ñ(SSS\u0001BBB) =\u0000g\n2µBsbˆx;(transverse)\n(14)\nfor gradients either longitudinal or transverse to BBB0. Our\nchoice of Zeeman-only ﬁeld along ˆ zallows the spin dynamicsto be trivial when enforcing semiclassical spins to be in one of\ntwo states SSS=~\n2(0;0;s)where s=\u00061. The charge and spin\ndynamics are determined by solving the equations of motion:\nFFFL+FFFSG=mdvvv\ndt=\u0000eB0vvv\u0002ˆz\u0000g\n2µBsbˆz;(longitudinal)\n(15)\nFFFL+FFFSG=mdvvv\ndt=\u0000eB0vvv\u0002ˆz\u0000g\n2µBsbˆx:(transverse) (16)\nIn either case, the constant force acts just like a spin-\ndependent effective constant electric ﬁeld. In the longitudinal\ngeometry, consisting of a constant force, the charge carrier ac-\ncelerates indeﬁnitely. By considering damping (to be done in\nnext section), this unphysical behavior is avoided. The system\nof equations for the transverse model can be solved exactly for\nany initial starting place and electron velocity in a way that\nmirrors the classical Hall effect calculation except now with\na spin-dependent electric ﬁeld. For simplicity we express the\nsolution for an electron starting at the origin with no initial\nvelocity, v0=0,\nrrr(t) =\u0012\n\u0000srSG(1\u0000coswct);srSG(\u0000wct+sinwct);0\u0013\n(17)\nvvv(t) =\u0012\n\u0000srSGwcsinwct;srSGwc(\u00001+coswct);0\u0013\n(18)\nwhich carve out cycloidal skipping orbits as shown in Fig-\nure 1. The period is T=2p=wc. The periodicity of the skip-\nping orbits is `=\u00002prSGs. From this solution, it is clear\nthat opposite spins will separate from one another along the\ny-axis. The average speed along the y-axis is vavg=`=T=\n\u0000gµBbs=2eB0=\u0000rSGwcs(and zero in x) which is identi-\ncal to the quantum calculation. This same average speed re-\nmains regardless of the initial position and velocity of the elec-\ntrons. For an unpolarized electron spin system, the behavior4\nis reminiscent of the spin Hall effect where a spin current is\nformed. However here, unlike the for the spin Hall effect, a\nlongitudinal charge current and its concomitant dissipation is\nunnecessary24.\nFor non-ballistic transport, in the spirit of the Drude model\nwe express the spin and charge dynamics in either the longi-\ntudinal or transverse geometry as\ndppp\ndt=\u0000e\u0012\nEEEe f fs+ppp\u0002BBB0\nm\u0013\n\u0000ppp\nt(19)\nwith Ee f f;i=gµB\n2e¶BZ;z\n¶ribeing an effective electric ﬁeld gener-\nated from a general Zeeman-only ﬁeld. This effective ﬁeld is\nconstant and uniform though for the linear gradient assumed\nthus far which ensures analytic solutions. A solution is read-\nily found for each spin orientation in z,s, in the steady state\nwhich gives for an unpolarized electron ensemble the second\nrank tensor of the spin current, ji;z:\nji;z=gµB\n2esii¶BZ;z\n¶ris+gµB\n2eeizksik¶BZ;z\n¶rks (20)\nwith the conductivity tensor\nˆsssc=0\n@sxx\u0000syx0\nsyxsxx 0\n0 0 szz1\nA (21)\nwhere\ns0=neµ;sxx=s0\n1+w2ct2;syx=sxxwct;szz=s0(22)\nandBBB0=B0ˆz. From this it is apparent that the charge current\nis zero, jc=j++j\u0000=0 but the spin current, js=j+\u0000j\u00006=0,\nis not.\nV . DISCUSSION\nNow the nuclear ﬁeld of Eq. (2) is used for the Zeeman-\nonly ﬁeld and we make estimates of the spin current. In the\nlongitudinal conﬁguration, with ˆB0jjˆz, Eq. (20) reduces to\njjjs=2s0EEEe f f=neµg\u0003µB\neB2\n0\nB2\n0+xB2\n`bnbˆB0=nµg\u0003µBbˆB0:\n(23)\nThis longitudinal spin current is plotted in Figure 2. The width\nof the curve in Figure 2 is governed by the local ﬁeld, B`.\nNuclear ﬁeld gradients could be produced in a variety of\nways. The simplest manner would be for the nuclear ﬁeld\nto be graded by the inhomogeneous electron spin polariza-\ntion arising from electron spin diffusion. In GaAs, the largest\npossible nuclear ﬁeld is \u001917 T which would correspond\nto efﬁcient dynamic nuclear polarization from highly polar-\nized electrons.16,21In practice, the maximum nuclear ﬁeld is\nsmaller; Chan et al. found it near 5 T.25At low temperatures,\nthe spin diffusion length in doped GaAs is on the order of 10\nµm. By ignoring additional nuclear spin diffusion, the decay\nlongitudinalB0||ˆzjs,z-2-10120246810B0(T)js(A/cm2)js,xjs,y== 0FIG. 2: Spin current density components versus applied magnetic\nﬁeld calculated in n-GaAs for a nuclear hyperﬁne gradient longitudi-\nnal (ˆz) to the applied ﬁeld. Parameters: t=0:4 ps, n\u00191016cm\u00003,p\nxB`=100 mT, bn=\u00001 T,b=10\u00003nm\u00001(corresponds to b\u0019\u00001\nmT/nm in a large ﬁeld), and g\u0003=\u00000:44.\njs,xjs,yjs,ztransverseB0||ˆz\n-2-1012-4-20246B0(T)js(A/cm2)\nFIG. 3: Spin current density components versus applied magnetic\nﬁeld calculated in n-GaAs for a nuclear hyperﬁne gradient transverse\n( ˆx) to the applied ﬁeld. Parameters: t=0:4 ps, n\u00191016cm\u00003,p\nxB`=100 mT, bn=\u00001 T,b=10\u00003nm\u00001(corresponds to b\u0019\u00001\nmT/nm in a large ﬁeld), and g\u0003=\u00000:44.\nof nuclear ﬁeld follows that of the electron spin. If we take\nthe maximum nuclear ﬁeld slightly above 5 T, the 1 =eﬁeld is\nabout 2 T over 10 µm which leads to b\u00190:2 mT/nm. Further\ncontrol of bmay be possible by controlling the electron spin\ndiffusion length with an electric ﬁeld.26,27\nTo ﬁnd the size of longitudinal spin current to be expected\ninn-GaAs, we estimate the the conductivity to be s0=neµ=\n2000=(Wm) with n=1016cm\u00003andµ=104cm2/Vs. The\neffective ‘electric ﬁeld’ is determined by\nEe f f=g\u0003µBb\n2e=(\u00000:44)(9:3\u000210\u000024J=T)\n2\u00021:6\u000210\u000019Cb (24)\nwhich computes to be 1 :3\u000210\u00005bJ/(T C) where the g-\nfactor for GaAs g\u0003=\u00000:44. Using b=0:2 mT/nm, we ﬁnd\nEe f f\u00193 V/m. The spin current would be 2 s0Ee f f=1:2\nA/cm2which is comparable to values measured in the spin5\nHall mechanism.28,29\nLarger linear effective ﬁeld gradients may be possible by\noptically orienting16spin with an appropriate optical grating.\nThrough the process of dynamical nuclear polarization, an ef-\nfective ﬁeld is created parallel to the applied ﬁeld with a trans-\nverse [Figure 1(a)] or longitudinal geometry. The ‘slope’ of\nthe linear grating will dictate the strength of borb. After\ngenerating the nuclear ﬁeld and allowing the electronic spin\nto relax, an unpolarized pump can excite carriers that will un-\ndergo the dynamics described herein. A pure electron spin\ncurrent will cross the sample. Note that the spin current is in-\ndependent of the spin polarization. Kerr or Faraday rotation\nspectroscopy may then resolve the opposite spins on either\nside of the pump beam’s spot. An alternate method of mea-\nsurement, which may avoid charge recombination of spin car-\nriers, is, after preparing the nuclear ﬁelds in the same manner,\nto have a polarized pump beam generate an imbalance of con-\nduction electron spins which then result in a charge current,\nproportional to the injected spin polarization, to be measured\nat opposite contacts.\nOur focus has been on the longitudinal spin current (as op-\nposed to the transverse spin current) since it is able to achieve\nlarger values in a broad ﬁeld range. For completeness, we dis-\nplay the transverse spin current in Figure 3. There are two\nﬁeld scales present: the narrow width is \u0018B`and the larger\nwidth scales with the momentum relaxation rate.VI. CONCLUSION\nIn this article, we have demonstrated how nuclear ﬁelds,\nwhich are effective magnetic ﬁelds that do not affect orbital\nmotion when uniform, induce spin and charge currents when\ngraded. Signiﬁcant nuclear ﬁelds (on order of Tesla) are com-\nmonly created in doped GaAs which offers the chance to ob-\nserve the effects described here. Managing gradients of these\nﬁelds remains to be seen; we suggest an optical means by\nwhich dynamic nuclear polarization is ﬁltered across a sam-\nple by selecting an appropriate optical grating. Our estimate\nof A/cm2spin current is similar to spin Hall currents measured\ninn-GaAs.\nVII. ACKNOWLEDGEMENTS\nThis work was supported in part by the Center for Emergent\nMaterials, an NSF MRSEC under Award No. DMR-1420451.\nNJH acknowledges additional support from the National Sci-\nence Foundation under Grant Numbers DMR-2014786 and\nDMR-2152540.\n\u0003Electronic address: harmon.nicholas@gmail.com\n†Electronic address: michael˙ﬂatte@mailaps.org\n1D. D. Awschalom, N. Samarth, and D. Loss, eds., Semiconductor\nSpintronics and Quantum Computation (Springer Verlag, Heidel-\nberg, 2002).\n2D. D. Awschalom and M. E. 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Weiss, Physical Review B 86, 205204 (2012)."    },    {        "title": "1810.03185v4.Tri_spin_dynamics_in_alkali_metal_noble_gas_NMR_gyroscope.pdf",        "content": "Tri-spin dynamics in alkali metal-noble gas NMR gyroscope\nGuobin Liu\u0003\nDepartment of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.\n(Dated: February 1, 2019)\nAlkali metal-noble gas NMR gyroscope is widely used for precision rotation measurement in funda-\nmental and applied physics. By numerically simulating the alkali-nuclear-nuclear tri-spin dynamics,\nwe investigate the dependence of gyroscope response on alkali spin relaxation time and nuclear spin\nmagnetization. We found additional resonance peaks appear due to a new source of instability,\nnamely the inherent multistability of tri-spin dynamics. The numerical simulation results agree\nwell with the recent experiment, enabling a better understanding and exploitation of the gyroscope\nsignal.\nI. INTRODUCTION\nA class of precision measurement in physics relies on\nmeasuring the rotational inertial e\u000bect of spin preces-\nsion, i.e., the measured Larmor frequency is shifted to\n!0=!\u0006\nR, with!the intrinsic Larmor frequency and\n\nRthe inertial rotation rate of the experimental setup\ncarrying the spin. NMR signal of long lived nuclear\nspin polarization is utilized to extract !0. Such NMR\ngyroscope or comagnetometry has been widely used for\nexperimental test of Lorentz invariance [1{3], search for\nCP-violating electric dipole moment [4], anomalous spin\ncouplings beyond the standard model [5], precision rota-\ntion sensor [6] and as unit sensor in the global network\nsearching for axion [7].\nOne kind of NMR gyroscope uses alkali metal-noble\ngas dual-spin ensemble in a thermal vapor cell, where al-\nkali spin serves both as a medium for polarization and as\nan embedded probe of the noble gas nuclear spin (over\nall electron spin vanishes due to full shell in electronic\ncon\fguration). The gyroscope signal can be greatly en-\nhanced due to the Fermi contact between alkali metal\nelectronic and noble gas nuclear spins [5, 8].\nThe simplest alkali metal-noble gas ( A-N) gyroscope is\nbased on dual spin comagnetometry [5], which works in\na well compensated zero magnetic \feld. In this case, the\ngyroscope signal is insensitive to any ambient magnetic\n\feld and only sensitive to the earth rotation or anomalous\ninteractions coupling to nuclear or electron spins. How-\never, periodic calibration for the compensation magnetic\n\feld is needed to maintain a zero compensation point [3],\nmaking a continuous measurement impossible.\nSpatial and temporal variations of the external \feld\nB0in which the spins precess limit gyroscope sensitivity.\nBy introducing an additional noble gas, one can measure\nsimultaneously the precession signal of two nuclear spins\noccupying the same volume [6, 9, 10], thus \n Rcan be\ndetermined without knowing B0via\n\nR=\r1!20\u0000\r2!10\n\r1\u0000\r2; (1)\n\u0003Email: gbliu17@gmail.comwith\r1;2the gyromagnetic ratio of nuclear spin N1and\nN2, respectively. This alkali metal-noble gas-noble gas\n(A-N1-N2) tri-spin NMR gyroscope can work in a \fnite\nnonzero magnetic \feld and thus loose the requirement\non maintaining a well controlled zero ambient magnetic\n\feld, making a long term continuous measurement pos-\nsible [10]. The long term continuous measurement is im-\nportant for practical applications where the calibration of\ncompensation \feld is unavailable or increasing the com-\nplexity of the apparatus if di\u000ecult. An additional bene\ft\nof the tri-spin gyroscope is its measurement independence\non the orientation of the gyroscope [10], compared to the\ndual spin version [5].\nAn alkali spin magnetometry usually measures dc or\nslowly varying magnetic \feld. Steep Zeeman resonance\nis utilized to monitor the change of \feld within the res-\nonance linewidth. In tri-spin NMR gyroscope, however,\nthe signal \feld comes from two nuclear magnetizations\nin motion and alkali spin probes it via coupling to nu-\nclear spins. Besides, from classical point of view, alkali\nspin relaxation time determines the Zeeman resonance\nlinewidth \u0001 \u0017and thus the dynamic range of alkali spin\nmagnetometry \u0001 Bvia\n\u0001B=\u0001\u0017\n\rA=1\n\u0019\rATA: (2)\nWith typical alkali gyromagnetic ratio \rA\u00181 MHz/G and\nspin relaxation time TA\u00181 ms, the alkali spin magnetom-\netry has a \fnite dynamic range \u0001 B\u00180.3 mG. The signal\nof nuclear magnetization is typically at the same level or\nhigher [10, 11]. It is thus necessary to study the response\nof gyroscope and its dependence on alkali spin relaxation\ntime and nuclear magnetization.\nWe have noticed that recently Limes et al. observed\na multiple resonance spectrum, i.e., the so called \\cross-\nmodulation\" peaks in a Rb-He-Xe tri-spin system [10]\nand they concluded that a better understanding is neces-\nsary in order to reach the full sensitivity potential. Here\nwe try to understand the experimental phenomenon and\nrelated e\u000bects by a numerical simulation of the tri-spin\ndynamic system.arXiv:1810.03185v4  [physics.atom-ph]  31 Jan 20192\nII. COUPLED BLOCH EQUATIONS AND\nNUMERICAL SOLUTIONS\nWe simulate the spin dynamics of the A-N1-N2tri-spin\nsystem via coupled Bloch equations as was done for the\nA-Ndual-spin system by Kornack et al. [11].\nFirst we assume that there is a classical macroscopic\nmagnetization \feld Mfor both alkali vapor and noble\ngases after the spin-exchange optical pumping. Two fun-\ndamental magnetic interactions dominate the spin mag-\nnetization dynamics of the tri-spin system, the Zeeman\ninteraction between atomic spins and the bias \feld B0\nand spin-exchange collisional interaction between alkali\nvapor and noble gas spins. The \frst interaction is well\ndescribed by NMR Bloch equation. For the second one,\nSchaeferet al. have shown that EPR (electron param-\nagnetic resonance) and NMR frequency shift happen to\nalkali and noble gas spin resonance, respectively, due to\nthe short lived but enhanced spin-exchange interactions\n[8]. These frequency shifts can be simulated by introduc-\ning an additional e\u000bective magnetic \feld, a spin-exchange\n\feldBse. Due to the attractive force between alkali elec-\ntron and noble gas nucleus, Bseis enhanced compared to\nthe classical macroscopic magnetization \feld by a factor\n\u0015, thus we have\nBse=\u0015M=8\u0019\n3\u00140M; (3)\nwith\u00140the enhancement factor over the classical mag-\nnetic \feld due to the attraction of alkali electron wave\nfunction to noble gas nucleus [8, 11].\nBased on above assumption, we can thus add these\nspin-exchange \felds into the classical Bloch equation and\nform a set of coupled Bloch equations. We consider \frst\nthe basic con\fguration of tri-spin dynamics where all\nmagnetic \felds and magnetizations couple to each other.\nWe refer it as complete-coupling con\fguration, for which\nthe equations of motion can be written as\n@MA\n@t=\rA\nqMA\u0002[B0+\u00151MN1+\u00152MN2] +MA\n0^z\u0000MA\nqTA;\n@MN1\n@t=\r1MN1\u0002[B0+\u00151MA+\u00153MN2] +MN1\n0^z\u0000MN1\n[TN1\n2;TN1\n2;TN1\n1];\n@MN2\n@t=\r2MN2\u0002[B0+\u00152MA+\u00153MN1] +MN2\n0^z\u0000MN2\n[TN2\n2;TN2\n2;TN2\n1];\n(4)\nwhere\rwith subscripts are gyromagnetic ratios for cor-\nresponding spins. MAandMN1=N2are the alkali vapor\nand noble gas magnetizations, respectively. \u00151;\u00152and\n\u00153are enhancement coe\u000ecients for A-N1,A-N2andN1-\nN2spin couplings, respectively. We assume that alkali\nspin has the same timescale for both longitudinal and\ntransverse relaxation times, termed as alkali spin relax-\nation timeTA.qin the \frst equation is the slowing down\nfactor depending on the alkali spin polarization, ranging\nfrom 6 for low polarization to 4 for high polarization [12].\nIn the square brackets of last two equations, TN1=N2\n2 andTN1=N2\n1 represent the transverse ( Mx;y) and longitudinal\n(Mz) relaxation times, respectively.\nSo far, analytical solution does not exist for the three-\nbody dynamical system, except for some special cases\n[13]. We thus resort to numerical solutions and use the\nembedded solver ODE45 of MATLAB software to solve\nthe coupled Bloch equations. We assign speci\fc species\nfor the theA-N1-N2spin system, for example, rubidium-\n87 forA, helium-3 for N1and xenon-129 for N2. Typical\nvalues for gyroscope are used for simulation [10, 11]. For\nexample,3He/129Xe spin magnetization \feld felt by alkali\nspin is\u001820\u0016G-1 mG and87Rb spin magnetization felt by\nnuclear spin is\u001820\u0016G. We assume that N1andN2initial\nmagnetization \felds are equal, termed as \u0015MN\n0. We have\nchecked that the di\u000berence between N1andN2initial\nmagnetization only changes the relative strength of the\ntwo resonance signal but not the general results as will\nbe described later. The dc bias \feld B0is along the z\naxis. The initial He and Xe spin magnetizations are set\nalong and against the yaxis, respectively. The initial\nalkali spin magnetization is set along the zaxis. The\nT1andT2relaxation times are taken 1 hour and 1000 s\nfor He spin magnetization, 1000 s and 80 s for Xe spin\nmagnetization. The enhancement factor \u00140is taken as\n5, 500 and -0.011 for Rb-He [14], Rb-Xe [15] and He-Xe\n[16] spin exchange interaction, respectively.\n0 0.2 0.4 0.6 0.8 1\nTimes (s)-0.1-0.0500.050.1MxA(t) ( 7G)6MN\n0=20 7G 6MN\n0=200 7G 6MN\n0=500 7G\n0510152025303540\nFourier frequency (Hz)10-810-610-410-2MxA ( 7G)6MN\n0=20 7G\n6MN\n0=200 7G\n6MN\n0=500 7G7.88.516.2816.3(a)\n(b)\nFIG. 1. Gyroscope signal MA\nxversus time (a) and frequency\n(b) for di\u000berent initial nuclear magnetizations simulated for\nthe complete coupling tri-spin system, with alkali spin relax-\nation timeTA=1 ms, meaning a dynamic range \u0001 B=0.45 mG\ngiven\rA=0.7 MHz/G for87Rb, based on Eq. 2.\nThe sensing medium of NMR gyroscope is alkali spin\npolarization, whose transverse component is read out\nby optical method as the gyroscope signal [5, 10, 11].\nFig. 1(a) shows gyroscope signal MA\nx(t) over one second,\nfor di\u000berent initial nuclear magnetizations \u0015MN\n0. Due to3\na much shorter relaxation time, the alkali spin can adia-\nbatically follow up the precessing nuclear magnetization.\nThe alkali spin follows the slow precession of two nuclear\nspins around a dc bias \feld, resembling a dual-frequency\noscillation. The gyroscope signal increases with the nu-\nclear magnetization. A slow drift appears in the long\nterm evolution in Fig. 1(a), which corresponds to a dc\ncomponent in frequency spectrum shown in Fig 1(b). The\nspectrum is Fourier transformation of oscillation over 200\nseconds.\nThe main resonance peaks at around 8.21 Hz and\n16.29 Hz are the129Xe and3He spin resonances. Due\nto spin exchange interaction [8], the nuclear magneti-\nzation cause a frequency shift to alkali spin precession\nand vice versa. The Xe-peak is obviously shifted (rela-\ntive to intrinsic frequency of 5.895 Hz at B0=jB0j=5 mG)\ndue to the strong Rb-Xe spin exchange interaction with\n\u0014Rb\u0000Xe\n0\u0018500, while the He-peak is only slightly shifted\n(relative to 16.225 Hz at B0=5 mG) due to \u0014Rb\u0000He\n0\u00185.\nTheN1-N2spin coupling is negligible due to much\nsmaller\u0014Xe\u0000He\n0 . The positions of main resonance peaks\ndepend on nuclear magnetization (insets of Fig. 1(b)) and\nrelaxation time (see Fig. 7(b) in the Appendix). In con-\nclusion, the complete coupling tri-spin system is unsuit-\nable for precision NMR gyroscope.\nWe consider an ideal con\fguration of tri-spin dynam-\nics where: 1) Nuclear spins precess solely due to the bias\n\feld along the zaxis; 2) Alkali spin precesses solely due\nto the nuclear magnetizations. Technically, the NMR\nfrequency shift of nuclear spins caused by alkali spin can\nbe averaged out by the so called dynamical decoupling\nmethod [9, 10]. The precession of alkali spins around the\ndc bias \feld can also be averaged out by fast periodic\nmodulation on the optical pumping [10]. For this par-\ntial coupling con\fguration, a new set of coupled Bloch\nequations can be written as\n@MA\n@t=\rA\nqMA\u0002[\u00151MN1+\u00152MN2] +MA\n0^z\u0000MA\nqTA;\n@MN1\n@t=\r1MN1\u0002[B0+\u00153MN2] +MN1\n0^z\u0000MN1\n[TN1\n2;TN1\n2;TN1\n1];\n@MN2\n@t=\r2MN2\u0002[B0+\u00153MN1] +MN2\n0^z\u0000MN2\n[TN2\n2;TN2\n2;TN2\n1];(5)\nFig. 2 shows the gyroscope signal simulated for the par-\ntial coupling tri-spin system with the same parameters\nas for the complete coupling con\fguration. In temporal\nsignal, although the phase of dual-frequency oscillation\ndistorts as the nuclear magnetization increases, the over-\nall oscillation is dominated by two frequencies. This is\nproved by Fourier analysis of oscillations over 200 sec-\nonds, as shown in Fig. 2(b), two main resonance peaks\nindependent of nuclear magnetization exist at 5.895 Hz\nand 16.225 Hz (see inset). We also show that they are in-\ndependent of alkali spin relaxation time (see Fig. 8 in the\nAppendix). In conclusion, the partial coupling tri-spin\nsystem can ensure a precision NMR gyroscope based on\nEq. 1.Fourier analysis also shows that additional reso-\nnance peaks show up in the spectrum. A minor\n`associate' peak near the main Xe-peak appear \frst\nwhen the nuclear magnetization is small, i.e., when\n\u0015MN\n0=20\u0016G\u001c\u0001B=0.45 mG for the dashed black line in\nFig. 2(b). As the nuclear magnetization increases, more\npeaks appear and grow with increasing \u0015MN\n0. This could\nbe harmful for gyroscope. For example, as the nuclear\nmagnetization approaches the dynamic range of alkali\nmagnetometry, the associated peak next to main reso-\nnance becomes comparable in amplitude to each other.\nIn this case, it could be practically problematic to \fgure\nout the true value of !0as the peaks are closely spaced\nin spectrum.\n0 0.2 0.4 0.6 0.8 1\nTimes (s)-0.4-0.200.20.4MxA(t) ( 7G)6MN\n0=20 7G 6MN\n0=200 7G 6MN\n0=500 7G\n0510152025303540\nFourier frequency (Hz)10-710-510-310-1MxA ( 7G)6MN\n0=20 7G 6MN\n0=200 7G 6MN\n0=500 7G\n5.895.9 16.2216.23(a)\n(b)\nFIG. 2. Gyroscope signal versus time (a) and frequency (b)\nfor di\u000berent initial nuclear magnetizations simulated for the\npartial coupling tri-spin system, with alkali spin relaxation\ntimeTA=1 ms.\nWe have noticed that it is di\u000ecult to measure accu-\nrately the nuclear spin polarization without a compli-\ncated apparatus [14, 15]. Alternatively, as we have shown\nhere, \ftting simulated gyroscope signal to experimental\ndata could be a good method to estimate quantitatively\nthe nuclear magnetization strength or alkali spin relax-\nation time in situ.\nIII. ORIGIN OF ADDITIONAL RESONANCE\nPEAKS\nFrom our simulations we noticed that additional res-\nonance peaks grow in quantity and amplitude as the\nnuclear magnetization or alkali spin relaxation time in-\ncreases. However, it is still unclear why these additional\npeaks are created from the very beginning. We attribute\nit to the instability of three-body dynamics rooted in4\nthe tri-spin ensemble. To prove this, we look at the dy-\nnamical trajectories of alkali spin precession and compare\nthem between the complete and the partial coupling con-\n\fgurations.\nFor the complete coupling con\fguration, Fig. 3 shows\nthe trajectory of alkali spin precession in the initial two\nseconds. Although both bias \feld and nuclear magne-\ntization apply torques to the alkali spin, the former is\nstronger than the latter. Thus the alkali spin is only\nslightly tilted away from the quantum axis along the z\naxis. The fast precession of alkali spin around the quan-\ntum axis \feld is dominant. The torque applied by nu-\nclear magnetization is small, acting on the alkali spin as\na perturbation or weak modulation. As the two nuclear\nspins have di\u000berent gyromagnetic ratios, the modulation\nis dual-frequency. Thus a double-ring structure forms in\nthe trajectory, as shown in Fig. 3.\nFIG. 3. Trajectory of alkali spin precession simulated for the\ncomplete coupling tri-spin system, with nuclear magnetiza-\ntion\u0015MN\n0=0.1 mG and alkali spin relaxation time TA=1 ms.\nThe reasons that we show only the initial two seconds\nof trajectory are the following: 1) The nuclear spin pre-\ncesses a few cycles within one second, so two seconds are\nenough to show the trend of dynamics in general; 2) At\nlonger times, alkali spin trajectory will \fll up the whole\nspace due to its everlasting changing phase in tri-spin dy-\nnamics. This will smear the main structure of trajectory,\nhindering us from understanding the essences of di\u000berent\ntri-spin dynamics.\nFor the range of nuclear magnetization studied here,\ni.e.,\u0015MN\n0from 20\u0016G to 0.5 mG, the tilting angle of al-\nkali spin is small and linearly proportional to \u0015MN\n0. In\nthis regime, the structure of trajectory does not change\nas nuclear magnetization increases (see Fig. 9 in the Ap-\npendix). The gyroscope signal measuring the transverse\nalkali magnetization, is small and increases linearly withnuclear magnetization, consistent with the simulation re-\nsults shown in Fig. 1.\nIn contrast, for the partial coupling tri-spin dynamics,\nthe trajectory is very di\u000berent. Here the bias \feld no\nlonger applies any torque to the alkali spin. The nuclear\nmagnetization applies a torque dominating the motion of\nalkali spin, which at \frst is tilted at a large angle away\nfrom the quantum axis after which is growing into a non-\nlinear region. Then the alkali-nuclear tri-spin coupling\ndominates the spin dynamics.\nIt's well known that three-body dynamics has a com-\nplex behavior, resulting in unpredictable long term dy-\nnamics or even chaos (dynamically instable, accompany-\ning with bistability or multistability) in some cases [17].\nThe dominance of tri-spin dynamics leads to a similar\ncomplex trajectory of alkali spin precession, as shown\nin Fig. 4. First, more rings are present (up right inset)\nand secondly, as the nuclear magnetization increases, the\nstructure of trajectory changes dramatically (see Fig. 10\nin the Appendix). One can notice that the complexity\nof trajectory is positively correlated to the number of\nadditional resonance peaks in spectrum.\nFIG. 4. Trajectory of alkali spin precession simulated for the\npartial coupling tri-spin system, with nuclear magnetization\n\u0015MN\n0=0.1 mG and alkali spin relaxation time TA=1 ms.\nThe alkali spin relaxation time plays an even more sig-\nni\fcant role in the complexity of tri-spin dynamics. Fig. 5\nshows the structure of trajectory changing dramatically\nasTAincrease by an order of magnitude. For the same\nnuclear magnetization as in Fig. 4, but with ten times\nlonger relaxation time, the alkali spin can \rip many times\nacross thexyplane (the equatorial plane z=0, up left in-\nset). The multiple crossovers with the equatorial plane is\ncorrelated again to the creation of more additional reso-\nnance peaks, as shown in Fig. 8 (see the Appendix).\nBased on the above analysis, we will expect that the5\nquantity and amplitude of additional resonance peaks\ngrow with increasing tilting angle of alkali spin\n\u0012=\rA\u0015MN\nxy(t)TA=q: (6)\nHowever, in principle these additional resonance peaks\nare inherent to the tri-spin system. In fact, they ap-\npear even for very small tilting angle, where the nu-\nclear magnetization is far below the dynamic range deter-\nmined by the alkali spin relaxation time. For example,\nthe aforementioned `associate' peak shown in Fig. 2(b),\nwhere\u0012is 2\u000eat most (when N1andN2spin magneti-\nzations are along the same direction) with \u0015MN\n0=20\u0016G\nandTA=1 ms.\n-505\n1010MzA ( 7G)\n1015\nMyA ( 7G)0\nMxA ( 7G)20\n0\n-10-10-10010\nMxyA ( 7G)01020MzA ( 7G)\n-10010\nMxA ( 7G)-10010MyA ( 7G)\nFIG. 5. Trajectory of alkali spin precession simulated for the\npartial coupling tri-spin system, with nuclear magnetization\n\u0015MN\n0=0.1 mG and alkali spin relaxation time TA=10 ms.\nWhile the tri-spin dynamics is inherently complex to\nbe expressed in any analytic form, we try to \fnd out some\nempirical rules from numerical simulations. We noticed\nthat there are two kinds of additional resonance peaks in\nspectrum. One kind appears at higher frequencies. The\nother kind appears as an associate peak closely spaced\nto the main peaks and to the \frst kind of additional\npeaks. We investigate the dependence of multiple reso-\nnance peaks on the bias \feld B0=jB0j. Fig. 6 shows the\nsimulated spectrum versus B0from Eq. 5. From the mea-\nsured positions of multiple peaks, we found the following\nformulas\n\u0001 = (\r1\u0000\r2)B0;\n\u000e= (d\r1=\r2e\r2\u0000\r1)B0;(7)\nwhere \u0001 is the di\u000berential frequency between two main\nresonance peaks of nuclear spin and \u000eis the distance be-\ntween the main peaks and their nearest associate peaks.\nd\r1=\r2eis rounding up to the next integer of the ratio\n0 5 10 15 20 2510-710-510-310-1\nB0=2 mG\n0 10 20 30 40 50 6010-710-510-310-1MxA ( 7G)B0=5 mG\n0102030405060708090\nFourier frequency (Hz)10-710-510-310-1\nB0=8 mGFIG. 6. Dependence of gyroscope spectral response on bias\n\feld simulated for the partial coupling tri-spin system, with\nnuclear magnetization \u0015MN\n0=0.1 mG and alkali spin relax-\nation timeTA=1 ms.\nbetween nuclear gyromagnetic ratios (assuming \r1>\r2).\nOne may notice that once the gyromagnetic ratios of two\nnuclear species are determined, the ratio \u0001/ \u000eis also set-\ntled. In the case of Rb-3He-129Xe system, \u0001/ \u000e\u00197.1241,\nmeaning about six additional resonance peaks if equally\nspaced in between Xe and He main peaks.\nIV. RESTRICTED PLANAR MOTION AND\nMULTIPLE RESONANCE PEAKS\nBased on the partial coupling con\fguration, we further\nconsider a particular case where the alkali spin precession\nis restricted within the a plane due to some practical\nreasons. For example, the restricted planar motion of\nspin precession could be e\u000bectively realized, by a serie\nof experimental techniques such the synchronous optical\npumping, NMR spin locking, or a combination of them\nor their slightly modi\fcated versions.\nActually, we have noticed that the modulated optical\npumping and the pulse train technique in the experiment\nmay e\u000bectively realize such a restricted planar motion for\nalkali spin polarization and lead to its sensitivity mainly\nto the nuclear magnetization MN\ny[10]. Therefore, any\ntorque applied to alkali spin by nuclear magnetization\nlying in the xorzaxis will be averaged out e\u000bectively,\nso the \frst equation of Eqs. 5 can be simpli\fed by elim-\ninating terms containing MN\nxorMN\nz. The new set of6\nequations of motion now becomes\n@MA\nx\n@t=\u0000\rA\nqMA\nz(\u00151MN1y+\u00152MN2y)\u0000MA\nx\nqTA;\n@MA\ny\n@t=\u0000MA\ny\nqTA;\n@MA\nz\n@t=\rA\nqMA\nx(\u00151MN1y+\u00152MN2y) +MA\n0\u0000MA\nz\nqTA;\n@MN1\n@t=\r1MN1\u0002[B0+\u00153MN2] +MN1\n0^z\u0000MN1\n[TN1\n2;TN1\n2;TN1\n1];\n@MN2\n@t=\r2MN2\u0002[B0+\u00153MN1] +MN2\n0^z\u0000MN2\n[TN2\n2;TN2\n2;TN2\n1]:(8)\n0 5 10 15 20 25 30 35\nFourier frequency (Hz)10-510-310-1MxA ( 7G)\n0 5 10 15 20 25 30 35\nFourier frequency (Hz)10-510-310-1MxA ( 7G)(a) without noise\n(b) with noise\nFIG. 7. Gyroscope spectral response simulated for planar-\nrestricted-motion of the Rb-3He-129Xe tri-spin system. (a)\nand (b) are results without and with white Gaussian noise\n(signal to noise ratio is 106), respectively. The bias \feld is\nB0=5 mG. The He and Xe magnetization \felds are 0.5 mG\nand 1.5 mG, respectively. The alkali spin relaxation time is\n1 ms.\nAn example of the simulated Fourier spectrum by solv-\ning the above equations is shown in Fig. 7, where six ad-\nditional resonance peaks appear between the He and Xe\nresonance peaks, in agreement with previous prediction\naccording to Eq. 7. Especially, to eliminate the residual\nphase information in Fig. 7(a), we plot Fig. 7(b) a Fourier\nspectrum mixed with realistic level of noise in the exper-\niment [10]. When comparing the simulation spectrum\nwith the experimental one, one shall remember that there\nmight be slightly di\u000berence in the exact positions of the\nmain two nuclear spin resonance peaks as the value of\nmagnetic \feld B0in the experiment is not equal to 5 mG\nas it is in the simulation. Although the parameters can\nnot be exactly \ftted to the experiment (for example, the\nHe and Xe magnetization \felds might be a bit higher\nthan that expected in experiment), to a good con\fdencelevel, we can conclude that the multiple peaks spectrum\nagrees very well with the \\cross modulation\" spectrum\nobserved in experiment [10].\nInterestingly, the multi-frequency spectrum of the tri-\nspin system on the condition of restricted planar mo-\ntion resembles the multi-period solutions of the classical\nthree-body dynamics for planar orbiting planets in as-\ntrophysics [13]. In this sense, we can view the multiple\nfrequency spectral components as multiple orbits of spin\nprecession. For example, in Fig. 8, we plot the trajec-\ntory of alkali spin precession for the initial two seconds\nin a restricted xzplane, according to Eq. 8. One can \fnd\nclearly spaced clusters of spin precession orbits. These\nmultiple orbits correspond to spin precession with di\u000ber-\nent periods or frequencies than the two main resonance\nfrequencies, as already shown in Fig. 7. We may also\nnotice that the precession orbits in Fig. 8 show more pat-\nterns or asymmetry than that in Fig. 4 and it is in consis-\ntent with a richer spectral composition while comparing\nthe spectra.\n-15 -10 -5 0 510 15 20\nMxA ( 7G)-15-10-505101520MzA ( 7G)-20020\nMxA ( 7G)-101MyA ( 7G)\n-10022\nMzA ( 7G)-101MyA ( 7G)\nFIG. 8. Trajectory of alkali spin precession simulated for\nthe restricted planar motion of partial coupling con\fguration.\nTwo insets at bottom left and top right corners show a zero\nMA\nycomponent due to the restricted alkali spin precession\nwithin the xzplane. The He and Xe initial magnetization\n\felds and the alkali spin relaxation time are the same as in\nFig. 7.\nThe multi-frequency spectrum and multiple orbits of\nspin precession represent a new kind of instability, which\nis in fact a multistability with in\fnite number of spin\nprecession orbits due to the everlasting changing mode\nof motions in three-body dynamics. As long as we work\nwith such an A-N1-N2tri-spin system in the partial cou-\npling con\fguration (in order to remove the noble gas\nNMR shift due to alkali metal polarization), we can not\nremove the additional peaks in the multi-frequency spec-7\ntrum from the root. However, we would like to point out\nthat the presence of the instability does not necessarily\nmean the tri-spin system is no more suitable for preci-\nsion measurement. Actually, to the best accuracy of the\nnumerical simulation, there is no evidence that the main\ntwo nuclear resonance peaks shift due to the presence of\nthe multiple resonance peaks.\nV. DISCUSSION AND CONCLUSION\nThe agreement between our numerical simulation and\nexperimental results leads us to the discovery of a new\nsource of dynamical instability in the alkali metal-noble\ngas comagnetometry or gyroscope. Di\u000berent from the A-\nNdual-spin comagnetometry or gyroscope working at a\ncompensated zero magnetic \feld, where the nonlinearity\nor dynamical instability happens due to the perturbation\nof an external large angle excitation pulse [11], here in\ntheA-N1-N2tri-spin system, the nonlinearity lies in the\ninternal instability of three-body dynamics.\nThe methods dealing with the instability are di\u000berent.\nFor the dual-spin gyroscope, one shall avoid or shield\nperturbation by external intense pulse of electromagnetic\n\feld in order to not driving the dual-spin system into its\ndynamically instable regime. For the tri-spin gyroscope,\nthe only concern is that one shall make a compromise\nbetween large enough gyroscope signal and small enough\ntilting angle \u0012according to Eq. 6. As long as we can dis-\ntinguish the main two nuclear spin resonance peaks out\nof the spectrum with crowded multiple peaks, the tri-spin\nensemble serves as a good system for precision rotation\nmeasurement according to Eq. 1. In this sense, the nu-\nmerical simulation answers the concern quantitatively.\nIn conclusion, we have studied the spin dynamics at\ndi\u000berent coupling con\fgurations for the alkali metal-\nnoble gas tri-spin NMR gyroscope. We report a new\nsource of instability for the tri-spin NMR gyroscope, orig-\ninating from the instability of three-body dynamics. The\nnumerical simulation results agree well with recently re-\nported experimental spectrum and enable a quantitative\nunderstanding of its complex structure. These \fndings\ncan be helpful for exploiting the full potential of the tri-\nspin NMR gyroscope.\nThe author would like to thank Dong Sheng and Mark\nLimes for helpful discussions. The author would also like\nto thank Andrei Ben-Amar Baranga for kind suggestions\nduring the revision of the manuscript.\nVI. APPENDIX\nWe also investigate the gyroscope response at di\u000berent\nalkali spin relaxation times. Fig. 9 and 10 show gyroscope\nsignal simulated for the complete and the partial coupling\ntri-spin systems, respectively.\n0 0.2 0.4 0.6 0.8 1\nTimes (s)-0.1-0.0500.050.1MxA(t) ( 7G)TA=0.1 ms TA=1 ms TA=10 ms\n0510152025303540\nFourier frequency (Hz)10-710-510-310-1MxA ( 7G)TA=0.1 ms TA=1 ms TA=10 ms\n88.216.2816.3(a)\n(b)FIG. 9. Gyroscope signal versus time (a) and frequency (b)\nfor di\u000berent alkali spin relaxation times simulated for the\ncomplete coupling tri-spin system, with nuclear magnetiza-\ntion\u0015MN\n0=0.1 mG.\n0 0.2 0.4 0.6 0.8 1\nTimes (s)-0.4-0.200.20.4MxA(t) ( 7G)TA=0.1 ms TA=1 ms TA=10 ms\n0510152025303540\nFourier frequency (Hz)10-710-510-310-1MxA ( 7G)TA=0.1 ms TA=1 ms TA=10 ms\n5.895.9\n16.2216.23(a)\n(b)\nFIG. 10. Gyroscope signal versus time (a) and frequency\n(b) for di\u000berent alkali spin relaxation times simulated for the\npartial coupling tri-spin system, with nuclear magnetization\n\u0015MN\n0=0.1 mG.\nWe also simulated the trajectories of alkali spin pre-\ncession at stronger nuclear magnetization. Fig. 11 and\n12 show the trajectories for the complete and the partial\ncoupling tri-spin systems, respectively.8\nFIG. 11. Trajectory of alkali spin precession simulated for the\ncomplete coupling tri-spin system, with nuclear magnetiza-\ntion\u0015MN\n0=0.5 mG and alkali spin relaxation time TA=1 ms.\nFIG. 12. Trajectory of alkali spin precession simulated for the\npartial coupling tri-spin system, with nuclear magnetization\n\u0015MN\n0=0.5 mG and alkali spin relaxation time TA=1 ms.\n[1] S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab,\nand E. N. Fortson, Phys. Rev. Lett. 57, 3125 (1986).\n[2] C. Gemmel, W. Heil, S. Karpuk, K. Lenz, Y. Sobolev,\nK. Tullney, M. Burgho\u000b, W. Kilian, S. Knappe-\nGr uneberg, W. M uller, A. Schnabel, F. Seifert,L. Trahms, and U. Schmidt, Phys. Rev. D 82, 111901\n(2010).\n[3] M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin,\nand M. V. Romalis, Phys. Rev. Lett. 107, 171604 (2011).\n[4] M. A. Rosenberry and T. E. Chupp, Phys. Rev. Lett. 86,9\n22 (2001).\n[5] T. W. Kornack, R. K. Ghosh, and M. V. Romalis, Phys.\nRev. Lett. 95, 230801 (2005).\n[6] E. A. Donley, 2010 IEEE Sensors , 17 (2010).\n[7] D. F. Jackson Kimball, D. Budker, J. Eby, M. Pospelov,\nS. Pustelny, T. Scholtes, Y. V. Stadnik, A. Weis, and\nA. Wickenbrock, Phys. Rev. D 97, 043002 (2018).\n[8] S. R. Schaefer, G. D. Cates, T.-R. Chien, D. Gonatas,\nW. Happer, and T. G. Walker, Phys. Rev. A 39, 5613\n(1989).\n[9] A. Korver, D. Thrasher, M. Bulatowicz, and T. G.\nWalker, Phys. Rev. Lett. 115, 253001 (2015).\n[10] M. E. Limes, D. Sheng, and M. V. Romalis, Phys. Rev.\nLett.120, 033401 (2018).\n[11] T. W. Kornack and M. V. Romalis, Phys. Rev. Lett. 89,253002 (2002).\n[12] I. M. Savukov and M. V. Romalis, Phys. Rev. A 71,\n023405 (2005).\n[13] M. \u0014Suvakov and V. Dmitra\u0014 sinovi\u0013 c, Phys. Rev. Lett. 110,\n114301 (2013).\n[14] M. V. Romalis and G. D. Cates, Phys. Rev. A 58, 3004\n(1998).\n[15] Z. L. Ma, E. G. Sorte, and B. Saam, Phys. Rev. Lett.\n106, 193005 (2011).\n[16] M. Limes, N. Dural, M. Romalis, E. Foley, T. Kornack,\nA. Nelson, and L. Grisham, arXiv preprint (2018),\narXiv:1805.11578.\n[17] R. Wiebe and L. Virgin, Chaos: An Interdisciplinary\nJournal of Nonlinear Science 22, 013136 (2012)."    },    {        "title": "1509.07949v1.Spin_Dynamics_and_Relaxation_in_Graphene_Dictated_by_Electron_hole_Puddles.pdf",        "content": "Spin dynamics and relaxation in graphene dictated by electron-hole puddles\nDinh Van Tuan1, Frank Ortmann,2, Aron W. Cummings1, David Soriano1and Stephan Roche,1;3\n1ICN2 - Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Spain\n2Institute for Materials Science, Dresden Center for Computational Materials Science,\nTechnische Universit ¨at Dresden, 01062 Dresden, Germany\n3ICREA, Instituci ´o Catalana de Recerca i Estudis Avanc ¸ats, 08070 Barcelona, Spain\n(Dated: October 13, 2018)\nThe understanding of spin dynamics and relaxation mechanisms in clean graphene and the upper time and\nlength scales on which spin devices can operate are prerequisites to realizing graphene spintronic technolo-\ngies. Here we theoretically reveal the nature of fundamental spin relaxation mechanisms in clean graphene\non different substrates with spin-orbit Rashba ﬁelds as low as a few tens of \u0016eV . Spin lifetimes ranging from\n50 picoseconds up to several nanoseconds are found to be dictated by substrate-induced electron-hole charac-\nteristics. A crossover in the spin relaxation mechanism from a Dyakonov-Perel type for SiO 2substrates to a\nbroadening-induced dephasing for hBN substrates is described. The energy dependence of spin lifetimes, their\nratio for spins pointing out-of-plane and in-plane, and the scaling with disorder provide a global picture about\nspin dynamics and relaxation in ultraclean graphene in presence of electron-hole puddles.\nPACS numbers: 72.80.Vp, 73.63.-b, 73.22.Pr, 72.15.Lh, 61.48.Gh\nThe tantalizing prospect of graphene spintronics was initi-\nated by Tombros and coworkers [1], who ﬁrst reported long\nspin diffusion length in large area graphene. The small spin-\norbit coupling (SOC) in carbon, plus the absence of a hyper-\nﬁne interaction, suggested unprecedented spin lifetimes ( \u001cs)\nat room temperature (from \u0016s to ms) [2, 3, 4, 5, 6, 7].\nHowever, despite signiﬁcant progress in improving\ngraphene quality, resolving contact issues, and reducing sub-\nstrate effects [1, 8, 9, 10, 11, 12, 13, 14, 15], the measured\n\u001csare orders of magnitude shorter, even for high-mobility\nsamples. Extrinsic sources of SOC, including adatoms\n[16, 17, 18, 19] or lattice deformations [20, 21], have been\nproposed to explain this discrepancy. Moreover, the na-\nture of the dominant spin relaxation mechanism in graphene\nis elusive and debated. The conventional Dyakonov-Perel\n(DP) [22] and Elliot-Yafet (EY) [23] mechanisms, usually\ndescribing semiconductors and disordered metals, remain\ninconclusive in graphene because neither effect can con-\nvincingly reproduce the observed scaling between \u001csand\nthe momentum relaxation time \u001cp[8, 11]. Although gen-\neralizations of both mechanisms have been proposed, they\ndo not allow an unambiguous interpretation of experiments\n[6, 20, 21, 24, 25].\nIt should be noted that the achieved room-temperature\nspin lifetime in graphene is already long enough for the ex-\nploration of spin-dependent phenomena such as the spin Hall\neffect [26, 27], or to harness proximity effects as induced for\ninstance by magnetic oxides [28] or semiconducting tung-\nsten disulphide [29]. However, a comprehensive picture of\nspin dynamics of massless Dirac fermions in presence of\nweak spin-orbit coupling ﬁelds is of paramount importance\nfor further exploitation and manipulation of spin, pseudospin\nand valley degrees of freedom [7, 30, 31, 32].\nIn this study, we show numerically that a weak uniform\nRashba SOC (tens of \u0016eV), induced by an electric ﬁeld orthe substrate, yields spin lifetimes from 50 ps up to sev-\neral nanoseconds. The dominant spin relaxation mechanism\nis shown to be dictated by long range potential ﬂuctuations\n(electron-hole puddles) [33]. For graphene on a SiO2sub-\nstrate, such disorder is strong enough to interrupt the spin\nprecession driven by the uniform Rashba ﬁeld, resulting in\nmotional narrowing and the DP mechanism. We also ﬁnd the\nratio\u001c?\ns=\u001ck\ns'1=2, demonstrating the anisotropy of the in-\nplane Rashba SOC ﬁeld. For the case of a hexagonal boron\nnitride (hBN) substrate, the role of electron-hole puddles is\nreduced to an effective energy broadening and the spin life-\ntime is limited by pure dephasing [34, 35]. These situations,\nhowever, share a common ﬁngerprint – a M-shape energy\ndependence of \u001csthat is minimal at the Dirac point. Taken\ntogether, our results provide deeper insight into the funda-\nmentals of spin lifetimes in graphene dominated by electron-\nhole puddles.\nResults\nDisorder and Spin dynamics.\nElectron-hole puddles are real-space ﬂuctuations of the\nchemical potential, induced by the underlying substrate,\nwhich locally shift the Dirac point [33, 36, 37]. Since\nmeasured transport properties usually result from an aver-\nage around the charge neutrality point, it is generally dif-\nﬁcult to access the physics at the Dirac point. As shown\nby Adam and coworkers [33], electron-hole puddles can be\nmodeled as a random distribution of long range scatterers,\nV(r) =PN\nj=1\u000fjexp[\u0000(r\u0000Rj)2=(2\u00182)], where\u0018= 10\nand30nm denote the effective puddle ranges for SiO2and\nhBN substrates, respectively [36, 38], and \u000fjis randomly\nchosen within [\u0000\u0001;\u0001]. Based on experimental data, typicalarXiv:1509.07949v1  [cond-mat.mes-hall]  26 Sep 20152\nimpurity densities are ni= 1012cm\u00002(0.04%) for SiO2\nandni= 1011cm\u00002(0.004%) for hBN substrates [36, 39].\nFrom such information, we can tune \u0001to obtain suitable\ndisorder proﬁles for the onsite energy of the \u0019-orbital. Fig.\n1 (main frame) shows the onsite energy distribution corre-\nsponding to hBN and SiO2substrates, where the Gaussian\nproﬁles give standard deviations of \u001b= 5:5and56meV , re-\nspectively. This allows us to extract \u0001 = 50 meV for SiO2\nand\u0001 = 5 meV for hBN. The inset of Fig. 1 shows the en-\nergy landscape for a sample with 0.04% Gaussian impurities\n(SiO2case).\nFIG. 1: Onsite energy distribution of the carbon atoms in the\ngraphene sample, which mimics the chemical potential induced by\nhBN (green) and SiO 2(black) substrates together with their Gaus-\nsian ﬁtting lines. Inset: Real space vizualization of the energy\nlandscape for a graphene sample with 0:04% Gaussian impurities\n(SiO 2case).\nTo fully characterize the role of electron-hole puddles, we\nevaluate the transport time \u001cpusing a real-space order-N ap-\nproach, which computes the diffusion coefﬁcient D(E;t).\nWe extract\u001cpfrom the saturation of D(E;t)since\u001cp=\nDmax(E)=2v2\nF(E)[40]. For numerical convenience, the\ncalculations are made using \u0001 = 0:1\r0(with\r0the nearest\nneighbor hopping parameter), thus in absence of intervalley\nscattering [41], while the ﬁnal values for hBN and SiO2sub-\nstrates are extrapolated from the numerical results using the\nscaling law\n\u001cp\u0018p\n\u0019n\u0003\u0018\nK0e\u0019n\u0003\u00182\nI(\u0019n\u0003\u00182); (1)\nwhereI1(x)is the modiﬁed Bessel function of the ﬁrst\nkind,K0= 40:5ni(\u0001=t)2(\u0018=p\n3a)4is a dimensionless pa-\nrameter dictating the strength of the Gaussian potential, and\nthe carrier density n\u0003is modiﬁed from the pristine graphene\n0 50 100 150 200\nt (ps)-1-0.8-0.6-0.4-0.200.20.40.60.81Pα (t)\n-400 -200 0 200 400\nE (meV)0.1110100100010000τp (ps)α=||, SiO2\nα=⊥,\nα=⊥, hBNSiO2hBN\nTΩ=π h_ /λR\na) b)SiO2FIG. 2: (a) Transport times for graphene on SiO 2and hBN sub-\nstrates (solid black and red curves, respectively). The dashed line\nshows the spin precession time. (b) Time-dependent spin polariza-\ntion for out-of-plane (solid red line) and in-plane (solid black line)\nspin injection for the SiO 2substrate, plus the ﬁts to the exponential\ndamping (dashed lines). The blue curves show the same informa-\ntion for the hBN substrate with out-of-plane injection.\ndensitynbyn\u0003=jnj+K0\n2\u00192\u00182[33, 42, 43]. The computed\n\u001cpare shown in Fig. 2(a) for both substrates. For SiO2,\u001cp\nis on the order of a few ps, while for hBN \u001cpis more than\ntwo orders of magnitude larger. The spin precession time,\nT\n=\u0019\u0016h=\u0015R, is shown for comparison.\nSpin dynamics and lifetimes in the presence of electron-hole\npuddles.\nWe now analyze the spin dynamics for puddles corre-\nsponding to the SiO2and hBN substrates. The blue curve in\nFig.2(b) shows the time-dependent spin polarization for the\nhBN substrate ( ni= 0:004% ) at the Dirac point for an initial\nout-of-plane polarization, PhBN\n?(t)(see Methods). The po-\nlarization exhibits oscillations with period T\n=\u0019\u0016h=\u0015R'\n55ps, corresponding to the spin precession induced by the\nRashba ﬁeld. Simultaneously, the polarization decays in\ntime, and by ﬁtting PhBN\n?(t) = cos (2\u0019t=T \n)e\u0000t=\u001cs, both\nT\nand the spin relaxation time \u001cscan be evaluated.\nFig.2(b) also shows PSiO 2\u000b(t)for the SiO2substrate (ni=\n0:04%) with initial spin polarization in-plane ( \u000b=k) and\nout-of-plane ( \u000b=?). In contrast to the hBN case, for\nwhichPhBN\n?(t)exhibits signiﬁcant precession, the disorder\nstrength of electron-hole puddles for SiO2is sufﬁcient to in-\nterrupt spin precession. As a result, the polarization for SiO2\nis better ﬁt with PSiO 2\nk=?(t) =e\u0000t=\u001cs. The absence of preces-\nsion forPSiO 2\n?(t)compared to PhBN\n?(t)is consistent with\nthe ratio between transport time and precession frequency,\nsince\u001cSiO 2p=T\n\u001c1whereas\u001chBN\np=T\n>1.\nTo scrutinize the origin of the dominant relaxation mecha-\nnism, we ﬁrst examine the spin lifetimes \u001csfor the SiO2case3\n50100150200τs⊥ (ps)\n-400 -200 0 200 400\nE (meV)0200400600800τs⊥ (ps)\n0.004%\n0.016%\n-400 -200 0 200 400\nE (meV)100150200250300350400τs|| (ps)\n0.04%\n0.08%\n0.16%c) a)\nb)SiO2 hBN ⊥\n||⊥\nFIG. 3: Spin lifetimes for out-of-plane (a) and in-plane (b) spin\ninjection for SiO 2substrate at impurity densities of 0.04% (black\nsolid curves), 0.08% (red dashed curves), and 0.16% (blue dotted\ncurves ). (c) Spin lifetime with out-of-plane spin injection for the\nhBN substrate at impurity densities of 0.004% (black curve) and\n0.016% (red curve).\nwhen rotating spin polarization (out-of-plane vs. in-plane),\nand varying the density of impurities ( 0:04%,0:08%, and\n0:16%). Fig.3 shows the extracted \u001csin the out-of-plane (a)\nand in-plane (b) cases. The energy dependence of \u001csex-\nhibits an M-shape increasing from a minimum at the Dirac\npoint, with a saturation and downturn of \u001csforE\u0015200\nmeV . The values of \u001csrange from 50 to 400 ps depending on\nthe initial polarization and impurity density. We observe an\nincrease of\u001cswithni, which shows that a larger scattering\nstrength reduces spin precession and dephasing, resulting in\na longer spin lifetime, as described by the so-called motional\nnarrowing effect [44]. Additionally, the ratio \u001c?\ns=\u001ck\ns(not\nshown) changes from 0.3 to 0.45 when niis varied from\n0:04% to0:16%. Such behavior is expected when enhanced\nscattering drives more randomization of the direction of the\nRashba SOC ﬁeld, which ultimately yields \u001c?\ns=\u001ck\ns= 0:5in\nthe strong disorder limit [2, 3]. These results are fully con-\nsistent with the DP spin relaxation mechanism [20, 21, 44].\nFig.3(c) shows \u001c?\nsfor the hBN substrate ( ni= 0:004%\nand0:016% ) where a similar M-shape is observed. While\n\u001c?\ns(BN)is similar to \u001c?\ns(SiO 2)near the Dirac point, it is\nmuch larger at higher energies, reaching nearly 1 ns (for\n\u0015R= 37:4\u0016eV). A striking difference is that the scaling\nof\u001cswithniis opposite to that of the SiO2case, with an in-\ncrease in puddle density resulting in a decrease in \u001cs, which\nindicates a different physical origin. For hBN, this behav-\nior is reminiscent of the EY mechanism, but we will argue\nbelow that its origin is a different one.Crossover in spin relaxation behavior for hBN and SiO 2\nsubstrates.\nFig.4 provides a global view of our results, where we plot\n\u001csvs.1=\u001cpfor the SiO2and hBN substrates (black and\nred symbols respectively) at the Dirac point and at E=\n\u0000200 meV (closed and open symbols respectively). For\nlow defect densities (hBN substrate), \u001csdecreases strongly\nwith decreasing \u001cp. However, with increasing defect den-\nsity ( SiO2substrate) this trend reverses and \u001csscales al-\nmost linearly with 1=\u001cp, according to the DP relationship\n\u001cs=\u0017\u0001(T\n=2\u0019)2=\u001cp. AtE=\u0000200meV ,\u0017= 1, ﬁtting\nthe usual DP theory. At the Dirac point, the scaling is some-\nwhat weaker, with \u0017= 1=4. These results are reminiscent of\nthose summarized in Fig. 5(a) of Drogeler et al. [13], where\nspin lifetimes of graphene devices on SiO2scaled inversely\nwith the mobility, while devices on hBN appear to show the\nopposite trend.\nWhile the SiO2results of Fig.4 show DP behavior, the\nnature of the spin relaxation for weak electron-hole pud-\ndles is less clear. The fact that \u001csand\u001cpdecrease together\nsuggests the EY mechanism, but we ﬁnd \u001cs\u0014\u001cpnear the\nDirac point and \u001cs\u001c\u001cpat higher energies. This contrasts\nwith the usual picture of EY relaxation, where charge car-\nriers ﬂip their spin when scattering off impurities, giving\n\u001cs=\u001cp=\u000b, where\u000b\u001c1is the spin ﬂip probability [6].\nInstead, this situation matches that described in Ref. [44];\nwhen\u001cp> T \n, the spin precesses freely until phase infor-\nmation is lost during a collision, in analogy to the collisional\nbroadening of optical spectroscopy. More collisions result\nin a greater loss of phase, reducing \u001cswith decreasing \u001cp.\nWe verify this by removing the real-space disorder (setting\n\u0001 = 0 ) and modeling the electron-hole puddles with an ef-\nfective Lorentzian energy broadening \u0011\u0003. The results are\nshown in Fig.4 (main frame, blue dashed line), where we\nplot\u001csvs.\u0011\u0003atE=\u0000200 meV (top axis). For small\n\u0011\u0003, the scaling matches well with the real-space simulations\nof hBN, indicating that the puddles can be represented as\na uniform energy broadening (See supplementary material).\nLarger values of \u0011\u0003lead to stronger mixing of different spin\ndynamics and \u001cssaturates at very large \u0011\u0003. There, the scal-\ning of\u001csvs.\u0011\u0003clearly fails to replicate the DP behavior seen\nin the real-space simulations, since the effective broadening\nmodel does not induce the momentum scattering necessary\nfor motional narrowing [44].\nWe ﬁnally explain the downturn of \u001csat the high energy\nwings of the M-shaped \u001csbehavior in the hBN case. We\ncompare the spin dynamics in the TB model (Eq. (2) in\nMethods) and the low-energy model in the absence of pud-\ndles ( \u0001 = 0 ). In this regime \u001cp\u001dT\n, and spin dephas-\ning and relaxation are driven by a combination of energy\nbroadening and a nonuniform spin precession frequency. For\nthe TB model, spin dynamics are calculated with the real-\nspace approach and with a standard k-space approach and4\nFIG. 4: Low-energy spin lifetimes versus 1=\u001cp(for initial out-of-\nplane spin polarization). Squares (circles) are for graphene on hBN\n(SiO 2) substrate. Closed (open) symbols are for spin relaxation at\nthe Dirac point (at E=\u0000200meV). The blue dashed line shows\nthe spin lifetime assuming only energy broadening (top axis). Inset:\nspin lifetime in absence of puddles computed using the TB model\nin real space (red circles) or k-space (blue solid line), and the low-\nenergy model in k-space (green dashed line), with \u0011= 13:5meV .\ngive identical \u001cs(inset of Fig.4, red circles and blue solid\nline), indicating the equivalence of the real- and k-space\napproaches in the clean limit when accounting for the full\nTB Hamiltonian. We observe that while for all models, the\nspin lifetime shows a minimum at the Dirac point (in agree-\nment with experimental data, and explained by a strong spin-\npseudospin coupling [34, 35]), spin transport simulations\nwith the widely used low-energy Hamiltonian H(0)(see\nMethods forH(0)and green dashed line in Fig. 4 inset for re-\nsults) clearly cannot capture the saturation and downturn of\n\u001cs(E), i.e. its full M-shape. To qualitatively reproduce the\nM-shape of\u001cs(E), the ﬁrst-order term of the Rashba Hamil-\ntonian,\u0015Ra\n2[kx(\u001bxsy+\u001bysx) +ky(\u001bxsx\u0000\u001bysy)], needs to\nbe included inH(0). This term introduces stronger dephas-\ning at higher energy, driven by the anisotropy of the Rashba\nspin-orbit interaction [35].\nIn addition to their different energy dependence, the TB\nand low-energy models also yield very different spin life-\ntimes. A value of \u001cs= 10 ns is obtained at the Dirac point\nfor the low-energy model, which is two orders of magnitude\nlarger than\u001csfrom the TB Hamiltonian, indicating a strong\nspin dephasing induced by the high-order k-terms. Inter-\nestingly, by studying the changes of \u001cs(E)with respect to\nthe Rashba SOC strength, we observe the scaling behavior\n\u001cs(E)\u0019\f(E)T\n\u0019\f(E)\u0019\u0016h\n\u0015R, meaning the spin relaxes af-\nter a ﬁnite number of precession periods \f(\f'4:5close tothe Dirac point), see Supplementary material. This suggests\nthat dephasing is the limiting factor of spin lifetimes in the\nultraclean case. We ﬁnally note that by taking \u0015R= 5\u0016eV\n(electric ﬁeld of 1 V/nm [4]), a spin lifetime of \u001cs'1:4ns\nis deduced at the Dirac point, whereas at higher energies \u001cs\ncould reach about 7 ns.\nDiscussion\nOur results show a clear transition between two different\nregimes of spin relaxation, mediated solely by the scattering\nstrength of the electron-hole puddles. For hBN substrates,\nspin relaxation is dominated by dephasing arising from an\neffective energy broadening induced by the puddles, and \u001cs\nscales with \u001cp. In contrast, for SiO2substrates dephasing\nis limited by motional narrowing, leading to a DP regime\nwith\u001cs/1=\u001cp. Remarkably, both regimes exhibit similar\nvalues of\u001csat the Dirac point and a similar M-shape energy\ndependence (Fig.3), making it a signature of spin relaxation\nin graphene for all puddle strengths. The crossover between\nboth mechanisms occurs when \u001cp'T\n, which might have\nbeen realized in some experiments. This could explain some\nconﬂicting interpretations of experimental data in terms of\neither Elliot-Yafet or Dyakonov-perel mechanisms [11].\nOur ﬁndings suggest alternative options for determining\nthe spin relaxation mechanism in graphene from experimen-\ntal measurements. Indeed, the typical approach, to examine\nhow\u001cpand\u001csscale with electron density and to assign either\nthe EY or DP mechanism accordingly, is not always appro-\npriate. For example, the EY mechanism in graphene is given\nby\u001cs/E2\nF\u0001\u001cp, such that\u001csand\u001cpwould scale oppositely\nwith respect to electron density if \u001cp/1=EF[6]. Similarly,\nfor our results the scaling of \u001cpand\u001cswith energy suggest\nan EY mechanism near the Dirac point and a DP mechanism\nat higher energies, but Figs. 3 and 4 indicate a richer behav-\nior. Therefore, to determine the spin relaxation mechanism\nit would be more appropriate to study how \u001csand\u001cpscale\nwith defect density or mobility at each value of the electron\ndensity.\nFinally it should be noted that our simulations are per-\nformed using a constant Rashba spin-orbit coupling interac-\ntion,\u0015R, which is different from the experimental situation\nwhere\u0015Rwill be increased at higher charge density owing\nto larger applied external electric ﬁeld. This might explain\nwhy, especially for hBN substrate, the simulations show a\nlarger variation of \u001csin energy than the gate voltage depen-\ndent spin lifetimes reported in the experiments [13, 14].5\nMETHODS\nModel of homogeneous SOC and electron-hole puddles.\nThe tight-binding (TB) Hamiltonian for describing spin\ndynamics in graphene is given by\nH=\u0000\r0X\nhijicy\nicj+i2p\n3VIX\nhhijiicy\ni~ s\u0001(~dkj\u0002~dik)cj\n+iVRX\nhijicy\ni~ z\u0001(~ s\u0002~dij)cj; (2)\nwhere\r0is the nearest-neighbor \u0019-orbital hopping, VIis the\nintrinsic SOC, and VRis the Rashba SOC. In the low-energy\nlimit, this Hamiltonian is often approximated by a contin-\nuum model describing massless Dirac fermions, H(0)=\n\u0016hvF~ \u001b\u0001~k+\u0015I\u001bzsz+\u0015R(~ \u001b\u0002~ s)z, wherevFis the Fermi\nvelocity, \u0016h~kis the momentum, ~ s(~ \u001b)are the spin (pseu-\ndospin) Pauli matrices, \u0015R=3\n2VR, and\u0015I= 3p\n3VI.\nThe value\u0015I= 12\u0016eV is commonly used for the intrin-\nsic SOC of graphene [4] while the Rashba SOC is electric\nﬁeld-dependent. Here, we let \u0015R= 37:4\u0016eV , taken from an\nextendedsp-band TB model for graphene under an electric\nﬁeld of a few V/nm [4, 5]. Higher-order SOC terms in the\ncontinuum model beyond H(0)allow an extension to higher\nenergy [45].\nSpin dynamics methodology.\nThe time-dependent spin polarization of propagating\nwavepackets is computed through [35]\n~P(E;t) =h\t(t)j~ s\u000e(E\u0000H) +\u000e(E\u0000H)~ sj\t(t)i\n2h\t(t)j\u000e(E\u0000H)j\t(t)i;(3)\nwhere~ sare the Pauli spin matrices and \u000e(E\u0000H)is the\nspectral measure operator. The wavepacket dynamics are\nobtained by solving the time-dependent Schr ¨odinger equa-\ntion [40], starting from a state j\t(t= 0)iwhich may have\neither out-of-plane ( z-direction) or in-plane spin polariza-\ntion. An energy broadening \u0011is introduced for expand-\ning\u000e(E\u0000H)through a continued fraction expansion of\nthe Green’s function [40], and mimics an effective disorder.\nThis method has been used to investigate spin relaxation in\ngold-decorated graphene [35]. 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Lohmann, J. H. Smet,\nK. von Klitzing, and Y . A., Nature Physics 4, 144 (2008).\n[37] A. Deshpande, W. Bao, F. Miao, C. N. Lau, and B. J. LeRoy,\nPhys. Rev. B 79, 205411 (2009).\n[38] Y . Zhang, V . Brar, C. Girit, V . Zettl, and M. Crommie, Nature\nPhysics 5, 722 (2009).\n[39] J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod,\nA. Deshpande, K. A. Watanabe, T. Taniguchi, P. Jarillo-\nHerrero, and B. J. LeRoy, Nature Materials 10, 282 (2011).\n[40] S. Roche, Phys. Rev. B 59, 2284 (1999).\n[41] F. Ortmann, A. Cresti, G. Montambaux, and S. Roche, EPL\n(Europhysics Letters) 94, 47006 (2011).\n[42] A. Rycerz, J. Tworzydo, and C. W. J. Beenakker, EPL (Euro-physics Letters) 79, 57003 (2007).\n[43] J. W. Kłos and I. V . Zozoulenko, Phys. Rev. B 82, 081414\n(2010).\n[44] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323\n(2004).\n[45] P. Rakyta, A. Korm ´anyos, and J. Cserti, Phys. Rev. B 82,\n113405 (2010).\nADDENDUM\nThis work has received funding from the European Union\nSeventh Framework Programme under grant agreement\n604391 Graphene Flagship. S.R. acknowledges the Span-\nish Ministry of Economy and Competitiveness for funding\n(MAT2012-33911), the Secretaria de Universidades e In-\nvestigacion del Departamento de Economia y Conocimiento\nde la Generalidad de Catalu ˜na and the Severo Ochoa Pro-\ngram (MINECO SEV-2013-0295). F.O. would like to ac-\nknowledge the Deutsche Forschungsgemeinschaft (grant OR\n349/1-1). Inspiring discussions with Sergio O. Valenzuela,\nShafﬁque Adam, and Jaroslav Fabian are deeply acknowl-\nedged."    },    {        "title": "1402.2345v1.Dynamical_decoupling_design_for_identifying_weakly_coupled_nuclear_spins_in_a_bath.pdf",        "content": "Dynamical decoupling design for identifying weakly coupled nuclear spins in a bath\nNan Zhao,1, 2, 3J¨org Wrachtrup,2and Ren-Bao Liu1\n1Department of Physics and Centre for Quantum Coherence,\nThe Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China\n23. Physikalisches Institut and Research Center SCOPE,\nUniversity of Stuttgart, Pfa \u000benwaldring 57, 70569 Stuttgart, Germany\n3Beijing Computational Science Research Center, Beijing 100084, China\nIdentifying weakly coupled nuclear spins around single electron spins is a key step of implementing quantum\ninformation processing using coupled electron-nuclei spin systems or sensing like single spin nuclear magnetic\nresonance detection using diamond defect spins. Dynamical decoupling control of the center electron spin with\nperiodic pulse sequences [e.g., the Carre-Purcell-Meiboom-Gill (CPMG) sequence] has been successfully used\nto identify single nuclear spins and to resolve structure of nuclear spin clusters. Here, we design a new type\nof pulse sequences by replacing the repetition unit (a single \u0019-pulse) of the CPMG sequence with a group of\nnonuniformly-spaced \u0019-pulses. Using nitrogen-vacancy center system in diamond, we show that the designed\npulse sequence improves the resolution of nuclear spin noise spectroscopy, and more information about the\nsurrounding nuclear spins is extracted. The principle of dynamical decoupling design proposed in this paper\nis useful in many systems (e.g., defect spin qubit in solids, trapped ion and superconducting qubit) for high-\nresolution noise spectroscopy.\nPACS numbers: 76.60.Lz, 03.65.Yz, 76.30.-v, 76.30.Mi\nI. INTRODUCTION\nCoupled electron-nuclear spin systems are important plat-\nform for quantum information processing1–5. Single electron\nspins are promising quantum processors because of their ad-\ndressability and controllability6,7. Nuclear spins are regarded\nas ideal quantum memories since they are less sensitive to the\nenvironmental noise and have longer coherence time8. Cou-\npled electron-nuclear spin systems have the advantages of\nboth ingredients. However, in many solid state systems, a\nlarge number of nuclear spins around the electron spin usu-\nally form a spin bath, serving as a decoherence source of elec-\ntron spins, rather than a kind of resource. Individual nuclear\nspins have to be resolved, otherwise they can hardly be used as\nquantum memories. In this sense, identifying single nuclear\nspins in a spin bath is highly desirable.\nSingle nuclear spins can be resolved through the splitting\nof the electron spin resonance (ESR) spectrum due to cou-\npling to the nuclear spins. The resolution of the ESR spectrum\nsplitting is limited by the transition linewidth. Usually, only\nthe nuclear spins strongly coupled to the electron spin (with\nthe coupling strengths much larger than the linewidth) can\nbe well resolved1–5. Resolving the weakly coupled nuclear\nspins (with coupling strengths comparable to the linewidth) is\na challenging task. This greatly limits the potential of using\nthese nuclear spins as resources.\nThe limitation on resolution by ESR linewidth can be over-\ncome by actively controlling the electron spins via, for exam-\nple, continuously driving spectroscopy9–11and dynamical de-\ncoupling (DD) control12,13. In particular, DD control of elec-\ntron spins was proposed to be a powerful tool to detect small\nnuclear spin clusters12. Under DD control, single nuclear spin\nclusters around center electron spins manifest themselves as\ncharacteristic ﬁngerprint oscillations on the electron spin co-\nherence. With this discovery, nuclear magnetic resonance can\nbe realized at the single-molecule level12. Very recently, ap-plying the standard Carre-Purcell-Meiboom-Gill (CPMG) DD\nsequence on the electron spins of nitrogen-vacancy (NV) cen-\nters, people have successfully demonstrated the identiﬁcation\nof weakly coupled individual nuclear spins13–15, and the re-\nsolving of the structure of single nuclear spins cluster16.\nIn addition to the widely used periodic CPMG control se-\nquence, various di \u000berent types of DD control sequence were\ndesigned for di \u000berent purposes17. For example, the Uhrig’s\nDD (UDD) sequence with nonuniformly-spaced pulses were\nproposed18and experimentally19,20shown to be the opti-\nmal sequence to protect the short-time center spin coher-\nence. Also, various concatenated or nested schemes based on\nCPMG and UDD are developed for di \u000berent purposes17, such\nas protecting spin coherence of multi-qubits21. In this paper,\naiming at identifying single nuclear spins and improving the\nresolution of nuclear spin noise spectroscopy, we generalize\nthe standard CPMG sequence and design a new control pulse\nsequence. With the designed sequence, more nuclear spins in\nthe spin bath can be resolved in comparison to the standard\nCPMG sequence.\nIn order to demonstrate the application of our designed DD\nsequence, we focus on the identiﬁcation of single nuclear\nspins around NV center electron spins in diamond, which is\na promising solid-state system in quantum information pro-\ncessing and nano-scale magnetometry22. We show that solely\nincreasing the CPMG control pulse number does not help re-\nsolve more nuclear spins. Instead, in order to achieve high res-\nolution, in the designed pulse sequence, we replace the CPMG\nrepetition unit (a single \u0019-pulse) by a group of \u0019-pulses. The\npulse timing structure within each repetition unit provides ad-\nditional degrees of freedom to tailor the corresponding noise\nﬁlter function in the frequency domain. Thus, according to\nthe features of the detected spin noise, the ﬁlter function can\nbe ﬁne-tuned to resolve more nuclear spins. The principle\nof the DD sequence design can be used in the general noise\nspectroscopy in other systems such as trapped ions19and su-arXiv:1402.2345v1  [cond-mat.mes-hall]  11 Feb 20142\n0.000 .010 .020 .030.00.51.0 \nN = 1 \n Coherence(\na)B = 10 Gauss0\n.00 .51 .01 .5-101B\n = 10 Gauss \nN = 50(\nb) \n CoherenceA\nB C 0.00.20.40.60.00.51.0 \n 0\n.20.30.40.5-101(\nc)CoherenceA\n0\n.350.400.450.50(d) T\nime t (ms)B0\n.40.81.2(e)C\nFIG. 1: (a) The spin coherence under Hahn echo control in a\nB=10 Gauss magnetic ﬁeld. The inset shows the coherence col-\nlapse and revival in a longer time scale, and the main panel zooms in\nthe ﬁrst collapse. (b) The spin coherence under 50-pulse CPMG con-\ntrol. Only the ﬁrst collapse is shown. Oscillations in green, red, and\nblue (denoted by A, B, and C in turn) are caused by three individual\n13C spins with hyperﬁne coupling strengths \u0018100 kHz. Their contri-\nbutions to the coherence are singled out in (c)-(e) with corresponding\ncolors.\nperconducting qubits23.\nThis paper is organized as follows. Section II describes the\nNV center spin coherence under DD control sequence and an-\nalyzes the semi-classical nuclear spin noise spectrum. Section\nIII demonstrates the improved resolution of the designed DD\nsequence. We conclude the paper in Section IV.\nII. NV CENTER ELECTRON SPIN COHERENCE UNDER\nDD CONTROL\nWe consider the NV center system as an example to demon-\nstrate the principle of DD design for high-resolution spin noise\nspectroscopy and nuclear spin identiﬁcation. The NV center\nelectron spin in a13C nuclear spin bath is described by13,24\nH=HNV+Hint+Hnuc; (1)\nwhere the NV center Hamiltonian HNVis\nHNV= \u0001S2\nz\u0000\reB\u0001S\u00113X\n\u000b=1!\u000bj\u000bih\u000bj: (2)\nHere, Sis the spin-1 operator of the electron spin, \u0001 =\n2\u0019\u00022:87 GHz is the zero ﬁeld splitting, Bis the applied mag-\nnetic ﬁeld, and \re=\u00001:76\u00021011rad s\u00001T\u00001is the electron\nspin gyromagnetic ratio. The zdirection is chosen along the\nN-V axis (the [111] direction). In the eigen-representation,HNVis diagonalized with the eigen-frequencies !\u000band the\ncorresponding eigenstates j\u000bi.\nThe Hamiltonian of13C nuclear spins bath is\nHnuc=\u0000\rCB\u0001X\njIj+Hdd; (3)\nwhere Ijis the jth nuclear spin, \rC=6:73\u0002107rad s\u00001T\u00001\nis the gyromagetic ratio of13C, and Hdddescribes the dipole-\ndipole interaction between13C nuclear spins. The electron\nspin couples to the13C nuclear spins through the hyperﬁne\ninteraction\nHint=S\u0001X\njAj\u0001Ij=X\n\u000b;jj\u000bih\u000bj\nA(\u000b)\nj\u0001Ij; (4)\nwhereAjis the hyperﬁne interaction tensor of the jth13C,\nandA(\u000b)\nj=h\u000bjSj\u000bi\u0001A jis the hyperﬁne ﬁeld felt by the jth\nnuclear spin for the electron being in state j\u000bi. In the second\nequation of Eq. 4, we have neglected the electron spin ﬂipping\nterms between di \u000berent eigenstatesj\u000biandj\fi, since the elec-\ntron spin is hardly ﬂipped by the nuclear spins due to the large\nenergy mismatch ( \u0018GHz) compared with the typical hyper-\nﬁne coupling strength ( <MHz).\nIn the NV center system described by the Hamiltonian\n(1), single nuclear spin detection has been realized in di \u000ber-\nent magnetic ﬁeld regimes13–15. In this paper, we will focus\non how to improve the resolution in the weak ﬁeld regime\n(B\u001810 Gauss, in diamond samples with natural abundance\n13C isotope). In this regime, single nuclear spin precession in-\nduces coherence collapse and revival e \u000bect in the Hahn echo.\nThe dipolar coupling Hddbetween nuclear spins causes the\nenvelope decay of the revival peaks [see Fig 1(a)]13,24. Within\nthe ﬁrst collapse [for t<30\u0016s in Fig. 1(a)], the dipolar cou-\npling can be neglected and the nuclear spins precess indepen-\ndently. In this case, the electron coherence between two eigen-\nstatesj\u000biandj\fiofHNVis expressed as13,24\nL\u000b\f(t)=Y\njLj;\u000b\f(t)\n=Y\njTr\u0014\n\u0001\u0001\u0001e\u0000iH(\u000b)\nj\u001c2e\u0000iH(\f)\nj\u001c1eiH(\u000b)\nj\u001c1eiH(\f)\nj\u001c2\u0001\u0001\u0001\u0015\n;(5)\nwhere Lj;\u000b\f(t) is the contribution of the jth nuclear spin to the\ndecoherence, and\nH(\u000b=\f)\nj=\u0000\rCh(\u000b=\f)\nj\u0001Ij (6)\nis the conditional Hamiltonian of the jth nuclear spin in e \u000bec-\ntive ﬁeld h(\u000b=\f)\nj=B\u0000A(\u000b=\f)\nj=\rC.\nFingerprint oscillations within the ﬁrst coherence collapse\nemerge when increasing the CPMG control pulse number14.\nFigure 1(b) shows the calculated NV center electron spin co-\nherence, in a randomly generated nuclear spin bath conﬁgura-\ntion, under 50-pulse CPMG control sequences in a magnetic\nﬁeld B=10 Gauss along zdirection. Under CPMG con-\ntrol, the ﬁrst collapse time increases linearly as increasing the\npulse number N13. Meanwhile, in contrast to the smooth de-\ncay within\u001830\u0016s, strong characteristic oscillations appear3\n01 02030405060708010-410-2100102(a) S(/s61559 )F\nrequency f (kHz)ABCED\n0\n5 01 001 502 000246810(\nb) \n F50(/s61559 t ) /s61559\n tX1030\n2 0040060010-2101104 10-410-2100102\nFIG. 2: (Color online) (a) The noise spectrum of a nuclear spin bath\ncoupled to an NV center spin. Circle symbols represent the noise fre-\nquency and amplitude pairs ( \u0017j;wj) in Eq. 7. The solid curve is ob-\ntained by replacing the \u000e-function in Eq. 7 by a Lorentzian line-shape\nwith a phenomenological broadening \u00180:1 kHz (corresponding to\nthe electron spin relaxation T1&1 ms). A large number of13C spins\ncontribute to the quasi-continuous band (the grey shadow region)\naround the13C Larmor frequecy fL=10:7 kHz at B=10 Gauss.\nSeveral13C spins close to the NV center contributes to the discrete\npeaks. Three peaks, denoted by A (green), B (red), and C (blue)\nin turn, with frequency higher than fLcorrespond to the oscillations\nshown in Fig. 1(c)-(e). Discrete peaks D and E close to the noise\nband are not resolved by the standard CPMG sequence by by the de-\nsigned DD sequence (see Fig. 3). (b) The ﬁlter function of F50(!t)\n50-pulse CPMG control sequence. The inset shows a global view in\nlogarithm scale.\non the decay proﬁle [see the oscillations in green, red, and\nblue in Fig. 1(b)]. These oscillations are caused by three13C\nnuclear spins close to the NV center, whose contributions are\nsingled out in Figs. 1(c)-(e). The oscillation features (e.g., the\npositions and the depths of the coherence dips) contain the in-\nformation of the hyperﬁne coupling, and have been used to\nresolve the single nuclear spins14.\nThe underlying physics of the coherent oscillations can be\nunderstood in semi-classical picture with nuclear spin noise\nspectrum S(!). To calculate the noise spectrum, we deﬁne\nthe averaged bath Hamiltonian H(j)\nbath=1\n2\u0010\nH(\u000b)\nj+H(\f)\nj\u0011\nand the\nnoise operator ˆbj=H(\u000b)\nj\u0000H(\f)\njof the jth13C bath spin. The\ntwo eigen-frequencies and the corresponding eigenstates of\nH(j)\nbathare denoted by \u0017(1;2)\njandj (1;2)\nji, respectively. In general,\nthe bath Hamiltonian H(j)\nbathand the noise ˆbjare not commuta-\ntive. The noise operator ˆbjwill induce transition between the\neigenstatesj (1;2)\nji. Thus, the noise spectrum is calculated by\nS(!)=X\njwj\u000e\u0010\n!\u0000!j\u0011\n; (7)\nwhere wj=jh (1)\njjˆbjj (2)\njij2is the noise amplitude, and !j=\n\u0017(2)\nj\u0000\u0017(1)\njis the noise frequency. Figure 2(a) shows the cal-culated nuclear spin noise spectrum of the same spin bath\nas that used in Fig. (1). The noise frequency is determined\nby both of the intrinsic Lammor frequency of13C nuclear\nspins in the given magnetic ﬁeld B, and the back-action aris-\ning from the hyperﬁne coupling of each nuclear spins to the\ncenter electron spin14,25,26. Particularly, for the center elec-\ntron spin being prepared in the superposition state of j0iand\nj+1i, the jth13C nuclear spin contributes noise with frequency\n!j=j\u0000\rCB+Aj=2j. The13C nuclear spins far from the NV\ncenter electron spin have hyperﬁne coupling strengths much\nsmaller than the applied magnetic ﬁeld, i.e., jAjj \u001c j\rCBj.\nThey produce nuclear spin noise with approximately the same\nfrequencies, and form a quasi-continuous noise band around\nthe Larmor frequency (see the grey shadow region in Fig 2).\nThe few13C nuclear spins close to the NV center (with dis-\ntance of several Å) have hyperﬁne coupling strengths (in the\norder of\u0018100 kHz) comparable or larger than the applied\nmagnetic ﬁeld strength (but not strong enough to cause ESR\nspectrum splitting). Thus, they produce nuclear spin noise\nwith frequencies signiﬁcantly di \u000berent from the Larmor fre-\nquency, and form discrete spectral lines. These nuclear spins\ncan be resolved by measuring the NV center electron spin co-\nherence under DD control.\nIn the semi-classical picture, the coherence of a two-level\nsystem in a time-dependent noise ﬁeld is approximately ex-\npressed as27\nL(t)\u0019exp\"\n\u00001\n2Z1\n0d!\n\u0019S(!)\n!2FN(!t)#\n; (8)\nwhere the noise ﬁlter function FN(!t) is the Fourier trans-\nform of the modulation function associated with the DD con-\ntrol scheme. Figure 2(b) shows the ﬁlter function of the\n50-pulse CPMG control. The periodic applied pulses in the\ntime domain give rise to sharp peaks at frequencies !N;k=\n(2k\u00001)N\u0019=t(fork=1;2;:::). As increasing the total evolu-\ntion time t, the ﬁlter function peaks sweep from the high fre-\nquency side. Whenever the ﬁlter function peaks overlap with\na discrete peak in the noise spectrum, there appears a coher-\nence dip according to Eq. (8). When the ﬁrst peak of the ﬁlter\nfunction touches the quasi-continuous noise band, the electron\nspin coherence collapses.\nIII. IDENTIFICATION OF SINGLE NUCLEAR SPINS\nThe results presented in Fig. 1 seem to imply that increasing\nthe coherence time by DD control would beneﬁt the single\nnuclear spin identiﬁcation. To check this, we compare the\ndecoherence behavior under 50-pulse CPMG control with that\nunder 150-pulse control, which has 3 times longer coherence\ntime than the former (see Fig. 3). However, the nulcear spin\nresolution is not improved. The same three nuclear spins as in\nthe 50-pulse case are resolved from the coherence oscillation\nfeatures.\nIndeed, the unnecessarily strong peak in the ﬁlter function\nFN(!t) prevents the further improvement of the resolution.\nHere, we should emphasize that Eq. (8) is obtained by keep-\ning the noise correlation only to the second order and, essen-4\n02004000\n2004000\n5 01 001 5002004000200400 \n (\na) \n(c)Fileter function(\nd)/s61559\n t / (2π)x50-\nq0 p- pq 0.00 .51 .01 .52 .0-101A\n' \n(g)CoherenceE\nD CAB0\n.00 .51 .01 .52 .0-101(\nh)T\nime (ms)EDC ABx50/s61556\n2 /s615562/s61556/s61556 -\n1/30 1 /3(b)0\nr - rx500.02 .04 .06 .0-101 \n(f) A\nBC\n10-410-2100 \n501 003 01 5(e) \n noise amp.2\n0f\nrequency (kHz)AB\nCD E \nFIG. 3: (Color online) (a) The ﬁlter function F50(!t) of 50-pulse CPMG control. (b) The ﬁlter function F150(!t) of 150-pulse CPMG control.\nThe 150-pulse sequence can be regarded as 50 repetitions of 3-pulse unit shown in the inset. (c) The ﬁlter function of a modiﬁed 50 \u00023-pulse\nsequence. The repetition unit is shown in the inset with parameters r=0:31. (d) The same as (c) but for a modiﬁed 50 \u00025-pulse sequence. The\nparameters are p=0:209 and q=0:384. (e) The noise spectrum in reciprocal axis. (f)-(h) The electron spin coherence under the DD control\nwith sequencies shown in (b)-(d) in turn.\ntially, is a perturbation treatment valid only in the weak noise\ncase. When the ﬁlter function peak is too strong (i.e., the ﬁl-\ntered noise S(!)FN(!t)=!2\u001d1, like in the 150-pulse CPMG\ncase), the electron spin coherence will obviously deviate from\nEq. 8. The central dip is deformed and the side dips become\nstrong. In this case, a single nuclear spin produces a broad-\nened and highly oscillating structure in the electron spin co-\nherence. The interference of oscillating structures blurs the\nﬁngerprint features of individual nuclear spins to be resolved.\nTo solve this problem, we design pulse sequences simul-\ntaneously with high frequency selectivity and moderate peak\nstrength of the ﬁlter function. To this end, a group of \u0019\npulses are used as the repetition unit, instead of a single one\nin CPMG sequence. We consider an N\u0002Msequence ( N\nrepetitions of an M-pulse unit), whose pulses are applied at\ntm;n=[(2n\u00001)=2+rm](t=N), where n=1;2;:::; Nand\nrm2[\u00001\n2;1\n2] for m=1;2;:::; Mis the relative position of\nthemth pulse within a repetition unit.\nThe principle of the designed pulse sequence can be under-\nstood through the analogy with the optical grating e \u000bect. The\nperiodic Nrepetitions in time domain gives rise to the peak\nstructure in the frequency domain described by\n˜FN(!t)=sin2!t\n2\ncos2!t\n2N: (9)\nThe structure within each repetition unit, which is local in time\ndomain, imposes a slow modulation function gM(!t) of thepeak structure in frequency domain. The ﬁlter function is ex-\npressed as\nFN\u0002M(!t)=gM(!t)˜FN(!t); (10)\nFor the standard CPMG sequence ( M=1),gM=1(!t)=\n16 sin4[!t=(4N)].\nTuning the relative positions rmwithin the repetition unit\nmodiﬁes the strength of the ﬁlter function peaks. Taking the\nsymmetric 3-pulse repetition unit for example, with \u0000r1=\nr3\u0011randr2=0, we have\ngM=3(!t)=16\u0012\ncos2!t\n4N\u0000cosr!t\nN\u00132\n: (11)\nAs a special example, the 150-pulse CPMG sequence can be\nregarded as 50 repetitions of a 3-pulse unit with r\u0003=1=3. In\nthis case, the pulses are indeed equally spaced, and one can\ncheck that the zero points of gM=3(!t) with r\u0003=1=3 precisely\ncancel the extra peaks of F50(!t) leaving only those peaks\nfulﬁl the peak condition of F150(!t) [i.e.,!t=(2\u0019)=150\u0002\n(2k\u00001)=2, see Fig. 3(b)].\nSlightly shifting the positions rmfrom the CPMG values\n(e.g., r\u0003=1=3 in the M=3 example above), we obtain a mod-\niﬁed pulse sequence with both high frequency selectivity and\nmoderate height of the ﬁlter function [see Figs. 3(c)]. With\nthis modiﬁed pulse sequence, as shown in Fig. 3(g), two more\nnuclear spins (denoted by D and E) around the NV center are5\nresolved. As the frequencies of the noise produced by nuclear\nspins D and E are too close to the quasi-continuous noise band\n[around the Larmor frequency \u001810 kHz for B=10 Gauss, see\nFigs. 2(a) and 3(e)], nuclear spins D and E cannot be resolved\nby the strong ﬁlter function peaks of the standard CPMG se-\nquence. While the much weaker and narrower ﬁlter function\npeak of the designed pulse sequence is able to resolve much\nﬁner structure close to the noise band edge.\nMore pulses in a repetition unit provides more degree of\nfreedom to design the noise ﬁlter function. For example, the\nstrong peak around !t=(2\u0019)\u001975 in Fig. 3(c) causes a broad\nand strongly oscillating structure, denoted by A0in Fig. 3(g).\nPractically, this highly oscillating structure A0may interfere\nthe identiﬁcation of signals from other nuclear spins. Using\na symmetric 5-pulse repetition unit shown in Fig. 3(d), with\nrelative pulse positions \u0000r1=r5=q,\u0000r2=r4=pand\nr3=0, one can eliminate the unwanted strong peak around\n!t=(2\u0019)\u001975, and get a much cleaner high-resolution ﬁnger-\nprint structure of ﬁve nuclear spins [see Fig. 3(h)].\nIV . CONCLUSION\nIn this paper, we investigate resolving weakly coupled sin-\ngle nuclear spins from a nuclear spin bath. A new type of DDcontrol pulse sequence of central electron spin is designed.\nApplying the designed pulse sequence in NV center system,\none can improve the resolution of the noise spectroscopy and,\nin comparison to the standard CPMG sequence, more nuclear\nspins around the NV center can be identiﬁed as resource for\nquantum information processing. The principle of designing\nDD control pulse sequence is widely applicable in other sys-\ntems, such as trapped ions and superconducting qubits, for\nhigh-resolution noise spectroscopy.\nAcknowledgement\nThis work is supported by Hong Kong RGC /CRF\nCUHK4 /CRF/12G and CUHK Focused Investments Scheme.\nN.Z. acknowledges NKBRP (973 Program) 2014CB848700,\nand NSFC No. 11374032. J.W. acknowledges funding from\nthe EU via SIQS and SQUTEC as well as the DFG via\nFOR1493 and SFB /TR21 and BMBF via Q.COM.\n1L. Childress, M. V . Gurudev Dutt, J. M. Taylor, A. S. Zibrov,\nF. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, Science\n314, 281 (2006).\n2M. V . G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze,\nF. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, Sci-\nence316, 1312 (2007).\n3P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe,\nS. Yamasaki, V . Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup,\nScience 320, 1326 (2008).\n4L. Jiang, J. S. Hodges, J. R. Maze, P. 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Das Sarma, Physical\nReview B 77, 174509 (2008)."    },    {        "title": "1501.00057v1.Seeing_spin_dynamics_in_atomic_gases.pdf",        "content": "October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nChapter 1\nSeeing spin dynamics in atomic gases\nDan M. Stamper-Kurn\nDepartment of Physics, University of California, Berkeley, California 94720, USA\nMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,\nCalifornia 94720, USA\nThe dynamics of internal spin, electronic orbital, and nuclear motion states of\natoms and molecules have preoccupied the atomic and molecular physics commu-\nnity for decades. Increasingly, such dynamics are being examined within many-\nbody systems composed of atomic and molecular gases. Our \fndings sometimes\nbear close relation to phenomena observed in condensed-matter systems, while on\nother occasions they represent truly new areas of investigation. I discuss several\nexamples of spin dynamics that occur within spinor Bose-Einstein gases, high-\nlighting the advantages of spin-sensitive imaging for understanding and utilizing\nsuch dynamics.\n1. Atomic materials science\nIt is conventional to begin a book chapter such as this with a coherent framing\nof the progress within a scienti\fc \feld, perhaps portrayed as setting up the next\nclear question that must be answered or an opportunity that must be exploited.\nHowever, it has always struck me that such a presentation belies the true and\nexciting nature of scienti\fc work, in which discoveries arise from unframed curiosity\nand in which the intellectual random walks of many individuals cause the bounds\nof our knowledge and our sense of what is worth knowing to di\u000buse outward. Read\nas a whole, the chapters of this book support this view of science as a di\u000busing\nendeavor. One may try to tie all the work described herein as a coherent body of\nwork (about coherence). But maybe it is best to appreciate the diverse manners in\nwhich the \feld of AMO science is evolving, all the fun that people are having in\ntheir laboratories and o\u000eces, and all the places we may go in the coming years.\nThis Chapter is about spin dynamics in quantum degenerate gases. I shall\nemphasize the role of high-resolution direct imaging to visualize such dynamics.\nDirect imaging provides a great deal of detailed information about the gases being\nprobed. While we have learned a few ways in which some portion of that information\ncan be analyzed, there still remains a wealth of information to be mined from\ndetailed images. Above all, however, the direct images of spin dynamics have the\n1arXiv:1501.00057v1  [cond-mat.quant-gas]  31 Dec 2014October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n2 Dan M. Stamper-Kurn\nvalue of being beautiful, of portraying the science at hand as an appealing subject\nto study.\nNow, to keep with convention, let me reframe the study of spin dynamics in\nquantum gases in a coherent framework.\nSimple, discrete quantum systems, along with methods that make use of reso-\nnance and coherence to manipulate and measure such systems, have been the bread\nand butter of atomic physics for a century. In experiments on atomic beams and\nvapor cells, and then certainly on trapped ions, the focus has often been on the in-\nternal dynamics within single atoms. Precise measurements of these dynamics have\nallowed us to test the principles by which atoms are constructed (e.g. quantifying\nhigh-order processes in quantum electrodynamics), or to characterize the conditions\nto which the atoms are exposed (e.g. sensing magnetic \felds).\nIncreasingly, however, atomic physics is giving access to many-body systems in\nwhich the atoms evolve no longer in isolation, but rather as a collection of mutually\ninteracting quantum objects. In other words, we are moving from the study of\natoms to the study of materials. The focus of materials science is typically di\u000berent\nthan that of atomic physics: whereas the atomic physicist may appreciate a precise\ncharacterization of the full dynamics of a simple quantum system, the material\nscientist may focus instead on general characterizations of thermodynamic ground\nstates and of small excitations atop such equilibria. Precise characterization of\nmaterials is often unwarranted since material samples are imperfect and di\u000ber from\none another. Characterizing far-from-equilibrium states is unnecessary since most\nmaterials cannot typically be driven very far from equilibrium.\nNow that atomic physicists are creating materials-like quantum systems under\nhighly controlled conditions, both scienti\fc approaches are bene\fcial and applicable.\nIt is believed that atomic systems, serving as \\quantum emulators\" of paradigmatic\nmany-body quantum systems, making use of the tremendous control, cleanliness,\nand powerful detection capabilities o\u000bered by atomic physics, may allow us to de-\ntermine better the principles underlying the materials science of both existing and\ntheorized matter. Conversely, one expects that the properties of complex materials\nrealized by these atomic systems will make it possible for us to improve our perfor-\nmance in the typical tasks of an atomic physicist, vis a vis precision measurement\nand sensing.\nThis duality is exhibited by the study of magnetism in quantum gases, which is\nthe topic of this Chapter. In particular, I focus on magnetic phenomena in spinor\nBose-Einstein gases, ones composed of bosonic atoms that are free to occupy the\nvarious Zeeman sublevels of a manifold of spin states. Following the example of\nmaterials science, I describe the propagation of small-amplitude spin modulations\nin a ferromagnetic condensate (Sec. 3) in terms of the dispersion of magnon excita-\ntions, and the spin dynamics of non-equilibrium quantum \ruids (Sec. 4) in terms of\nspontaneous symmetry breaking and phase transitions. Following the example of\natomic physics, I describe the propagation of magnons (Sec. 3) as a promising formOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 3\nof atomic interferometry, the spin-mixing instability (Sec. 4) in terms of the gener-\nation of spin-nematicity squeezed states, and the combination of Larmor precession\nand high-resolution, high-sensitivity imaging (Secs. 2, 5) as a potentially powerful\nmagnetic \feld sensor.\nThis Chapter is not meant to be a comprehensive review of studies of mag-\nnetic phenomena in quantum gases, or even speci\fcally of research on spinor Bose-\nEinstein gases. Such reviews are provided elsewhere (e.g. in Refs. 1{3). Rather, the\nintention is to highlight some of the new research directions in materials science and\natomic physics that have been identi\fed in recent years and that should, I believe,\nbe taken up (by the reader!) in the coming years.\n2. Imaging methods\nOne of the standard ways to diagnose an ultracold gas is to let it out of its trapping\ncontainer, allow it to expand a while, and then to take a picture that reveals the\nspatial distribution of the expanded cloud. This technique is favored because the\nexpanded gas can be made far larger than the imaging resolution, so as to avoid\nimaging aberrations. Also, the expanded gas can be made to have a low optical\ndensity. One can illuminate the sample with light that is resonant with an atomic\ntransition, and then quantify the resulting shadow image to determine faithfully the\ncolumn density distribution. This distribution carries information on an amalgam of\nproperties of the gas { such as the one-particle distribution and two-particle correla-\ntions in position and momentum space, the interaction energy, angular momentum,\nvorticity, etc. { depending on the time of \right. That is, the spatial distribution at\nzero time of \right is obviously the in-trap spatial distribution, but understanding\nwhich property is revealed by which feature in the time-of-\right distribution for\nlonger expansion times depends on assumptions about the behavior of the gas upon\nexpansion. For gases that are structured and correlated with greater complexity,\nthe connection between time-of-\right spatial distributions and in-situ properties is\nless straightforward.\nThe time-of-\right imaging method can be made spin sensitive by applying a\nmagnetic \feld gradient during the time of free expansion, emulating Stern and Ger-\nlach's early experiment in which atoms in di\u000berent Zeeman states were de\rected\nto di\u000berent spatial regions on a detector [4]. Separating the di\u000berent Zeeman com-\nponents from one another requires a non-zero time of \right, so that, aside from\nthe overall population distribution among the Zeeman sublevels, all other informa-\ntion about the gas is ambiguous. In particular, the spatial spin distribution can be\ndetermined only with poor spatial resolution.\nMy group has developed imaging methods that allow for the spin distribution of\na quantum gas to be measured directly, in-situ, with high spatial resolution and high\nspin sensitivity. Both methods make use of the atomic physics of single atoms to\ncharacterize the magnetic structure of a many-body system: The \frst method reliesOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n4 Dan M. Stamper-Kurn\non optical birefringence, while the second relies on the spin-selectivity of microwave\ntransitions between di\u000berent hyper\fne states within the electronic ground state.\n2.1. Spin-sensitive dispersive in-situ imaging\nBirefringent materials are ones for which the real optical susceptibility is a tensor,\nrather than a scalar; in other words, light passing through the material acquires a\nphase shift that depends on the polarization of the light, and, for this reason, may\nemerge from the material with a di\u000berent polarization than it had before entering\nthe material.\nOne distinguishes between linear and circular birefringence. In the case of linear\nbirefringence, if we represent the polarization of light in a particular basis of lin-\near polarizations, the optical susceptibility is a diagonal matrix, meaning that light\nentering the material with either of these linear polarizations acquires a polarization-\ndependent phase shift but does not change in polarization. For example, the bire-\nfringent materials used in optical waveplates are linearly birefringent, de\fned by\northogonal \\fast\" and \\slow\" axes. In the case of circular birefringence, the basis\npolarizations that diagonalize the susceptibility tensor are the left- and right-handed\ncircular polarizations. Circular birefringence occurs in situations that break time-\nreversal symmetry, e.g. in optical isolators that use strong magnetic \felds and the\nFaraday e\u000bect.\nA spin-polarized atomic gas can exhibit both linear and circular birefringence.\nFor example, consider a gas fully polarized in the jmJ= 1=2isublevel (with respect\nto some spatial quantization axis) of a J= 1=2 ground state. The oscillator strength\nfor an optical transition to either a J0= 1=2 orJ0= 3=2 excited state di\u000bers for\n\u001b+and\u001b\u0000polarized light; this can be seen by examining the values of Clebsch-\nGordan coe\u000ecients. Thus, the dispersive phase shift imposed on light that passes\nthrough such an atomic gas, and that is detuned from the optical transition, is\ndi\u000berent for the two circular optical polarizations. This di\u000berence quanti\fes the spin\npolarization of the atomic gas. As another example, consider a gas fully polarized\nin thejmJ= 0isublevel of a J= 1 ground state, and probed near a transition\nto aJ0= 1 excited state. This gas does not couple to \u0019-polarized light (linear\nalong the quantization axis), but does couple to the orthogonal linear polarization.\nThus, such a gas exhibits linear birefringence which, in this case, identi\fes the spin\nquantization axis.\nIt can be shown that circular birefringence reveals the spin-vector moments\nwhile linear birefringence reveals the spin-quadrupole moments of the atoms being\nprobed [5]. These moments yield the magnetization vector and nematicity tensor\nof the material comprised of such atoms. The spin-vector moments are associated\nwith coherences between Zeeman sublevels that di\u000ber by \u0001 m= 1 in their magnetic\nquantum number, while the spin-quadrupole moments are associated with \u0001 m= 2\ncoherences. That is why I could use the example of a J= 1=2 ground state to\ndiscuss circular birefringence, but needed a J= 1 ground state, which can haveOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 5\nx z \ny \n+ linear \nphas e \nplate polarizer \n1\n2 14 1\n12 ′=2 \n=1 (a) (b) (c)\nFig. 1. We employ circular birefringence to image the magnetization of an F= 1 spinor gas. (a)\nThe relative oscillator strengths for circular polarized light driving an optical transition from the\nF= 1 ground state to an F0= 2 excited state are shown. Atoms polarized in the jmF= +1i\nsublevel, quantized along the direction of the optical helicity, interact more strongly with the\nlight. A dispersive image taken with light near this transition therefore contains information on\nthe magnetization of the gas along that direction. The dispersive signal is turned into an image\nin one of two standard ways. (b) One way is to convert the phase shift on a circularly polarized\nlight \feld into an intensity image using a phase plate placed in the Fourier plane of the image,\nthis being a form of phase contrast imaging [8]. (c) A second way is to illuminate the sample with\nlinear polarized light. The circular birefringence causes a rotation of the linear polarization, and\nthis rotation is analyzed by passing the light through a linear polarizer before the camera.\n\u0001m= 2 coherences, to exemplify linear birefringence.\nMy group made use of the circular birefringence of87Rb atoms in the F= 1\nhyper\fne ground state in order to image their magnetization [6]. Polarized light\ndetuned by several hundred MHz from the D1 optical transitions was sent along\nthe thin axis of a scalene ellipsoidal optical trap containing a gas of several million\natoms, which were typically deep within the quantum degenerate regime (Fig. 1).\nIn one approach, we imaged with a brief pulse of circular polarized light, and used\nphase-contrast imaging to convert the spatially varying phase shift on the probe light\ninto an intensity image. A single image yielded information on both the density and\nthe column-integrated magnetization of the gas along the probe axis. To isolate the\nmagnetization signal, and also characterize the projections of the magnetization\nalong all axes, we used a combination of rf pulses and Larmor precession to rotate\nthe magnetization (Fig. 2). In a second approach, we used linearly polarized light\nand quanti\fed its polarization rotation using a linear polarizer before the camera.\nThe image now measured only the magnetization. Again, a sequence of rf-pulses\nwas used to bring each of the magnetization components into view [7].\nThe quality of data obtained by this method is exempli\fed by Fig. 3, which\nreveals the response of a quantum degenerate gas to being quenched across a\nparamagnet-to-ferromagnet phase transition [9]. The physics revealed by these im-\nages is discussed below in Sec. 4. Here, let me just emphasize how eye-opening\nthese images were. Prior studies of spin-mixing dynamics in rubidium spinor\ngases [10, 11], relying on Stern-Gerlach time-of-\right imaging, had revealed damped\noscillations in the Zeeman-state populations, with very little spatial resolution. The-October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n6 Dan M. Stamper-Kurn\nt=0 spin along x  \nt=0 spin along y  \nt=0 spin along z  \nπ/2 \nπ/2 \nπ/2 \ntime time \nFig. 2. All three components of the vector magnetization can be quanti\fed by taking a sequence\nof imaging frames while the gas magnetization precesses in an applied magnetic \feld, and then\nis re-oriented using rf pulses. Shown is the expected image frame sequence for a gas initially\nmagnetized along each of the cardinal directions. The \frst set of panels shows the expected signal\nduring nine equally spaced image frames taken during Larmor precession. The image signal from\ntransversely magnetized sample shows a temporal oscillation, the phase of which determines the\ndirection of the magnetization within the transverse plane. Following a \u0019=2 rf pulse, which causes\nthe longitudinal magnetization to be rotated into the transverse plane, a second set of several image\nframes can be analyzed to quantify that transverse magnetization strength. A two-dimensional\nimage of the vector magnetization is reconstructed from such an imaging sequence. The entire\nsequence occurs within just a few ms, faster than the dynamical timescales for spin transport and\nmixing dynamics.\norists guessed that the damping was due to some sort of multi-mode spatial dynam-\nics [12], but it was impossible to tell from the data whether that was the proper\nexplanation. In any case, such multi-mode e\u000bects did not seem like something in-\nteresting to examine further, but rather were seen as something to try to eliminate\nby working with smaller and more tightly con\fned samplesa. In contrast, our high-\nresolution imaging of spin distributions revealed a rich dynamics: the development\nof spin domains with magnetization lying in all directions transverse to the magnetic\n\feld direction, separated by snaking domain walls and pierced by spin vortices.\nIndeed, the progress of ultracold atomic physics over the past couple of decades\nhas shown, time and time again, that improved imaging methods allow us to exam-\nine interesting physical phenomena whose study was previously inaccessible, and,\nindeed, inconceivable. Examples of such leaps include the \frst realization of non-\naThe strategy of restricting the sample geometry to achieve strictly single-mode dynamics has\nbeen pursued and has turned out to be very bene\fcial for isolating and controlling the many-\nbody-quantum nature of spin-mixing dynamics; see Refs. 13{15 for some examples.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 7\n(b) 60 90 120 150 210 180 30 ms \nMX\nMY\nFig. 3. The transverse magnetization of a spinor Bose-Einstein condensate was measured at a\nvariable time (indicated at top of image) after it was quenched across a paramagnet-to-ferromagnet\nphase transition. The quantum degenerate gas was initially allowed to equilibrate in the jmF= 0i\nstate at a high quadratic Zeeman shift qwhere that state is the equilibrium spin state. The\nquench was performed by rapidly lowering the quadratic Zeeman energy to near zero, encouraging\nthe formation of a transversely magnetized ferromagnetic condensate. The data are encoded in\na color scheme (see color wheel at lower left) where the transverse magnetization orientation is\nindicated by the hue, and its magnitude by the color saturation. The image of each separate\ncondensate has a width of about 30 microns. The magnetization appears over 10's of ms after the\nquench, resulting in the formation of some regions with small neighboring spin domains of opposite\norientation (highlighted by the yellow box and magni\fed for view for the latest time image) and\nalso large domains of common orientation (green box). The \fgure is adapted from Ref. 9.\ndestructive imaging [16] that allowed for studies of bosonic stimulation in the for-\nmation of a Bose-Einstein condensate [17], high-resolution objectives that allowed\nfor the realization of Josephson junctions formed by optical potentials [18] and ob-\nservations of scale invariance [19] and quantum criticality [20] in two-dimensional\ngases, and the recent studies of dynamics and magnetism in the single-site-resolving\nquantum gas microscope [21, 22]. I am certain this trend will continue.\n2.2. Absorptive spin-sensitive in-situ imaging\nDispersive imaging has been billed as a form of \\non-destructive imaging,\" implying\nthat imaging the gas does not deposit so much energy so as to utterly destroy\nit. Perhaps a better moniker is \\less-destructive imaging,\" for the act of imaging\nnecessarily disturbs the gas. There is a tradeo\u000b between how much information oneOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n8 Dan M. Stamper-Kurn\nwants to extract the gas and how much one wants to preserve the gas for continued\nevolution and imaging. The o\u000b-resonant probe light used in dispersive imaging\nperturbs the gas in several ways. Characterizing the vector magnetization, and\neven more so the tensor nematicity, of a gas requires a sequence of several images,\nand, thus, that each image be non-destructive enough to leave enough signal for the\nsubsequent image. Therefore, there is an upper bound on the signal to noise with\nwhich these characteristics can be quanti\fed. These bounds were explored in my\ngroup's work on spatially resolved magnetic \feld sensing, which is discussed further\nin Sec. 5.\nThe limitations we identi\fed led us to consider an alternative imaging scheme:\nabsorptive spin-sensitive in-situ imaging (ASSISI). The idea here is to analyze the\npopulation of atoms in one set of atomic levels jAiby promoting a small fraction\nof them to unoccupied atomic levels jBi, and then to image those promoted atoms\nselectively with probe light that is resonant with levels jBibut far detuned from\nlevelsjAi. The promoted atoms are allowed to scatter very many photons, allowing\none to obtain a low-noise image of their distribution. After the image is taken, the\nimaged atoms are expelled from the trap, leaving the remaining sample of atoms\nrelatively unperturbed. The application of this scheme to imaging the jF= 1i\nhyper\fne states of87Rb is described in Fig. 4. The signal-to-noise ratio of this\nimaging method is found to be limited by shot-noise \ructuations in the number of\npromoted atoms, and vastly improved over the quality of dispersive imaging.\nAnother important bene\ft of ASSISI is that one can fully characterize the atomic\nspin state. In particular, the dispersive imaging method described previously per-\nmits measurements of the spin-quadrupole moment only through the linear birefrin-\ngence of the atomic gas. For alkali atoms, this linear birefringence is weak except for\nlight very close to atomic resonances, where the absorption of the probe light causes\nsevere problems. In contrast, ASSISI permits a complete in-situ \\Stern-Gerlach\"\nanalysis of the Zeeman sublevels along any axis. Performing such analyses along\nseveral axes permits one to determine the atomic spin state completely, including\nthe \u0001m= 2 coherences associated with the spin-quadrupole moments, and even\nhigher-order coherences in the case of high-spin gases.\n3. Coherent propagation of magnons\nNow I turn to the \frst of three examples where high-resolution in-situ imaging has\nbeen useful for elucidating magnetic phenomena in quantum gases, and where such\nphenomena have been examined both from the viewpoint of materials science and of\natomic physics. We start with understanding how magnetic excitations propagate\nin such gases.\nUltracold spinor Bose-Einstein gases will condense into a state that is both mag-\nnetically ordered and also super\ruid. After all, it is the magical nature of Bose-\nEinstein condensation to cause particles to occupy the lowest-energy single particleOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 9\nF = 2\nF = 1\n2\n4\n6\n1\n3\n5microwave selection optical probe\nFig. 4. Absorptive spin-selective in-situ imaging of an F= 1 rubidium gas. Left: The ground\nelectronic state has two hyper\fne states, denoted by quantum numbers F= 1 andF= 2. Here\nthe Zeeman sublevels within each manifold are separated by the linear Zeeman shift, which is\nshown out of scale to the very large energy separation between hyper\fne spin manifolds. Our task\nis to image a gas that occupies the F= 1 levels. This is accomplished in six steps. First, a short\nmicrowave pulse is used to drive a small fraction of atoms from one of the F= 1 magnetic sublevels\nto the unoccupied F= 2 state. Second, an optical probe measures the in-situ column density of\nthe selected atoms. In the process the atoms are optically pumped, so that the absorption image is\nquantitative, and also the atoms are heated su\u000eciently so that they are expelled from the trap after\nimaging. This selection/probe sequence is repeated three times to gather three image frames. The\nimages show a standing wave of magnons in a ferromagnetic initial state. The scale bar indicates\na 50\u0016m distance.\nstates with a macroscopic occupancy, even at temperatures Twhere the energy dif-\nference between the lowest and next-lowest energy is much smaller than the thermal\nenergy available kBT. Given this tendency, one expects the spinor Bose-Einstein\ncondensates to adopt the single-particle spin state with the lowest energy. Absent\nany external spin-dependent in\ruences such as applied magnetic \felds, the domi-\nnant spin-dependent energy that remains is the contact interaction, which describes\nthe e\u000bect of the short-range interaction potential between the neutral atoms collid-\ning at very low incident energy. This interaction can be expressed by the operatorOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n10 Dan M. Stamper-Kurn\nC(Fi;Fj)\u000e3(ri\u0000rj) whereiandjare the indexes of the two atoms colliding, Fi;j\nare their spin operators and ri;jtheir positions. We know that C(Fi;Fj) must\nbe rotationally symmetric. Here I have neglected the magnetic dipole interactions,\nwhich are treated as a long-range interaction and are rotationally symmetric only\nunder rotations both in position and spin space.\nTo keep the discussion focused, I will concentrate on the speci\fc example of\ntheF= 1 spinor gas of87Rb, for which one \fnds C(Fi;Fj) =c(1)\n0+c(1)\n1F1\u0001F2,\nwithc(1)\n1<0b. Such an interaction favors ferromagnetic ordering, in which all\natoms' spins are commonly oriented so as to minimize the contact interaction energy.\nFor this reason, we may denote the F= 187Rb quantum gas as a ferromagnetic\nsuper\ruid.\n3.1. Nambu-Goldstone bosons of magnetically ordered super\ruids\nSystems that spontaneously break a continuous symmetry commonly possess gap-\nless bosonic collective excitations, known as Nambu-Goldstone bosons. To visualize\nsuch excitations, we consider an equilibrium state in which a continuous symmetry\nSis broken; e.g. a ferromagnet with uniform magnetization (Fig. 5a). If one applies\nan in\fnitesimal operation within Sthroughout the system, the energy is unchanged.\nNow consider that one transforms the system by applying an in\fnitesimal symmetry\noperation that varies in space with a well-de\fned wavevector k. Such a transfor-\nmation de\fnes a bosonic collective excitation: the Nambu-Goldstone boson. This\nexcitation may be expected to raise the energy of the system, since now the order\nparameter is nonuniform. However, in the limit k!0, the transformed system\nis nearly uniform, so the rise in energy should tend to zero, implying the boson is\ngapless. The Nambu-Goldstone theorems do not hold for systems with long-range\ninteractions, e.g. for charged super\ruids and for the broken electroweak symmetry.\nIn recent years, the Nambu-Goldstone theorems have been re-examined in an\nattempt to reconcile the phenomena of symmetry breaking in relativistic and non-\nrelativistic scenarios: In particle physics, excepting the Higgs mechanism, each\nbroken symmetry leads to a gapless linearly dispersing bosonic mode. In contrast,\nin material systems, not every broken symmetry produces a new boson, and many\nof these bosons disperse quadratically, rather than linearly. The mathematical\nconditions leading to these di\u000berences have only recently been identi\fed [23, 24].\nNow we return to considering the ferromagnetic super\ruid realized by the F= 1\n87Rb gas. It breaks three symmetries: a combination of gauge symmetry and ro-\ntational symmetry, and the symmetries of rotations about the transverse xandy\naxes. While one might naively expect the ferromagnetic state to carry three Nambu-\nbThe quantity c(1)\n1is determined by di\u000berences in s-wave scattering lengths for collisions in di\u000berent\nangular momentum channels, which are in turn determined by the particular form of the atom-atom\ninteraction potential, which is in turn determined by comparison to a variety of measurements.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 11\n50µm\nm=−1\nm=0\nm=1(d) (e)\nm=−1\nm=0\nm=1(a)\n(b)\n(c)\nFig. 5. Magnon schematic (left): (a) A material equilibrates in a uniform broken symmetry\nstate (e.g. a ferromagnet with magnetization pointing up). (b) Applying a uniform symmetry\ntransformation (e.g. rotating the magnetization) does not change the free energy. (c) This suggests\nthat an inhomogeneous transformation of the system, which varies only over long length scales,\ncosts little extra energy. This excitation is the Nambu-Goldstone boson, which, in the absence of\nlong-range interactions, is gapless at the limit of zero wavevector. Magnon images (right): Zeeman-\nstate populations are characterized in-situ by a sequence of microwave excitation and resonant\noptical absorption imaging. (d) A localized coherent wavepacket is created using the vector ac\nStark shift of light focused tightly within the center of the condensate. (e) A spatial grating\nof magnons with well de\fned wavevectors is created using two beams of light that form a high-\ncontrast interference pattern at the location of the trapped gas. For weak magnon excitations atop\na longitudinally polarized gas, the mF= 0 population is representative of the magnon population.\nGoldstone bosons, in fact there are just two. The breaking of gauge/rotation sym-\nmetry (rotation about zgives a phase shift) is the origin of the gapless, linearly\ndispersing phonon. The consequence of the breaking of transverse rotational sym-\nmetries is di\u000berent. Because the generators of the two broken symmetries have a\ncommutator with a non-zero expectation value in the ground state [23], the two\nsymmetries generate only a single Nambu-Goldstone boson, and the dispersion re-\nlation of this boson is quadratic with wavevector, i.e. E(k) =~2k2=(2m\u0003). This\nboson is the ferromagnetic magnon. It corresponds to rotating the uniform ground\nstate by an in\fnitesimal angle \u0012about the transverse axis n\u001e=xcos\u001e+ysin\u001e\nwith\u001e=k\u0001r. Under mean-\feld theory, the e\u000bective mass m\u0003is equal to the atomic\nmass.\n3.2. Coherent atom optics with magnon excitations\nMagnons can be written onto a spinor Bose gas by realizing a spatially dependent\nrotation of the gas magnetization. Such rotation can be e\u000bected using the vector\nac Stark shift of an o\u000b-resonant light \feld [25], which imposes an e\u000bective magnetic\n\feld oriented along the helicity of the light \feld and is proportional in strength to\na product of the light ellipticity and intensity.\nTo see how this light-induced energy shift creates magnon excitations, consider\nthe e\u000bect of light with \u001b+circular polarization along the xaxis, with an intensity\npattern ofI(r). The vector ac Stark shift creates an e\u000bective inhomogeneity in the\nmagnetic \feld \u0001 B(r)/I(r)x. Suppose a fully polarized ferromagnetic condensate\nis prepared with its magnetization uniform, and oriented transversely to a magnetic\n\feld along x. The vector ac Stark shift now causes the condensate magnetizationOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n12 Dan M. Stamper-Kurn\nto precess by an additional angle \u0001 \u001e/I(r) in the transverse plane. This inho-\nmogeneous spin texture represents a coherent wavepacket of magnons (realized in\nRef. 26). Similarly, consider that the ferromagnetic condensate is prepared with\nuniform longitudinal polarization along a magnetic \feld oriented perpendicular to\nx. If the intensity or polarization of the incident light is now modulated at the\nLarmor precession frequency, the atoms will undergo Rabi oscillations due to the\napplied light \feld, again creating a coherent magnon excitation.\nThe magnons produced in this way were characterized by our ASSISI method,\nas shown in Figs. 5. To interpret these images, we recall that a rotation by the\nangle\u0012about the n\u001eaxis transverse to the magnetization of a ferromagnetic spinor\ntransforms the condensate wavefunction, to lowest order in \u0012, as\n =pn0\n@1\n0\n01\nA!pn0\n@1\nei\u001e\u0012=p\n2\n01\nA: (1)\nTherefore, to lowest order, a magnon excitation atop a condensate in the jmF= +1i\ninternal state is just a single atom in the jmF= 0istate. Therefore, an image of the\nmF= 0 atomic population is e\u000bectively an image of the (small) magnon density.\nConsider that one prepares a spatial grating of magnons using the vector ac\nStark shift of light beams intersecting with wavevector di\u000berence q. One thereby\nprepares a coherent magnon wave with the initial wavefunction\n M(t= 0)/1 +\u0011cos(q\u0001r) = 1 +\u0011\n2\u0000\neiq\u0001r+e\u0000iq\u0001r\u0001\n(2)\nwhere we have assumed the ferromagnetic condensate to be stationary and uniform.\nThe factor\u0011accounts for the visibility of the light interference pattern. This density-\nmodulated magnon wave is a superposition of magnons at three wavevectors: k2\nf0;q;\u0000qg. After evolving for a time t, the energy di\u000berence between stationary\nand moving magnons causes the magnon density distribution to become\nj M(t)j2/1 +\u00112cos2(q\u0001r) + 2\u0011cos\u0012E(q)\u0000E(0)\n~t\u0013\ncos(q\u0001r) (3)\nWe observe the amplitude of the density modulation at wavevector qis temporally\nmodulated at the Bohr frequency ( E(q)\u0000E(0))=~, providing an interferometric\nmeasurement of the magnon dispersion relation. Such modulations are shown in\nFig. 6. By Galilean invariance, the contrast modulation frequency is Doppler free.\nContrast interferometry was demonstrated in a scalar Bose-Einstein condensate,\nwhere the signal was strongly sensitive to density-dependent shifts of the Bragg\nresonance frequency [27].\nFrom a materials science perspective, a precise measurement of the magnon dis-\npersion relation is a precise test of quantitative many-body theories that describe\nthe interacting ferromagnetic Bose gas. A recent theoretical calculation [29] that\ngoes beyond mean-\feld theory to consider the e\u000bects of phonon-magnon interac-\ntions predicts an increase in the magnon mass above the bare atomic mass by aOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 13\n0 20 40 80 10005101520250 20 40 60 80 100\n−0.06−0.04−0.020.00\n60\nTime (ms)Contrast(a)\nB\n(c)\n(b)\n(ω(k)−ω(0))/2 π  (Hz)ω/ωfree − 1\ns-wave MFTs-wave MFT\nk/2π (µm−1)0 50 100 150 200 2500.0 0.1 0.2 0.3 \n N4)m  (10\nF=0ω/2π  (Hz)\n0 4 8 12202224\nFig. 6. In our magnon interferometer, a standing wave of magnons is imprinted onto the con-\ndensate, being composed of magnon waves at three distinct wavevectors. Free propagation of the\nmagnon waves causes the contrast of the spatial modulation of the magnon density at a wavevec-\ntorqoscillates sinusoidally at frequency 2( E(q)\u0000E(0))=~. (a) Images of the mF= 0 population\nduring this free propagation shows high initial contrast, diminished contrast after half a temporal\ncycle, and the high contrast obtained after a full temporal cycle (with fringes \u0019out of phase with\nthe initial pattern). (b) The contrast shows clear temporal oscillations. (c) From these oscilla-\ntions, we obtain the magnon recoil energy, shown in closed black circles. The recoil frequency\nis corrected for a frequency shift with the number of magnons produced in the interferometer,\nwith the measured recoil energy being the value obtained by interpolation to zero density. Data\nare compared to the theoretical prediction of mean \feld theory (red line) that the magnon recoil\nenergy should equal the free-atom recoil energy. Our data show the magnon mass to be slightly\nlarger than the bare atomic mass. Figure reproduced from Ref. 28.\nfractional amount of about 3 \u000210\u00003. In our experiment, we \fnd the magnon mass\nto be heavier than the atomic mass by a fractional amount of 3 \u000210\u00002. The ori-\ngin of this discrepancy is unknown, though one should note that the theoretical\ncalculations were performed for a uniform zero-temperature gas with no magnetic\ndipole interactions, whereas the samples used in our experiment were inhomoge-\nneous, \fnite-sized, at non-zero temperature, and possessed discernable magnetic\ndipolar interactions (as were also seen in the experiment). Given the precision of\nthe magnon contrast interferometer and the \rexibilty of the experimental setup, it\nis clear there will be scienti\fc payo\u000bs in continuing this investigation: probing gases\nof variable size and geometry, isolating the e\u000bects of dipolar interactions, varying\nthe temperature in a controlled manner, probing di\u000berent initial states, and so on.\nFrom an atomic physics perspective, the remarkable observation is how close\nis the magnon mass to the atomic mass: In a gas with a density on the order of\n1014cm\u00003, interactions appear to shift the magnon recoil frequency only at the\nprecent level, with the absolute value of the shift being less than a Hertz. In\ncontrast, the density-induced shifts in the Bragg resonance frequency would beOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n14 Dan M. Stamper-Kurn\norders of magnitude larger.\nIndeed, the properties of ferromagnetic magnons have enticing implications for\natom interferometry. Their gapless nature implies that magnons are created without\na density-dependent interaction energy shift. With the magnon mass being just\nabout equal to the atomic mass, magnon excitations propagate just about like free\nparticles. Thus, an interferometer based on the free propagation of magnons within\na ferromagnetic spinor Bose-Einstein condensate at equilibrium (equal chemical\npotential across the condensate) is equivalent to an atom interferometer using free\nparticles at zero gravity and in vacuum.\n4. Spin-mixing instability in a spatially multi-mode system\nThe Section above illustrates how the study of atomic materials can complement\nthat of solid-state materials by allowing and justifying precise quantitative tests of\ntheory. Another advantage o\u000bered to the study of materials science is the study\nof materials far from equilibrium, which is possible in atomic gases owing to the\nvast separation of timescales between the equilibration time (from just below a\nmillisecond to much longer) and the shortest time (e.g. microseconds) in which the\nsystem Hamiltonian can be varied or the atomic internal state can be modi\fed.\nOne scenario considered for such non-equilibrium dynamics is the response of\nan initially equilibrated many-body system to a rapid change of the system Hamil-\ntonian. Similar to quenching a hot iron bar by submersing it in water, here we\nconsider a quench of a quantum system to a condition that favors a di\u000berent equi-\nlibrium state. The many-body system is thus uniformly in a non-equilibrium state.\nThe deviations from equilibrium can propagate spatially across the system, allow-\ning correlations and entanglement to span a distance that grows with time. The\ndeviations can also serve to nucleate instabilities [30].\nSeveral works by my group have examined the quench of a degenerate spinor\nBose-Einstein gas across a symmetry breaking phase transition. In particular, we\nconsider the87RbF= 1 spinor gas for which the spin-dependent contact interac-\ntions favor ferromagnetism, as discussed above. Atop these interactions, we add a\nsingle-particle energy term, a quadratic Zeeman energy qF2\nzalong an experimentally\nselected axis z. For large positive q, this term favors a condensate fully polarized in\nthejmF= 0istate quantized along the zaxis. The Hamiltonian of the system is now\nsymmetric under rotations about zand also under uniform gauge transformations\n(multiplication of the system wavefunction by a uniform complex phase). For large\npositiveq, the non-magnetized equilibrium state breaks just the gauge symmetry, in\nestablishing a complex, scalar super\ruid order parameter. For small positive q, the\nferromagnetic state is magnetized transverse to the zaxis, and thus breaks the ax-\nisymmetry of the Hamiltonian as well as gauge symmetry. These states of di\u000berent\nbroken symmetry are separated by a phase transition, which can be traversed by\nchangingq. An additional change of symmetry occurs for q<0, where the favoredOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 15\nlongitudinally magnetized state breaks the discrete Z2time-reversal symmetry but\nretains axisymmetry.\nLet us focus on a quench from the non-magnetized, axisymmetric phase at large\npositiveqto the symmetry broken ferromagnetic phase at small positive q. How\ndoes a system quenched through this transition dynamically break the previously\nunbroken axisymmetry?\n4.1. The Kibble-Zurek picture of inhomogeneous symmetry breaking\nIn the study of materials one encounters many cases of systems that undergo\nsymmetry-breaking transitions, typically after being rapidly cooled (conventionally\nquenched). Considering systems as conventional as ferromagnets or supercooled\nwater, we observe that symmetry breaking occurs locally. Small regions of the ma-\nterial undergo the phase transition independently, and these regions then serve to\nnucleate larger regions of commonly broken symmetry. Where these regions abut,\nwe observe defects { domain walls in magnets or crystal defects in ice { that signify\na con\rict between the choices of broken symmetry in di\u000berent regions, and that\nheal only very slowly as these con\ricts are resolved.\nThe same phenomena are understood to occur in quantum materials undergo-\ning quantum phase transitions. The picture often invoked to describe the symmetry\nbreaking dynamics is called the \\quantum Kibble-Zurek mechanism,\" [31{33], fol-\nlowing earlier considerations for classical thermal phase transitions [34, 35]. This\npicture divides the quench dynamics into three periods in time. In the \frst, a\nquantum system quenched across a phase transition will form regions of commonly\nbroken symmetry with a size that shrinks as the quench rate is increased, with the\nexact relation between size and rate determined by the character of the dispersion\nrelation for excitations near the phase transition point. Second, the jumbled sym-\nmetry breaking will result in a tangle of defects of various types that depend on the\ngeometry of the order parameter manifold. Third, these defects will slowly coarsen\nand heal over long times as the system becomes ordered at longer length scales.\nSo far, the predictions for the second and third periods of evolution have been\nexamined in spinor Bose-Einstein condensates that are su\u000eciently large in either\none or two dimensions so that it makes sense to invoke the Kibble-Zurek mecha-\nnism to describe their evolution. The spatially inhomogeneous symmetry breaking\nfollowing a very rapid (essentially instantaneous) quench across the aforementioned\nparamagnet-to-ferromagnet transition is shown in Fig. 3 [9]. Regions of transverse\nmagnetization appear some tens of milliseconds after the quench. Regions of dif-\nferently oriented magnetization are joined either by extended regions in which the\nmagnetization gradually varies from one orientation to another, or by sharp lines\nof apparently unmagnetized gas (although it is possible the gas within these lines\nremains fully magnetized, with the magnetization varying over a length scale be-\nlow our imaging resolution). The dynamics of symmetry breaking also give rise toOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n16 Dan M. Stamper-Kurn\ntopological defects: polar-core spin vortices that, in fact, turn out to be the only\ntopologically distinct vortex for the F= 1 ferromagnetic super\ruid. The quantum\nquench in a polar F= 1 spinor condensate has also been studied [36].\nAs for the third period of the Kibble-Zurek chronology, several elements of coars-\nening dynamics have been examined. Early studies [37, 38] shed light on the growth\nof spin domains in one-dimensional spinor Bose-Einstein condensates through two\ndistinct mechanisms: Just as the normal and super\ruid components of super\ruid\nhelium show distinct transport characteristics, here the non-condensed component\nof the gas allows for coarsening through thermal activation and re-equilibration,\nwhile the condensed component of the gas allows for domain growth through coher-\nent quantum tunneling across domain walls.\nMy group examined the process of coarsening in two dimensional systems. We\nexamined the evolution of an initially hot and unmagnetized gas after it was cooled\nacross the Bose-Einstein condensation phase transition and allowed to equilibrate\nfor several seconds. Regions of magnetized gas were observed after crossing the\ntransitionc. In contrast with the \u001810 ms timescale for the initial formation of spin\ndomains, here we observe a seconds-long timescale for domain coarsening. One\ndisappointing conclusion of this study is that the equilibration timescales for the\n87RbF= 1 spinor Bose-Einstein gas are so long as to be experimentally inaccessible\nfor spatially extended gases.\n4.2. Symmetry breaking through parametric ampli\fcation of spin\n\ructuations\nLet us return to the question of how an isolated system spontaneously breaks sym-\nmetry. If the initial state of the system is symmetric, and the Hamiltonian of the\nsystem conserves that symmetry at all times in the evolution, then the \fnal state of\nthe system is perforce symmetric. In that case, it stands to reason that the states\nwe observe at late times after the quench are indeed symmetric superpositions of\nstates of macroscopically broken symmetry.\nQuantum optics o\u000bers a candidate for such a superposition state: squeezed\nvacuum. Applying a phase-sensitive parametric ampli\fer to the vacuum state causes\none quadrature of the electromagnetic \feld to be ampli\fed. Both the vacuum state\nand the squeezed vacuum state are symmetric, in that quadrature measurements\nare equally likely to give positive or negative values. However, measurements on\ncAs far as we could tell, the condensation of the gas and the formation of magnetized spin domains\noccurred simultaneously. However, according to our experimental protocol, for each gas sample we\nwere able to measure either the presence of a condensate (in time-of-\right images) or the presence\nof transverse magnetization (in in-situ images). It would be interesting to perform both probes in a\nsingle experimental realization so that one can resolve better whether Bose-Einstein condensation\noccurs temporally before the formation of magnetized domains. I suspect that it does.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 17\nthe squeezed state are likely to yield either a large positive or large negative value,\ndeviating from the vacuum levels by potentially macroscopic amounts.\nPerhaps, then, the dynamics of a many-body quantum system following a quan-\ntum quench may be described as the parametric ampli\fcation of initially symmetric\nand microscopic vacuum \ructuations. Questions arise: What is being ampli\fed?\nWhat is being squeezed? Is the ampli\fer quantum-limited? What is the gain of\nthis ampli\fer? If this ampli\fer is applied to a spatially extended condensate, the\nampli\fer might be expected to act simultaneously but separately on \ructuations\nin di\u000berent portions of the condensate. If so, how are di\u000berent spatial patterns\nof \ructuations a\u000bected by the ampli\fer? In other words, what is the spatial gain\nspectrum of this ampli\fer?\nAnswers have been provided to several of these questions. It is understood that\nthe spin \ructuations in this case { ampli\fcation through the spin-mixing instability\nof a gas initially in the jmF= 0iquantum state { exist in two independent planes,\nrepresenting two di\u000berent polarizations of magnon excitations of the initial state\n[39, 40]. Each plane is spanned by two observables: one component of the tranverse\nspin and one component of the spin-quadrupole tensor, serving as the independent\nand non-commuting quadratures equivalent to the electric \feld quadratures of light\nor the position and momentum quadratures of a mass on a spring. The transverse\nspin quadrature can be measured directly using circular birefringence dispersive\nimaging [6, 9, 39]. The spin-quadrupole moment can be measured by \frst rotating\nit onto the transverse-spin quadrature by brie\ry applying a strong quadratic Zeeman\nshift, as has been demonstrated in single-mode experiments probing this instability\n[13].\nFor spatially extended systems, one can characterize the gain of this ampli\fer in\nmomentum space, just as one characterizes an ampli\fer that acts on a continuous\nsignal by its gain spectrum in frequency space. Through a linear stability analysis\nof the Gross-Pitaevskii equation that describes the coherent evolution of an initially\nuniform spinor Bose-Einstein condensate, one \fnds the magnon dispersion relation\nfor thejmF= 0iinitial state, given as E2\ns= (\u000fk+q)(\u000fk+q\u0000jc(1)\n1jn) where\n\u000fk=~2k2=2mis the free particle kinetic energy at wavenumber kandnis the\ncondensate density. For q < q 0= 2jc(1)\n1jn, there is a range of wavenumber where\nE2\ns<0 indicating a regime of parametric instability with a temporal gain given\nbyp\njE2sj=~. The colored spectrum of this ampli\fer can be interpreted as de\fning\nparticular localized spatial modes that are ampli\fed with highest gain [40, 41].\nRough features of this gain spectrum, and how it tunes with the quadratic Zeeman\nshift, were observed in Ref. 39 (Fig. 7).\nThe noise characteristics of this parametric ampli\fer have been examined in\nsingle-mode spinor systems, where it has been shown to yield highly spin squeezed\nstates [13]. My group attempted to characterize the noise limits of this parametric\nampli\fcation in a spatially extended spinor gas, using our high-resolution imaging\nto quantify the ampli\fer output after a variable time of ampli\fcation [39]. Un-October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n18 Dan M. Stamper-Kurn\nMy~\nt = 87 ms\n t = 157 ms\n-2 0 2 5 9 14 q / h =25 µm\nHzMx~\nFig. 7. An electronic ampli\fer has a spectrum, describing the fact that gain at particular temporal\nfrequencies is stronger than at other frequencies. Similarly, in a spatially extended spinor Bose-\nEinstein condensate, quenching to conditions that favor symmetry breaking into a ferromagnetic\nstates causes the condensate to act as a spin-\ructuation ampli\fer that has a gain that depends\non the spatial mode structure of the \ructuation. In particular, quenching the system to lower\nvalues of the quadratic Zeeman shift (images further to the left) causes spin \ructuations with\nsmaller wavelength to be ampli\fed more strongly. This spatial gain spectrum is visible in images\ntaken at variable times after the quench. The initial spin noise in the spinor condensate, which\nwe presume to be present at equal strength at all wavevectors (spatially white) is ampli\fed by a\nwavelength-dependent spin ampli\fer. The macroscopic magnetization spectrum produced by this\nquench shows the spatial lengths most favored by the ampli\fer. Each image here is a map of\nthe tranverse magnetization in a single condensate, at one of two di\u000berent times after the quench\n(indicated on the left) at several di\u000berent values of the quadratic Zeeman shift (indicated on the\nbottom). Figure taken from Ref. 39.\nfortunately, our uncertainty in the temporal gain of the ampli\fer, stemming from\nuncertainty in the scattering lengths of87Rb, did not permit us to place tight con-\nstraints on the ampli\fer noise.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 19\n5. Spatially resolved magnetometry with a quantum \ruid sensor\nFinally, let me consider the use of cold atomic gases for spatially resolved magne-\ntometry to study materials. At present, several technologies are used for spatially\nresolved magnetometry at micron length scales, including superconducting quantum\ninterference devices (SQUIDs), scanning Hall probe microscopes, magnetic force\nmicroscopes and magneto-optical imaging techniques [42]. Among these, the tech-\nnologies o\u000bering the highest \feld sensitivity for single-pixel measurements are the\nSQUID microscope and the scanning Hall probe microscope. SQUID sensors excel\nat measuring high-frequency signals, e.g. in the GHz range, where quantum-limited\nperformance has been achieved [43], equivalent to roughly S= 1 pT2=Hz sensitiv-\nity in a 100 \u0016m2measurement area. For low frequency signals, the performance of\nSQUID microscopes is compromised by 1 =f\rux noise, achieving S'(40 pT)2=Hz in\na 100\u0016m2measurement area [44, 45]. The scanning Hall probe microscope achieves\na worse sensitivity of S'(100 nT)2=Hz, but its high spatial resolution (optimal\naround 200 nm) makes it competitive with SQUID sensors at length scales at or\njust below one micron.\nBoth these technologies require the sensor to be scanned sequentially across the\n\feld of view of the microscope, reducing the amount of measurement time available\nat each resolved pixel. Thus, magnetic \feld images with many resolved pixels are\nacquired with correspondingly worse measurement sensitivity. This disadvantage\nis circumvented by the use of magneto-optical imaging, which relies on the linear\nFaraday e\u000bect (birefringence) of solid-state materials. Magneto-optical imaging\nallows for the simultaneous acquisition of a magnetic-\feld image across a wide \feld\nof view, with spatial resolution around 1 \u0016m matching the wavelength of light used\nfor imaging. The sensitivity of this technique in present implementations is lower, at\nsingle pixels, than the SQUID and scanning-Hall probes described above. However,\nthis worsening of sensitivity is made up by the parallelism of measuring across a\nwide \feld of view.\nThese magnetic microscopy methods are used in an extensive range of appli-\ncations. Scienti\fc applications include imaging vortex structures in superconduc-\ntors [42, 46] and resolving their temporal dynamics [47] and probing for spontaneous\nmagnetization in p-wave superconductors [48]. A major technological application is\nnon-destructive evaluation of metallic objects and microfabricated devices by means\nof detecting eddy currents or bound currents in stressed ferromagnetic materials,\nsusceptibility imaging, and detailed mapping of current \row [49].\nA cold-atom magnetometer is compatible with many of the applications dis-\ncussed above, and yet achieves a single-pixel sensitivity that is orders of magnitude\nbetter than the SQUID and scanning Hall probe sensors while maintaining the\nadvantage of detecting over a wide \feld of view in a single measurement. The pos-\nsibility of using spinor Bose-Einstein gases for high precision magnetometry was\nestablished by prior work of my group [26]. In that work, a dense, spin-polarizedOctober 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n20 Dan M. Stamper-Kurn\nBose-Einstein condensed gas of87Rb was con\fned in a near-planar geometry in\na far-o\u000b-resonant optical dipole trap. A measurement was begun by tipping the\nspin polarization to lie transverse to an applied magnetic \feld, with a spatially uni-\nform initial orientation. The gas was then allowed to undergo Larmor precession,\nat an angular frequency determined by the local magnetic \feld magnitude B(\u001a)\nas!L(\u001a) =\rB(\u001a) where\r=gF\u0016B=~is the atomic gyromagnetic ratiodand\u001a\nindicates position in the imaged 2D plane. After a measurement time \u001c, the Lar-\nmor precession phase \u001e(\u001a) was measured by high-resolution magnetization-sensitive\nimaging.\nLet me highlight several features of that work. First, the cold atom gas rep-\nresents an extended sensing \feld, providing a simultaneous spatial (2D) map of\nthe \feld magnitude. Compared with a scanning single sensor, such as a scan-\nning SQUID microscope, this simultaneous mapping rejects long-spatial-wavelength\ntemporal \feld \ructuations, which appear as a spatially uniform shift of the Lar-\nmor phase \u0016\u001e, to reveal static short-wavelength spatial inhomogeneities, which are\nmeasured as B(\u001a)\u0000\u0016B=1\n\r\u001c(\u001e(\u001a)\u0000\u0016\u001e). Such common-mode rejection bestows\nthe practical advantage of achieving superior magnetic \feld sensitivity even in the\nabsence of any magnetic shielding.\nSecond, the sensitivity of this proof-of-principle experiment already approached\nthe standard quantum limit for magnetic sensing. This limit derives from atomic-\nshot-noise limit to phase measurements, (\u0001 \u001e)2=N\u00001, where the number of atoms\nin a resolved pixel of area Ais given asN=n2DAwithn2Dbeing the areal atom\nnumber density. The measurement sensitivity is then given as\nS(A) =1\n\r21\nA1\nn2D\u001cD(4)\nThe sensitivity, with units [T2=Hz], scales inversely with the areal resolution, and\nimproves with higher atom column density ( n2D), single-measurement time \u001c, or\nduty cycle D\u00141 which is the ratio of the single-measurement time to the cycle\ntime between measurements.\nNow, consider implementing this measurement with a layer of ultracold atoms\ntrapped at micron distances from a mirror surface, where they serve as a sensor\nfor magnetic materials below, on, or above the surface (Fig. 8). For example, the\natoms might be trapped within an optical lattice trap, formed by light re\recting at\nshallow angles o\u000b the surface of the material being probed or o\u000b a thin \\cover slip\"\nplaced above the material. The quality of the optical trap is not critical: distortions\nof the potential by scattered light will distort the atomic density distribution, but\nwill not a\u000bect the Larmor precession of the trapped atoms.\nOnce placed within the surface-supported optical lattice trap, the spin-polarized\natoms can serve as a sensor of steady (dc) or alternating (ac) magnetic \felds. For\ndConventionally, \rwould be de\fned with a minus sign. We use \rhere to relate the Larmor\nfrequency directly to the magnetic \feld.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 21\nmagnetic sample\nmirror surface\noptical lattice\nlightatomic sensing medium∆B\nSide view Top view\nFig. 8. Magnetometery scheme. Side view: A mirror surface on a magnetic sample is\nused to produce a vertical optical lattice from laser light re\rected o\u000b the mirror surface.\nThe incident angle of the laser light controls the lattice spacing. Top view: Following\nLarmor precession for the single-shot measurement time, the orientation of the transverse\nmagnetization or nematicity is imaged at high spatial resolution across the atomic sample.\nLocal inhomogeneities of the magnetic \feld are visible in the inhomogeneous orientation\nof the atomic spin.\nthis, the atomic spin is prepared in a spatially uniform, transversely polarized,\ninitial state, allowed to evolve for a measurement time \u001cin the presence of the\n\feld produced by the sample, an applied bias \feld, and rf pulses. Last, the atoms\nare probed using state-sensitive imaging to reveal the phase of Larmor precession\nwith high spatial resolution, and the imaging data processed to derive the target\nmagnetic \feld.\nThe quantum-limited sensitivity to be expected from this device is derived from\nEq. 4. Consider that atoms are trapped in the optical-lattice site closest to the\nre\rective material surface, at a distance d= (\u0015=4) csc\u0012from the mirror surface,\nwhere\u0015is the wavelength of the light forming the lattice and \u0012is its angle of\nincidence. With \u0015on the order of 1 \u0016m,d\u0015250 nm. Let us assume a three-\ndimensional density of n3D= 3\u00021014cm\u00003, a thickness of d=4 for the lattice-\ncon\fned atomic layer, and single-shot measurement time of \u001c= 3 s. One then\nobtains\nS(A)'\u0012\n1:5pT2\nHz\u0013\n\u0002\u0012\u0016m2\nA\u0016m\nd1\nD\u0013\n(5)\nThis measurement sensitivity is far superior to existing dc SQUID magnetometers\n(limited at\u0018(10 nT)2=Hz) which are regarded presently as the most powerful\nmagnetometers at the micron length scale. Additional bene\fts of such a cold-atom\nmagnetometer include the following:\nAbsolute calibration: The magnetic sensitivity for this cold-atom device is based\ndirectly on atomic properties. In contrast, microfabricated SQUID magnetometers\nare subject to calibration errors and drifts. Magnetometers based on solid-state\nspin impurities are strongly in\ruenced by their inhomogeneous environments, so\nthat each impurity must be individually diagnosed.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n22 Dan M. Stamper-Kurn\nSimultaneous measurement across a wide spatial region: The cold-atom magne-\ntometer envisioned here excels at di\u000berential measurements of the magnetic \feld\nobtained in parallel in many resolved pixels of the sensor surface. This di\u000berential\nsensitivity provides three important advantages. First, long-wavelength magnetic\ndisturbances result in common-mode shifts of the Larmor phase and are thus re-\njected. Thus, the in\ruence of any source of stray magnetic \felds further than 100's\nof microns from the sensor is essentially eliminated, obviating the need for strict\ncontrols or shielding of the magnetic environment. Second, this feature obviates\nthe need to register precisely the positions of a magnetic object and the sensor;\nrather, the magnetic object is simultaneously located and measured by the cold-\natom sensor. Third, making simultaneous measurements over a total sensor area\nAtotat many resolved pixels, each of area A, provides an additional improvement\nin sensitivity, reducing Sin Eq. 4 by a factor A=Atot, compared to a single scanning\nsensor.\nOperation at high temperature: Atoms can be maintained at ultralow tempera-\nture even within micron distances of high-temperature objects [50]. Thus, a cold-\natom magnetometer can measure magnetic objects over a wide temperature range,\ne.g. for the study of magnetic phase transitions.\nOperation at high magnetic \feld: While high \felds disturb the operation of\nsuperconductor-based magnetic sensors, the immunity of our cold-atom magne-\ntometer to uniform bias \felds allows high sensitivity to be achieved even in a strong\nuniform applied magnetic \feld. This allows for sensitive detection of magnetic\nsusceptibility.\nLet us quantify the ideal sensitivity of such a cold-atom sensor for several ap-\nplications.\nDetecting currents: First, consider the task of detecting currents on or within\nthe sample, an essential component in many applications of magnetometry for non-\ndestructive evaluation of materials and devices, including detection of conducting\nmaterials by the generation of eddy currents, the identi\fcation of disturbances in\ncurrent \row through conductors, the assessment of material defects by the hysteretic\nmagnetization of stress-modulated materials, and the characterization of failures in\nintegrated circuits [49, 51]. A line current Irunning parallel to the mirror surface at\na depth ofrproduces a magnetic \feld of magnitude B=\u00160I=(2\u0019r). We compare\nthis \feld to the single-shot atom-noise-limited measurement sensitivity S(A)single =\n1\n\r21\nAn2D\u001c2. We consider the task of resolving features of the \feld produced by this\nwire (or many wires) with areal spatial resolution of A=d2set by the distance\nof the atoms from the mirror surface. As above, we take n2D=n3D\u0002d=4 and\nn3D= 3\u00021014cm\u00003, and single-shot measurement time \u001c= 3s. This gives a\nsignal-to-noise ratio of unity for a current of\nI= 4 pA\u0002\u0010\u0016m\nd\u00113=2\n\u0002\u0012r\n\u0016m\u0013\n(6)October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 23\nDetecting permeable materials: Second, consider the task of detecting a spher-\nical structure of radius R, composed of material with magnetic susceptibility \u001fm,\nand buried within the substrate at a distance r > R +dfrom the atomic sensor\nlayer. Let us assume the surrounding material has susceptibility \u001fm= 0. In an ap-\nplied magnetic \feld B0, the spherical structure acquires a uniform magnetization of\nM'(\u001fm=\u00160)B0. Outside the sphere, the induced magnetization produces a \feld\nequivalent to a magnetic dipole of moment m= (4=3)\u0019R3M, i.e. the \feld produced\nat the atomic gas above the mirror surface has magnitude B'(\u00160=4\u0019)jmj=r3.\nThis induced \feld is nearly uniform over an area A\u0018r2across the atomic gas;\ntaking this as the resolving area of the magnetometer, we conclude a single-shot\nmeasurement will detect a susceptibility of order\n\u001fm= 3\u000210\u00008\u0002\u0010\u0016m\nd\u00111=2\n\u0002\u0012r\n\u0016m\u00132\n\u0002\u0010\u0016m\nR\u00113\n\u0002\u0012G\nB0\u0013\n(7)\nThus, at just a few Gauss applied \feld, micron-scale structures with susceptibility\nbelow 10\u00008should be detectible; this is well below the typical susceptibility of\ndiamagnetic solids (10\u00005) and equal to those of diamagnetic gases (10\u00008).\nDetecting surface spins: Finally, we consider the task of locating and quantifying\nmagnetic particles on the mirror surface, at distance dfrom the atomic sensor. Let\nthese particles have a magnetic moment equal to Ne\u0016B(i.e.Necommonly aligned\nelectron spins). Using the areal resolution A=d2, the single-shot sensitivity is\nequivalent to\nNe= 0:9\u0002\u0012d\n\u0016m\u00133=2\n(8)\nThus, even nanoscale ferromagnetic particles should be detectible.\n6. Conclusion\nThere remain many things to be learned about magnetic phenomena in quantum\ngases, and methods of direct imaging may allow us to investigate them. One area\nthat is clearly ripe for investigation is the detailed dynamics of spin textures, do-\nmains, domain walls, and vortices. Our methods for imaging are now su\u000eciently\nre\fned that we should be able to investigate such dynamics by taking a set of im-\nages of a single gaseous sample as it evolves in time. It will be interesting to track\nthe motion of magnetic features so as to understand how they change and in\ru-\nence one another. Such investigations will reveal the basic elements of super\ruid\nhydrodynamics as it occurs in magnetic quantum gases, where the \row of mass\nand spin currents are interrelated. It will also be valuable to measure precisely the\ntemporal spin correlation function, which can be quanti\fed by examining a pair of\nconsecutive images taken of an evolving gas. For systems in thermal equilibrium,\nthis correlation function is related to the nature of spin excitations through the\n\ructuation-dissipation theorem.October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\n24 Dan M. Stamper-Kurn\nAnother open area of investigation is the e\u000bect of non-zero temperature on\nmagnetic order and dynamics. Much of the literature on spinor Bose-Einstein con-\ndensates considers their zero-temperature properties. At non-zero temperature, we\nknow that additional phenomena will arise, such as \\spin-locking\" [52] and spin\nwaves driven by the exchange interaction [53]. Both these phenomena have been\nobserved with spatial resolution in pseudo-spin-1/2 gases, but detailed investiga-\ntions in higher-spin spinor gases are lacking. There are several types of experiments\nwhere very high quality data are being produced on lowest-temperature gases, such\nas the studies of quantum spin mixing dynamics and our recent precision mea-\nsurement of the magnon dispersion relation. Extending these studies carefully to\nhigher temperature gases might help reveal essential di\u000berences stemming from the\ndynamics of non-condensed atoms.\nFinally, we still await applications of spinor-gas magnetometry to the studies of\nmagnetic materials. Our measurement of the magnetic dipolar interactions in the\nrubidium spinor Bose gas, identi\fed as a gap in the magnon excitation spectrum in\nRef. 28, was an application of spinor-gas magnetometry to measure magnetic gases\nthrough the magnetic \feld they generate. However, as sketched in this Chapter,\nthere are excellent prospects for studying solid-state materials by placing a spinor-\ngas sensor nearby. These applications will be boosted by quantum e\u000bects such as\nspin-nematic squeezing and by improvements in spin-sensitive imaging.\nAcknowledgements\nI thank the members of my research group who worked with me on spinor Bose-\nEinstein gases over the years. In particular, I acknowledge the \\E4 team\" of G.\nEdward Marti, Ryan Olf, Andrew MacRae, Fang Fang, and Sean Lourette for in-\nspiring a new set of ideas about how to improve spin-dependent imaging and how\nto characterize magnon excitations. This work is supported by the NSF, NASA and\nby the AFOSR's MURI program on advanced quantum materials.\nReferences\n[1] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev.\nMod. Phys. 80, 885, (2008).\n[2] I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulations with ultracold quan-\ntum gases, Nature Physics .8, 267, (2012).\n[3] D. M. Stamper-Kurn and M. Ueda, Spinor Bose gases: Symmetries, magnetism, and\nquantum dynamics, Rev. Mod. Phys. 85, 1191, (2013).October 9, 2018 4:31 World Scienti\fc Review Volume - 9.75in x 6.5in seeing_spin_dynamics_stamper_kurn\nSeeing spin dynamics in atomic gases 25\n[4] W. Gerlach and O. Stern, The directional quantisation in the magnetic \feld, Ann.\nPhys.-Berlin .74, 673, (1924).\n[5] D. Suter, The Physics of Laser-Atom Interactions . Cambridge Studies in Modern\nOptics, (Cambridge University Press, Cambridge, 2005).\n[6] J. Higbie, L. Sadler, S. Inouye, A. P. Chikkatur, S. R. Leslie, K. L. Moore, V. 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Lett. 89, 090402,\n(2002)."    },    {        "title": "1008.4030v2.Modeling_of_diffusion_of_injected_electron_spins_in_spin_orbit_coupled_microchannels.pdf",        "content": "arXiv:1008.4030v2  [cond-mat.mes-hall]  20 Nov 2010Modeling of diﬀusion of injected electron spins in spin-orb it coupled microchannels\nLiviu P. Zˆ arbo,1Jairo Sinova,2,1I. Knezevic,3J. Wunderlich,4,1and T. Jungwirth1,5\n1Institute of Physics ASCR, v.v.i., Cukrovarnick´ a 10, 162 5 3 Praha 6, Czech Republic\n2Department of Physics, Texas A& M University, College Stati on, Texas 77843-4242, USA\n3Department of Electrical and Computer Engineering Univers ity of Wisconsin-Madison Madison, Wisconsin 53706, USA\n4Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Ki ngdom\n5School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom\nWe report on a theoretical study of spin dynamics of an ensemb le of spin-polarized electrons\ninjected in a diﬀusive microchannel with linear Rashba and D resselhaus spin-orbit coupling. We\nexplore the dependence of the spin-precession and spin-diﬀ usion lengths on the strengths of spin-\norbit interaction and external magnetic ﬁelds, microchann el width, and orientation. Our results are\nbased on numerical Monte Carlo simulations and on approxima te analytical formulas, both treating\nthe spin dynamics quantum-mechanically. We conclude that s pin-diﬀusion lengths comparable or\nlarger than the precession-length occur i) in the vicinity o f the persistent spin helix regime for\narbitrary channel width, and ii) in channels of similar or sm aller width than the precession length,\nindependent of the ratio of Rashba and Dresselhaus ﬁelds. Fo r similar strengths of the Rashba and\nDresselhaus ﬁelds, the steady-state spin-density oscilla tes or remains constant along the channel for\nchannels parallel to the in-plane diagonal crystal directi ons. An oscillatory spin-polarization pattern\ntilted by 45◦with respect to the channel axis is predicted for channels al ong the main cubic crystal\ndirections. For typical experimental system parameters, m agnetic ﬁelds of the order of tesla are\nrequired to aﬀect the spin-diﬀusion and spin-precession le ngths.\nPACS numbers: 75.76.+j, 71.70.Ej, 61.43.Bn\nI. INTRODUCTION\nSpin-orbit (SO) coupling in vacuum is a relativistic\neﬀect in which the magnetic moment of a moving elec-\ntron couples to an external electric ﬁeld. The eﬀect can\nbe explained by recalling that the moving magnetic mo-\nment is seen in the laboratory frame as both magnetic\nand electric dipole moment and the electric dipole com-\nponent couples to the external electric ﬁeld. The correct\nmagnitude of the SO coupling term can be derived using\nthe Dirac equation for the moving particle. Owing to the\nband structure, the SO coupling for electrons in solids\ncan be enhanced by orders of magnitude with respect\nto the value computed in vacuum. This makes the SO\ncoupling-based eﬀects experimentally accessible and en-\nables the use of SO coupling as a tool for purely electrical\ngeneration and manipulation of spins in devices.1–4\nThe prototype spintronic device using SO coupling as\na spin control tool is the Datta-Das transistor.5It con-\nsists of a SO coupled channel connected to spin polarized\nsource and drain electrodes. Inside the channel, the elec-\ntron undergoes coherent spin rotations under the inﬂu-\nence of the SO ﬁeld which can be tuned electrically by an\nexternal gate. However, the simplicity of the Datta-Das\nconcept is deceptive whenever the channel is not one-\ndimensional. The main problem concerning spin trans-\nport in the channel of a Datta-Das device is that on one\nhand, the SO coupling strength in the channel has to be\nlarge enough to enable control of the electron spin. On\nthe other hand, however, a large SO coupling can lead\nto a faster spin relaxation via D’yakonov-Perel6mecha-\nnism than the electron dwell time in the channel if the\nchannel is not one-dimensional or ballistic. To overcomethe diﬃculty, it was proposed7,8to exploit the symmetry\narising from the interplay of Rashba9and Dresselhaus10\nSO ﬁelds in the two-dimensional electron gas (2DEG)\nformed in semiconductorheterostructures. This proposal\nhas opened a way to the Datta-Das transistor operating\nin a non-ballistic regime.7,8,11\nSO ﬁelds in the 2DEG act as momentum dependent\nmagnetic ﬁelds that couple to the electronic magnetic\nmoment. Impurities, phonons, or crystalline defects\ncan scatter the electrons which changes their momenta\nand, therefore, changes the eﬀective SO-induced mag-\nnetic ﬁeldactingonthe electronspin. Individual electron\nspins in the channel acquire diﬀerent phases with respect\nto eachother, resultingin the relaxationofthe total spin.\nThis is the qualitative picture of the D’yakonov-Perel re-\nlaxation.\nThe idea behind the non-ballistic Datta-Das spin tran-\nsistor is that one could tune the Rashba and Dres-\nselhaus SO coupling strengths to be equal, e.g. via\ngate voltage.12In this case, the orientation of the total\nRashba-Dresselhaus SO ﬁeld is independent of momen-\ntum and is parallel to one of the in-plane diagonal axes\n(that can be either [1 10] or [110] depending on the rel-\native sign of the Rashba and Dresselhaus ﬁelds) in the\n(001)-plane of the 2DEG in a cubic semiconductor. The\namplitude of the SO ﬁeld depends only on the momen-\ntum component perpendicular to the direction of the SO\nﬁeld.7,13This can lead to a path independent spin pre-\ncession of individual electron spins, and thus to a sup-\npression of the spin relaxation, in 2DEG channels ori-\nented perpendicularto the linearRashba-DresselhausSO\nﬁeld. Moreover, the Hamiltonian exhibits the U(1) sym-\nmetry which means that an in-plane spin state parallelto2\nthis SO ﬁeld direction is inﬁnitely long lived. This state\nwill be dephased if the cubic Dresselhaus term is present\nin the system.14–16Randomness in the SO coupling in-\nduced by remote impurities would cause additional spin\nrelaxation.17Nevertheless, inﬁnite spin lifetimes are still\npossible in SO coupled 2DEGsif the spatially varyingSO\nﬁeld can be described as a pure gauge and, thus, removed\nby a gauge transformation.18\nFurthermore, it was shown13that the many-electron\nsystem whose individual particles are described by the\nabove U(1) symmetric single-particle Hamiltonian dis-\nplays a SU(2) symmetry which is robust against both\nspin-independent disorder and electron-electron interac-\ntions. Owing to this symmetry, a collective spin state\nexcited at a certain wave vector would have an inﬁnite\nlifetime. Such a state is called the persistent spin helix13\n(PSH) and it has already been observed in transient spin\ngrating experiments.19,20\nIn another recent experiment,21spin polarized cur-\nrent passing through a micrometer-size 2DEG channel\nhas been detected by measuringthe SO coupling-induced\nHall signal. This is called the spin injection Hall eﬀect\n(SIHE). The fact that the SIHE observed in a diﬀusive\nchannel is robust against disorder and temperature ef-\nfects and that the estimated Rashba and Dresselhaus SO\ncouplings are similar in the 2DEG system employed in\nthe experiment leads to the question whether the PSH\nphysics is relevant to this transport experiment.\nIn this paper, we investigate theoretically spin dynam-\nics of electrons in the 2DEG channel in the PSH regime\nas well as in regimes of diﬀerent Rashba and Dresselhaus\nSO ﬁeld strengths. In the context of the above SIHE ex-\nperiment we point out in particular that spin diﬀusion\nlengths comparable to the spin precession length occur\nin channels whose widths are smaller or comparable to\nthespinprecessionlength, regardlessofthe ratiobetween\nthe Rashba and Dresselhaus SO coupling strengths. This\nis one of the several conclusions of the calculations pre-\nsented below which consider the dependence of the spin\ndiﬀusion characteristics on experimentally relevant sys-\ntem parameters such as the strengths of SO and external\nmagnetic ﬁelds, microchannel width, and orientation.\nIn our calculations we employ the non-interacting elec-\ntronapproximation22,23andconsiderthe diﬀusive regime\nin which the SO splitting is muchsmallerthan the energy\nlevel broadening due to disorder scattering, ∆ SO≪/planckover2pi1/τ.\nIn this approach, momentum and position of electrons\ncan be treated as classical variables. We emphasize that\nthe direct correspondence mentioned above between the\nsuppressed spin relaxation in the single-particle trans-\nport problem and the collective PSH state is valid in this\ndiﬀusive regime. Here the group velocity of an electron\nin the Rashba-Dresselhaus 2DEG can be approximated\nby its momentum divided by the mass. The spin pre-\ncession angle of such a particle depends only on the dis-\ntance traveled along the direction perpendicular to the\nSO ﬁeld.13The resulting spin density pattern of an en-\nsemble of injected electron spins then coincides with the\nFIG. 1: (Color online) (a) Representation of the our model\ndevice. Spin- ↑polarized particles are injected from the source\nelectrode in a Rashba and Dresselhaus SO coupled channel.\n(b) Schematic depiction of the EMC method. The time evo-\nlution of each particle belonging to the ensemble is sampled\nat equal intervals ∆ tcalled subhistories. The particle spin\nprecesses in the SO ﬁeld during the free ﬂight time, but is\nunaﬀected by collisions.\nspin density pattern of the PSH spin wave. The expres-\nsion of velocity of SO-coupled electrons contains terms\nproportional to SO coupling strength. For example, the\nvelocity along [1 10]-direction of Rashba and Dresselhaus\nSO coupled electrons in the PSH regime ( α=−β) is\nv110=/planckover2pi1k110/m∗±2β//planckover2pi1, whereα=−βare the Rashba\nand Dresselhaus SO coupling strengths. This means that\nin the opposite limit of strong SO coupling and weak dis-\norder,thevelocityandmomentumarenotsimplypropor-\ntional to each other and the direct link is lost between\nthe one-particle and collective physics in the regime of\nequal or similar Rashba and Dresselhaus ﬁeld strengths.\nThe paper is organized as follows: In Sec. II we intro-\nduce our method and discuss our approximations. In\nSec. IIA we outline the features of the Monte Carlo\nmethod we use in our simulations. In Sec. IIB we discuss\nthe single particle evolution of quantum spin in SO cou-\npled 2DEG and its dependence on crystalline direction\nof propagation, external magnetic ﬁeld and interplay of\nRashba and Dresselhaus SO couplings. In Sec. III we\nshow how the steady state spin density distribution of an\nensembleofelectronsinthe channelisaﬀected byvarying\nthe aboveparameters. Sec. IV givesthe main conclusions\nof our work.\nII. THEORETICAL MODEL\nWe are interested in the spin dynamics in the 2DEG\nchannel of an experimentally relevant spintronic model\ndevice which is schematically depicted in Fig. 1(a). The\ntypical device is of a few micrometers in size and this is\nconsiderably larger than the Fermi wave length in the3\n2DEG channel. It means that the quantum interfer-\nence eﬀects on the orbital motion of electrons can be\nneglected. In other words, it is suﬃcient to solve Boltz-\nmann transport equation (BTE) for this system, rather\nthan use a fully quantum mechanical treatment such as\nKeldysh formalism. On the other hand, we cannot ne-\nglect quantum mechanics of the spin dynamics since the\ntypical spin precession length in experiments21ranges\nfrom a few hundreds of nanometers to a few microns.\nTherefore, we employ the ensemble Monte Carlo24–27\n(EMC) method which is a well established tool in semi-\nconductor device simulations and can be extended to in-\nclude spin coherence28in a micrometer size device. The\nEMC method oﬀers a way to solve BTE that is beyond\nthe reach of drift-diﬀusion models. Drift-diﬀusion mod-\nels must rely on various approximations in order to avoid\nthe tremendous mathematical diﬃculties arising in BTE.\nTreatment of nonlinear terms, inclusion of diﬀerent scat-\ntering mechanisms, or of dissipation eﬀects require dras-\ntic approximationsso that the result ofthe drift-diﬀusion\ncalculation might not reﬂect anymore the features of the\ntheoretical model, but rather those of the mathematical\napproximations. Bycontrast, including disorder, dissipa-\ntion, temperature, transient or nonlinear eﬀects in EMC\nsimulations is straightforward and does not require fur-\nther approximations. In fact, state-of-the-art EMC sim-\nulations are often used to test the validity of the drift-\ndiﬀusion models. Inclusion of a various rangeof eﬀects in\nthe EMC is done without signiﬁcant changes in the com-\nputational complexity, thus making it more suitable for\ndevice simulation than other powerful quantum mechan-\nical techniques such as nonequilibrium Green function\n(NEGF) method. For example, including dissipative ef-\nfects in EMC has only a minor impact on the calculation\ncomplexity, while in the case of NEGF method it can re-\nduce the size of computationally accessible systems from\na few hundreds to a few tens of nanometers.\nIn our calculations, we make the following approx-\nimations: (i) Electron orbital degrees of freedom are\ndescribed by classical momentum and position and the\nspin degree of freedom by quantum-mechanical spin den-\nsity matrix. This semiclassical approximation is justiﬁed\nby the diﬀusive regime we consider. In this regime we\ncan approximate the electron velocity by momentum di-\nvided by mass. (ii) Interactions between electrons are\nneglected. (iii) We consider only short range impurity\nscattering. (iv) Temperature enters our simulations only\nthrough the Fermi-Dirac distribution function. (v) For\nsimplicity, we neglect the electrostatics of the channel.\nThis is justiﬁed since we are primarily interested in the\nspin dynamics of electrons. A small electric ﬁeld present\nin the channel is not expected to have an important in-\nﬂuence on the spin precession pattern of the electronic\nsystem.\nIn the next two subsections we brieﬂy outline the spin\ndependent EMC method and analyze the motion of a\nsingle particle in the SO ﬁeld.A. Spin Dependent Monte Carlo\nElectrons in the channel, shown in Fig. 1(a), can be\nmodeled asan ensemble of Nnoninteractingparticles. In\nthe EMC method, we track the individual motion of each\nparticle in the ensemble and we use the data to calculate\nan approximate particle distribution in phase space. As\nshown in Fig. 1(b), we divide the time of simulation in\nsmall time intervals ∆ tcalled subhistories. During the\nsubhistory, a particle moving in electromagnetic and SO\nﬁelds in the channel can be randomly scattered by im-\npurities and, in general, also by phonons or other scat-\ntering mechanisms that are present in the channel. The\ntime between collisions, called the “free ﬂight time”, is\nrandomly generated and depends on the scattering rate\ncorresponding to each type of collision.25Fig. 1(b) gives\nan intuitive picture of the time evolution of the ensemble\nof electrons. The semiclassical particle is described by\nits position r(t) and momentum k(t). As it was recently\nshown,22,28wecantreatspin-dependent phenomenaif, in\naddition to the semiclassical variables, we consider that\neach particle is described by a 2 ×2 spin density ma-\ntrix ˆρ(t). The spin polarization vector is then given by\ns= Tr[ˆρσ]. The propagation of a particle during the\nfree ﬂight is described by the equations of motion for its\nattached dynamical variables\nm∗d2r\ndt2=−e[E+vk×B], (1a)\nvk=1\n/planckover2pi1∇kEk≈/planckover2pi1k\nm∗, (1b)\nˆρ(t+δt) =e−i\n/planckover2pi1ˆHspin(k)δtˆρ(t)ei\n/planckover2pi1ˆHspin(k)δt,(1c)\nwhich must be integrated together to ﬁnd the particle\ntime evolution during free ﬂights. Here, vkis the parti-\ncle velocity, EandBare the electric and magnetic ﬁelds,\nm∗is the eﬀective mass of the particle, and Ekis the\nelectronic band dispersion in the 2DEG. Note that the\napproximation in Eq. (1b) means that we neglect any\ninﬂuence of the SO coupling on the trajectory of the\nsemiclassical particle,29. While equations (1a) and (1b)\ndescribe a semiclassical electron propagating in a solid,\nEq. (1c) describes the quantum mechanical evolution of\nits spin during short time δtwhich is controlled by the\nspin-dependent part of the Hamiltonian, ˆHspin. This\nHamiltonian includes the internal SO ﬁeld and the ex-\nternal magnetic ﬁeld, ˆHspin=ˆHSO+ˆHZ.\nAt the end of each subhistory we calculate the en-\nsemble averaged quantities of interest such as currents,\nchargeandspindensities. Afterthe simulationconverged\nand the system is in steady state we can use the subse-\nquent subhistories to compute time averaged values for\nthe physical quantities. In our case, the system is in\nsteady state when the ﬂux of electrons through the drain\nelectrode becomes constant.4\nB. Spin Dynamics in Rashba-Dresselhaus Field\nTheelectrongasin the heterostructurecanbe modeled\nby the Rashba and Dresselhaus SO coupled Hamiltonian\nˆH=ˆp2\n2m∗+α\n/planckover2pi1(ˆpyσx−ˆpxσy)+β\n/planckover2pi1(ˆpxσx−ˆpyσy).(2)\nHereβis the Dresselhaus SO coupling which, for simplic-\nity, is kept constant in our simulations, αis the exper-\nimentally adjustable12,30Rashba parameter, m∗is the\neﬀective mass of the 2DEG, and σx,σy, andσzare the\nPauli matrices. The crystalline axes labeled x,y, and\nzcorrespond to the [100], [010], and [001] directions, re-\nspectively, and the 2DEG lies in the xy-plane. We are ig-\nnoring the cubic Dresselhaus terms since the linear terms\naredominantfornottoohighcarrierconcentrations. The\neﬀect of the magnetic ﬁeld is included by making the\nsubstitution p→p−eAin the Hamiltonian (2) and by\nadding the Zeeman term,\nˆHZ=−1\n2gµBB·σ (3)\n=−1\n2gµB(B/bardblσ/bardbl+B⊥σ⊥+Bzσz),\nwhereBis the magnetic ﬁeld strength, Ais the cor-\nresponding vector potential, gis theg-factor, and µB\nis the Bohr magneton. We wrote the Zeeman Hamil-\ntonian in terms of the magnetic ﬁeld components B/bardbl\nwhich is parallel to the transport direction in the chan-\nnel,B⊥which is the in-plane magnetic ﬁeld component\nperpendicular to the current direction, and Bzwhich is\nthe out-of-plane component of the magnetic ﬁeld. The\nunit vectors corresponding to in-plane axes parallel and\nperpendicular to the transport direction are labeled by\nn/bardbl= (a,b,0) andn⊥= (b,−a,0) while nz= (0,0,1)\ncorresponds to the z-axis. For example, if the electronic\ntransport is along [1 10]-axis, a= 1/√\n2 andb=−1/√\n2.\nUsing this notation we can express the spin matrices as\nσ/bardbl=n/bardblσ=aσx+bσyandσ⊥=n⊥σ=bσx−aσy.\nNote that the above Hamiltonian (3) does not take into\naccount the change in the eﬀective mass or g-factor in\nthe 2DEG as a result of applying magnetic ﬁeld.31In what follows, we derive the spin precession length\nof an electron propagating in a straight line along an\narbitrary direction in the SO coupled 2DEG. As in the\nspin-dependent EMC approach described in the previous\nsection, the electron is a point particle whose spin rotates\ncoherently under the inﬂuence of a weak SO ﬁeld and\nthe applied external magnetic ﬁeld. We consider a spin-\n���electron (spin parallel to +ˆ z-direction) injected along\nan arbitrary direction n/bardbl= (a,b,0) and subject to both\nSO and magnetic ﬁelds. The electron spin is described\nby the spin density matrix ρ=1\n2(I2+sσ), where s=\n(sx,sy,sz) is the spin polarization vector and I2is the\n2×2 identity matrix. Initially, the electron has spin-\n↑, so its polarization vector is s= (0,0,1) and the spin\ndensity matrix is ρ0=1\n2(I2+σz).\nNext, we rewrite the SO part of the Hamiltonian (2)\nas\nˆHSO= Ωxσx+Ωyσy+Ωzσz, (4)\nwith Ω x=αky+βkx, Ωx= (αkx+βky) and Ω z= 0.\nWe label the unit vector parallel to Ω= (Ωx,Ωy,Ωz) by\nh= (hx,hy,hz). After a short time step δtduring which\nthe momentum is considered constant, we obtain with\nthe aid of Eq. (1c)\nρ(δt) =1\n2I2+1\n2cos(2Pδt)σz−1\n2sin(2Pδt)(hxσy−hyσx),\n(5)\nwhereP=1\n/planckover2pi1/radicalBig\nΩ2x+Ω2y. In order to study the single\nparticle spin precession we consider that the electron\nmomentum along the transport direction is constant.\nSuch assumption is true as long as there is no trans-\nverse external electric ﬁeld and no out-of-plane magnetic\nﬁeld. The condition that the spin ﬂips during the mo-\ntion of the electron is 2 Pt↑→↓=π, as seen from Eq. (5).\nThe spin precession length along the transport direction\nn/bardbl= (a,b,0) is computed as L↑→↓\nab0=vkt↑→↓. (Recall\nthatvk≈/planckover2pi1k/m∗in the diﬀusive, weak SO coupling\nregime.) Considering the eﬀects of the in-plane magnetic\nﬁeld, we obtain for the spin precession length\nL↑→↓\nab0=π/planckover2pi12\n2m∗/radicalBigg/parenleftbigg\nαb+βa+1\n2gµBB/bardbla+B⊥b\nk/parenrightbigg2\n+/parenleftbigg\n−αa−βb+1\n2gµBB/bardblb+B⊥a\nk/parenrightbigg2. (6)\nBy applying the spin precession length formula (6) we\ncan gain an intuitive understanding of the spin dynamics\nin the SO coupled heterostructure. Of particular interest\nis the case of α=−βand [110] channel orientation (orα=βand [110]-orientation)in which the PSHsymmetry\nis present. In this case, the spin precession depends only\non the distance traveled by electrons along the channel.\nFrom Eq. (6) we can immediately see that for α=−β5\nFIG. 2: (Color online) Spin density distribution /angbracketleftSz(r)/angbracketrightin a\n[110]-oriented 2DEG channel with ﬁxed Dresselhaus SO cou-\nplingβ=−2.0×10−12eV·m for diﬀerent values of Rashba\nSO coupling (a) α=−β, (b)α= 0, (c) α=−0.5βand (d)\nα=−1.5β. The light color refers to spin- ↑. The distance\nbetween the successive maxima of the spin distribution cor-\nresponds to twice the spin precession length computed from\nEq. (6). The spin diﬀusion length is inﬁnite in (a), while in\nall the other cases it exceeds the spin precession length, as\nexpected in narrow channel.\nand electron propagating along the [1 10]-direction, the\nspin precession length is the shortest while for spin prop-\nagating along the [110]-direction it is inﬁnite.\nIII. DISCUSSION OF THE EMC SIMULATIONS\nWe now employ the spin-dependent EMC method out-\nlined in Sec IIA to numerically simulate spin dynamics\nin the microchannel illustrated in Fig. 1(a). Spin- ↑elec-\ntrons are injected from the source electrode and propa-\ngate in the SO ﬁeld of the disordered 2DEG. Electrons\nthat reach the microchannel edge are reﬂected back with\nunchanged spin. An electron that re-enters the source or\nexits the drain is erased and replaced by a new spin- ↑\nelectron injected from the source.\nWe choose the 2DEG parameters that correspond to\nthe GaAs 2DEG of Ref. [21]. Our channel length is L≈\n3µm and the width will take values both smaller and\nlarger than the spin precession length which is of the\norder of a few hundred nanometers. Temperature of the\nelectron ensemble in all simulations is 300 K. Electron-\nphonon scattering is neglected. Disorder in the system\nis due to randomly placed point-like spinless impurities.\nThe electron-mean-free path is 26 nm. The electronic\ndensity of the 2DEG system is ne= 2.5×1012cm−2.\nThe corresponding Fermi wave length is λF≈8nm. The\nDresselhaus spin-orbit coupling is kept constant during\nsimulations at β=−2.0×10−12eV·m.\nThe electron ensemble consists of N= 130000 elec-\ntrons. We run the simulation until the system reaches\nFIG. 3: (Color online) Spin density distribution /angbracketleftSz(r)/angbracketrightin\nthe 2DEG channel with α=−βfor various orientation of\nthe transport axis: (a) [1 10], (b) [110] and (c) [100]. In the\nPSHregime, the relative spin orientations ofparticles inj ected\nat the source depends only on the initial distance between\nthem along the [1 10]-direction, i.e. the ﬁrst component of the\nrelative position vector of the two particles d= (d110,d110).\nIn panel (a) that distance is d110= 0, resulting in no spin\ndephasing, in panel (b) is d110=dand in panel (c) it is\nd110=d/√\n2, where dis the initial distance between the two\nparticles. Therefore,the spin relaxation in the [110] and [ 100]-\noriented channels depends on the channel width.\nsteadystateandafterthatweusethelast2000timesteps\nto calculate time averaged spin densities. The spin den-\nsities are normalized to the number of particles present\nin each grid cell such that the spin of a cell containing\nall spin-↑particles is 1. In what follows, we show how\nthe spin density distribution /angb∇acketleftSz(r)/angb∇acket∇ightin the channel is\naﬀected by changes in the width of the channel, the crys-\ntalline axis along which the transport takes place, and\nstrength of Rashba SO coupling and magnetic ﬁelds.\nIn general, the spin polarization along the channel is\nrandomized due to the D’yakonov Perel spin dephas-\ning mechanism which is dominant in GaAs heterostruc-\ntures. This eﬀect is visible, e.g., in Fig. 2(b). The spin-\ndiﬀusion length depends on the parameters of the mi-\ncrochannel. In the limiting case of equal Rashba and\nDresselhaus SO coupling strengths and, e.g., α=−β,\nthe PSH symmetry13arises in the [1 10]-oriented 2DEG\nchannel and the oscillatory dependence of /angb∇acketleftSz(r)/angb∇acket∇ighton the\ncoordinate along the channel is undamped, as shown\nin Fig. 2(a). In this case spin orientations of injected\nelectrons are not randomized by scattering. The spin-\ndiﬀusion length is inﬁnite and the spin-precession length\nis given exactly by Eq. 6. In Figs. 2(c) and (d) we show\nthat the PSH regime is robust against sizable changes in\ntheα/βratio.23\nWe next proceed to channels which are not oriented\nalong the [1 10]-direction. In Fig. 3 we compare results\nfor the [100], [110]-oriented channels with the [1 10] chan-6\nFIG. 4: (Color online) Spin density distribution /angbracketleftSz(r)/angbracketrightin\nthe channel as a function of its width for (a) α=−0.5βand\n[110]-injection, (b) α=−βand [100]-oriented channel, and\n(c)α=−βand [110]-injection direction. As expected from\ntheory, the larger channel widths lead shorter spin diﬀusio n\nlengths.\nnel, assuming α=−β. Fig. 3(a) shows that the spin\nprecesses fastest for the channel oriented along the [1 10]-\ndirection, while Fig. 3(b) showsno spin precessionforthe\n[110]-oriented channel, consistent with Eq. 6. The result\nin Fig. 3(c) for the channel oriented along the [100]-axis\nis less obvious, however, we can still use the spin preces-\nsionformula(6) tounderstandthe 45◦rotatedoscillatory\npattern for this channel direction. Since the orientation\nof the spin depends on the distance the particle travels\nalongthe[1 10]-direction,weexpectthatapatternformed\nby averagingthe spin densities along a [110]-orientedline\nwould repeat itself with a 2 L↑→↑\n110period along the [1 10]-\nline. As sketched in Fig. 3, we can follow individual tra-\njectories of two electrons injected from the source. Let us\nconsider two particles and connect them by the relative\nposition vector dwhose length dis the initial distance\nbetween particles. We can decompose dalong [110]- and\n[110]-axes, d= (d110,d110). Ifα=−β, the diﬀerence be-\ntween the spin directions of the two particles when they\nmeet inside the channel is given by d110. The spin coher-\nence of the electron ensemble is therefore partially lost\nbecause of the initial distribution of d110’s of the injected\nelectrons. From this it is apparent that the spin-diﬀusion\nlengthscaleswith theratiooftheprecessionlengthtothe\nFIG. 5: (Color online) (a),(b) Pictorial explanation of the\nspin density patterns in the α=−βregime for electrons\ninjected along a [110]-oriented channel of width (c) L↑→↑\n110and\n(d) 2L↑→↑\n110.\nchannel width. We emphasize that all these arguments\nareindependent ofthe mean-free-path. Indeed, we would\nobtain the same steady state spin density distribution if\nthe channel in Fig. 3 were ballistic.\nThe dependence of the spin diﬀusion length in our\nensemble of electrons on the channel width is further\nquantiﬁed in Fig. 4. For the [1 10]-oriented channel the\nspin-diﬀusion length is inﬁnite as long as α=−βand\nit decreases with increasing width of the channel when\nRashbaandDresselhauscouplingstrengthsarenotequal,\nas shown in Fig. 4(a).22,32Forα=−β, the spin-diﬀusion\nlength is ﬁnite for channel orientations diﬀerent from\n[110]-direction and it again decreases with increasing\nchannel width. This is illustrated in Fig. 4(b) for the\n[110]-orientedmicrochannel and in Fig. 4(c) forthe [100]-\nchannel.\nWe now provide a more detailed understanding of the\nnumerical spin-density patterns obtained by the EMC\nsimulations, focusing on the α=−βcase and the [110]-\noriented microchannels. For ﬁxed and equal Rashba and\nDresselhaus coupling strengths ( α=−β), the spin orien-\ntation of an individual particle depends only on the dis-\ntance from the injection point along the [1 10]-direction.\nThespindensitypatternofanensembleofparticlesstart-\ningfromagivenpointdependsonlyonthestrengthofthe\nSOcoupling. Ourensembleaveragingprocedureamounts\ntosummingup allspin densitypatternsofparticlesstart-\ning from diﬀerent points along the source-channel inter-\nface. We use this idea to explain the spin density dis-\ntribution obtained for the [110]-oriented channel of two\ndiﬀerent widths, as shown in Fig. 5. The width of the\nchannel in Fig. 5(c) is equal to the spin precession length\nL↑→↑\n110. All spins starting at the source at point A shown\nin Fig. 5(a) generate a spin density pattern illustrated7\nFIG. 6: (Color online) Spin density distribution /angbracketleftSz(r)/angbracketrightin\nmagnetic ﬁeld for the SO coupled 2DEG channel oriented\nalong [110]-direction. The four panels show (a) Hanle eﬀect\nfor 1T in-plane ﬁeld and no SO coupling, spin precession for\n(b)α=−βand in-plane magnetic ﬁeld B/bardbl= 1T, (c) α=−β\nand in-plane magnetic ﬁeld B⊥= 1T (d) α=−βand out-of-\nplane magnetic ﬁeld Bz= 1T.\nin the corresponding column A in the ﬁgure. The spin\ndensity pattern generated by a spin starting at point B\nis the same, only shifted by the distance between A and\nB along the [1 10]-direction. All the other spin density\npatterns are shifted in the same manner. We can assign\nnumbers to the projection of spin along z-direction for\neach spin-density pattern, as shown in Fig. 5(b). Sum-\nming up these numbers we obtain qualitatively the same\ntransverse proﬁle of the spin density in the channel as in\nthe EMC simulation. This gives an intuitive explanation\nof the reduced mean spin polarization along the chan-\nnel edges for channel width smaller or equal to LSOseen\nin Fig. 5(c). We can use the same procedure to explain\nthe randomization of spins in the entire cross-section of\na wider, [110]-oriented channel shown in Fig. 5(d).\nAs apparent from the one particle formula (6), the\nmagnetic ﬁeld eﬀect on the spin dynamics depends on\nthe magnitude of the electron momentum and, there-\nfore, on the electron density in the channel. As shown in\nFig. 6(a), the momentum dependence leads to faster ran-\ndomizationofthe spin densitydistribution in the channel\nthan in the momentum independent case of Dresselhaus\ncoupled 2DEG channel shown in Fig. 2(b). We will ﬁrst\nanalyze the eﬀect of the in-plane magnetic ﬁeld on spin\nprecession. Our calculations in Fig. 6 show that in the\nα=−βcase and for the [1 10]-oriented channel, the spin\ndensity is only slightly aﬀected by the in-plane magnetic\nﬁeld parallel to the channel direction even at magnitudes\nof the order of Tesla. In-plane magnetic ﬁeld of the same\nmagnitudebutperpendiculartothechanneldirectionhas\na sizable eﬀect on the electronic spin density pattern. To\nunderstand the dependence on the in-plane ﬁeld orien-\ntation we rewrite Eq. (6) for α=−βand for the [1 10]-channel,\nL↑→↓\n110=π/planckover2pi12\n2m∗/radicalBig\n4β2−βgµBB⊥\nk+1\n4g2µ2\nBB2\n⊥+B2\n/bardbl\nk2.(7)\nFrom Eq. (7) we see that for ﬁelds ∼1T, the magnetic\nﬁeld component B⊥aﬀects the spin precession since the\nZeeman splitting is comparable to the Dresselhaus spin\nsplitting at the Fermi level. On the other hand, at higher\nmagnetic ﬁelds, the quadratic terms in magnetic ﬁeld\npresent in the denominator of Eq. (7) will dominate the\nlinear term and the eﬀect of magnetic ﬁeld becomes in-\ndependent of its orientation. At these high ﬁelds, the\nspin-diﬀusion length is limited primarily by the magnetic\nﬁeld.\nIV. SUMMARY\nWe used the spin-dependent EMC method to simulate\nspin dynamics of an ensemble of electrons in the 2DEG\nchannel with SO coupling. Our calculations were done in\nthe diﬀusive regime in which there is a correspondence\nbetween the long spin-diﬀusion length of an ensemble of\nnoninteracting electrons and the collective PSH state.\nOur numerical simulations and qualitative analyti-\ncal considerations show that the spin-precession pattern\nand the spin-diﬀusion length in the channel with equal\nstrengths of the Rashba and Dresselhaus SO ﬁelds de-\npends only on geometric factors, i.e., on the channel ori-\nentation and width. The presence of magnetic ﬁeld in\nthe channel suppresses the spin-diﬀusion length. How-\never, for the experimentally relevant system parameters\nthe ﬁelds magnitude must be of the order of a few Tesla\nfor the eﬀect to be observable. The presence of the\nPSH symmetry yields an oscillatory spin-density pattern\nwith inﬁnite spin-diﬀusion length in [1 10]-oriented chan-\nnels of an arbitrary width. The spin-diﬀusion length is\nstill comparable to the precession length for any ratio of\nthe Rashba and Dresselhaus ﬁelds as long as the channel\nwidth is comparable or smaller than the spin-precession\nlength. These predictions are independent of the scatter-\ning mean-free-path.\nAcknowledgments\nWe acknowledge support from EU Grant FP7-\n215368 SemiSpinNet, from Czech Republic Grants\nAV0Z10100521, KAN400100652, LC510, and Praemium\nAcademiae, and from NSF-MRSEC DMR-0820414,\nDMR-0547875, and SWAN-NRI. J. S. is a Cottrell\nScholar of Research Corporation. L.P.Z. would like to\nthank Max Fischetti for correspondence.8\n1I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.,\n76, 323 (2004).\n2J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nI. Zutic, Acta Phys. Slovaca, 57, 565 (2007).\n3J. Sinova and A. H. 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Nazarov\nKavli Institute of NanoScience, Delft University of Technology,\nLorentzweg 1, NL-2628 CJ, Delft, The Netherlands\n(Dated: March 23, 2022)\nWe show that the annihilation dynamics of excess quasi-particles in superconductors may result\nin the spontaneous formation of large spin-polarized clusters. This presents a novel scenario for\nspontaneous spin polarization. We estimate the relevant scales for aluminum, ﬁnding the feasibility\nof clusters with total spin S/similarequal104/planckover2pi1that could be spread over microns. The ﬂuctuation dynamics\nof such large spins may be detected by measuring the ﬂux noise in a loop hosting a cluster.\nVarious experiments using superconductors have been\ninterpreted in terms of a long-lived, non-equilibrium\nquasi-particle population that persists at low tempera-\ntures [1–21]. Such quasi-particles may be created, for ex-\nample, by Cooper pair breaking due to the absorption of\nstray photons or cosmic rays – the dominant mechanism\nis not clear at the moment. The bottleneck for their evac-\nuation is the two-particle recombination mediated by the\nelectron-phonon interaction. A simple balance predicts\na residual quasi-particle density n∼c0= (2A/¯Γ)1/2,\nwhereAis the rate of non-equilibrium generation of\nquasi-particles per unit volume, and ¯Γ is a material con-\nstant characterizing the inelastic quasi-particle relaxation\ndue to the electron-phonon interaction. The subject has\nattracted much interest recently as excess quasi-particles\nwill ultimately limit the performance of many supercon-\nducting devices [3, 5, 6, 9–11, 13]. Therefore one needs to\ndeepen earlier studies on quasi-particle relaxation as, e.g.,\nRef. [22]. Several strategies, such as quasi-particle trap-\nping in normal islands or vortices [1, 9, 12, 18] and quasi-\nparticle pumping with microwave pulse sequences [23],\ncan be used to evacuate quasi-particles from the region of\ninterest and lead to a better device performance. By con-\ntrast, unintentional trapping of quasi-particles in bound\nstates below the superconducting gap edge, present in\ndisordered superconductors, may slow down the relax-\nation dramatically at low concentrations [24] since the\nrecombination requires two quasi-particles and thus is ex-\nponentially suppressed for those in distant bound states.\nAll above considerations neglect the quasi-particle\nspin. We note the spin selectivity of the recombination\nprocess: in the absence of interactions violating spin con-\nservation, the recombination only proceeds if two quasi-\nparticles are in a spin-singlet state. In this Letter, we\nshow that this spin selectivity may become a mechanism\nof non-equilibrium spin polarization. The quasi-particles\nalign their spins forming a polarized cluster with greatly\nenhanced concentration, the number of particles in the\ncluster and its size being limited by spin relaxation pro-\ncesses. We derive the corresponding conditions for alu-\nminum, showing the feasibility of the clusters of ∼104quasi-particles that could be spread over microns. The\npolarization of the cluster slowly ﬂuctuates in time, and\nwe propose a simple setup where the resulting noise can\nbe utilized for the experimental observation of the phe-\nnomenon.\nA cluster consists of an ensemble of quasi-particles with\nmutually overlapping wavefunctions. In the presence of\nspin-singlet recombination, a cluster of Nquasi-particles\nis stable only if no pair of quasi-particles has an overlap\nwith a spin-singlet state. This is the case if the cluster\nis in a maximal spin state, with total spin S=N/2. Let\nus align the z-axis with the cluster polarization.\nIf a new quasi-particle is added to such a cluster, the\nnumber of quasi-particles changes by 1: N→N/prime=N+1,\nwhereas the total spin changes by ±1/2:S→S/prime=\nS±1/2 = (N±1)/2. Thez-projection of the spin is S/prime\nz=\nN/2+sz, wheresz=±1/2 is thez-projection of the spin\nof the incoming particle. Thus, if sz= 1/2, we obtain the\nmaximal spin state |S/prime= (N+1)/2,S/prime\nz= (N+1)/2/angbracketright. By\ncontrast, if sz=−1/2, there are two possible spin states:\n|S/prime= (N±1)/2,S/prime\nz= (N−1)/2/angbracketright. The relative probabil-\nities of these two possibilities are determined by the cor-\nresponding Clebsch-Gordan coeﬃcients, which are given\nin Sec. I of the Supplemental Material (SM) [25]. Note\nthat|S/prime= (N+ 1)/2,S/prime\nz= (N−1)/2/angbracketrightis also a maximal\nspin state, though its polarization is not along the z-axis\nanymore. Since the orientation of the incoming spin is\nrandom, the probabilities for being and not being in a\nmaximal spin state are thus given as [1 + 1 /(N+ 1)]/2\nand [1−1/(N+ 1)]/2, respectively. As a consequence,\nthe probability that the new cluster is stable is larger\nthan the probability that the new cluster can decay. This\nasymmetry thus favors the growth of spontaneously po-\nlarized clusters of quasi-particles. The polarization axis\nof such a cluster is not ﬁxed, but changes randomly and\nslightly with each new quasi-particle added.\nFrom this consideration, we construct a simple model\nfor the spin dynamics of excess quasi-particles. To do\nso, we consider Nquasi-particles in a volume V. We\nassume that the diﬀusion of the particles is suﬃciently\nfast that the spatial structure of their wavefunctions doesarXiv:1912.10661v1  [cond-mat.supr-con]  23 Dec 20192\nnot aﬀect the spin dynamics and concentrate on spin\neﬀects only. Let us consider clusters that are close to\nthe stable conﬁguration with maximal spin S=N/2.\nWe choose the instantaneous spin quantization axis such\nthatSz=Sand describe the cluster’s deviation from\nthe maximal spin state with the integer m=N/2−S,\nm/lessmuchN,S.\nWe consider four diﬀerent processes that can change\nthe state (N,m ) of the cluster:\n(1) Quasi-particle injection: Quasi-particles are in-\njected with a rate AVand arbitrary spin. Thus, half of\nthem are aligned with the polarization axis of the existing\ncluster, whereas half of them are antialigned. If the spin\nis antialigned, we ﬁnd that the probability of creating an\nadditional spin ﬂip, m→m+1, is (N−m)/(N−m+1).\nThe possible processes are thus ( N,m )→(N+1,m) with\nrateAV[1+1/(N−m+1)]/2 and (N,m )→(N+1,m+1)\nwith rateAV[1−1/(N−m+ 1)]/2.\n(2) Singlet annihilation: Such annihilation processes\nare possible only if the system is not in a maximal spin\nstate. At small concentration of spin ﬂips, m/lessmuchN, the\ncorresponding rate is, thus, proportional to m. In partic-\nular, the process ( N,m )→(N−2,m−1) happens with\nrate ¯Γ(N−m)m/V .\n(3) Spin ﬂips: Spin-orbit coupling admits for inelastic\nspin-ﬂips via the electron-phonon interaction. We as-\nsume that each spin may ﬂip independently. As for the\ninjection process, a spin ﬂip does not necessarily change\nthe total spin – however we will neglect the correspond-\ning 1/N-corrections to the rates. The rate for the process\n(N,m )→(N,m + 1) is then given as ( N−m)/τs, where\n1/τsis the spin ﬂip rate for a single spin. Similarly the\nprocess (N,m )→(N,m−1) has the rate m/τ s.\n(4) Triplet annihilation: In the presence of spin-orbit\ncoupling, pairs of quasi-particles may annihilate even\nwhen they are in a spin-triplet state. To account for\nsuch processes, we introduce a weak spin-independent\nannihilation, ¯Γt/lessmuch¯Γ. Taking into account all possible\norientations of the spins of the annihilated particles, this\nadds the following processes: ( N,m )→(N−2,m) with\nrate¯Γt(N−m)2/(2V) as well as ( N,m )→(N−2,m−1)\nwith rate ¯Γt(N−m)m/V and (N,m )→(N−2,m−2)\nwith rate ¯Γtm2/(2V).\nWith this, the dynamics are described by a master\nequation explicitly given in Sec. II of the SM [25]. As a\nﬁrst step, we derive the mean ﬁeld solutions for the most\nprobableNandm. The evolution equations for these\nquantities read:\ndN\ndt=AV−2¯Γ\nV(N−m)m−¯Γt\nVN2, (1)\ndm\ndt=AV\n2/parenleftbigg\n1−1\nN−m/parenrightbigg\n−¯Γ\nV(N−m)m (2)\n−¯Γt\nVNm+1\nτs(N−2m).Using ¯Γt/lessmuch¯Γ andm/lessmuchN, Eq. (1) yields the stationary\nsolution\nm0=AV2−¯ΓtN0\n2¯ΓN0. (3)\nSubstitution into Eq. (2) gives an equation for the aver-\nageNin the cluster:\n0 =AV\nN0−2\nτsN0−¯Γt\nVN2\n0. (4)\nWe can distinguish two regimes, depending on whether\nspin relaxation (SR) or triplet annihilation (TA) domi-\nnates. In the SR regime, Eq. (4) yields N(s)\n0=/radicalbig\nAVτ s/2,\nwhile in the TA regime, one ﬁnds N(t)\n0=/parenleftbig\nAV2/¯Γt/parenrightbig1/3.\nThe corresponding values for mare given as m(s)\n0=\n(V/¯Γτs)N(s)\n0andm(t)\n0= [(AV2¯Γ2\nt)1/3/2¯Γ]N(t)\n0, respec-\ntively. Comparing the two expressions for N0, we con-\nclude that the TA regime requires A > V/ (¯Γ2\ntτ3\ns), that\nis, a suﬃciently high injection rate at any given volume.\nThe above equations allow us to derive the require-\nments for the cluster to be highly polarized, that is,\nN0/greatermuchm0. Let us ﬁrst consider a small Asuch the\ncluster is in the SR regime. In this case, a suﬃciently\nsmall volume V/lessmuchVc≡¯Γτsis required. If at a given\nV <V cwe increase A, and therefore the number of par-\nticles in the cluster, we cross-over to the TA regime, and\na high polarization persists up to A/similarequalAc(Vc/V)2with\nAc≡¯Γ/(¯Γtτs)2. This requirement is convenient to ex-\npress in terms of the number of particles in the cluster,\nN/lessorsimilarNc≡¯Γ/¯Γt.\nNote that, in a polarized cluster, N0largely exceeds the\nvalueNunpol .=c0Vexpected for an unpolarized system.\nIt is constructive to express the concentrations as follows:\nin the SR regime,\nN(s)\n0/V=c0\n2ζ−1/2\ns, m(s)\n0/V=c0\n2ζ1/2\ns, (5)\nwithζs≡V/Vc/lessmuch1, and in the TA regime,\nN(t)\n0/V=c0\n2ζ−1/2\nt, m(t)\n0/V=c0\n2ζ1/2\nt, (6)\nwithζt≡(AV2/AcV2\nc)1/3/2/lessmuch1. The regions where a\npolarized state is expected are illustrated in Fig. 1, see\nalso Sec. III of the SM [25] for more details.\nLet us estimate the relevant material parameters ¯Γ,¯Γt,\nandτs. In aluminum, the phonon-assisted recombination\nrate for quasi-particles near the gap edge is character-\nized by ¯Γ/similarequal18 s−1µm3[26]. As to the triplet annihi-\nlation rate, it involves a phonon emission accompanied\nby a spin-ﬂip, and is estimated as ¯Γt∼α2\nsoK¯Γ, where\nαso∼10−2is the dimensionless spin-orbit strength, and\nthe suppression factor Kreﬂects the smallness of the mo-\nmentum transfer in the course of the emission. As such,\nKcrucially depends on the wave vector qof the phonon3\n1VVc1AAcunpolarized TA SR N⇠Nc\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>\nFIG. 1: Unpolarized and polarized regimes versus the cluster\nvolume and the injection rate, according to Eqs. (5) and (6).\nHereVc=¯Γτs,Ac=¯Γ/(¯Γtτs)2, andNc=¯Γ/¯Γt.\ninvolved that is set by the energy ∼∆ released, cq/similarequal∆,c\nbeing the sound velocity and ∆ being the superconduct-\ning gap. In the absence of disorder, K/similarequal(qa)2[27],abe-\ning the interatomic distance scale. With the disorder set-\nting a mean free path l,K/similarequal(ql)−1for 1/lessorsimilarql/lessorsimilar(l/a)2/3,\nK/similarequalqlforq/lessorsimilarl−1[28]. To have a disorder-independent\nestimation, we resort to the least suppressed case, K=1.\nThis gives ¯Γt∼α2\nso¯Γ∼10−4¯Γ.\nIt may seem that the relevant spin-ﬂip rate is deter-\nmined by elastic spin-orbit processes as it is usual in the\ncontext of spin transport [29], 1 /τso∼α2\nso(δ/epsilon1/∆)1/2/τel,\nwhereτelis the elastic scattering time, and δ/epsilon1/lessorsimilar∆ char-\nacterizes the energy window for the excess quasi-particles\nabove the superconducting gap [30]. However, this esti-\nmation holds for propagating electron waves rather than\nfor the localized states we are dealing with. As explained\nin [31], elastic spin-orbit interaction is ineﬃcient in relax-\ning the spin of localized states, not lifting the Kramers\ndegeneracy. Therefore the spin ﬂips should involve in-\nelastic processes. We assume that the dominant spin-\nﬂip process is the phonon emission/absorption in the\npresence of spin-orbit coupling. The corresponding rate\nis then estimated as 1 /τs∼α2\nso(δ/epsilon1/∆)7/2K/τ 0, where\nτ0∼400 ns in Al is the normal-state inelastic phonon\nscattering time at energy ∼∆ [22]. The ﬁrst and second\nsuppression factors reﬂect the smallness of the spin-orbit\ninteraction and the reduction of the density of states [22],\nand the factor Know corresponds to the energy trans-\nferδ/epsilon1/similarequalcq. As above, we resort to the least suppressed\nchoiceK=1. Even this choice gives very long spin-ﬂip\ntimes: atδ/epsilon1/similarequal0.1∆ we estimate τs/similarequal10 s.\nWith this, we estimate the critical volume Vc=τs¯Γ∼\n180µm3. This implies that the spin-polarized cluster can\nbe spread over micron lengths and Vcis not a very restric-\ntive parameter. A more severe restriction comes from\nthe triplet annihilation that sets the maximum number\nof particles in the cluster, Nc=¯Γ/¯Γt∼104. The criticalinjection rate, where the cross-over from spin-ﬂip limited\nto triplet-annihilation limited clusters size takes place, is\nthen estimated as Ac∼105s−1µm−3. (A similar injec-\ntion rate was reported in Ref. [17].) The quasi-particle\ndensity is enhanced compared to the unpolarized case, if\nV <V candA<A c(Vc/V)2.\nIt is important to note that the number of particles\nin the cluster strongly ﬂuctuates. The mean-ﬁeld solu-\ntion gives the most probable number of particles in the\ncluster,N0, while/angbracketleftN/angbracketrightdiﬀers from N0by a factor and\nthe ﬂuctuations/angbracketleft/angbracketleftN2/angbracketright/angbracketright=/angbracketleftN2/angbracketright−/angbracketleftN/angbracketright2are of the order\nN2\n0. To quantify the ﬂuctuations, we utilize a Fokker-\nPlanck equation, cf. Sec. II of the SM [25], that gives the\ndistribution function\nP(N) =CN2exp/bracketleftbigg\n−2¯Γt\n3AV2N3−2\nAVτ sN2/bracketrightbigg\n, (7)\nwhere the constant Censures the normalization. For the\nSR and TA regimes, this gives, respectively,\n/angbracketleftN/angbracketright(s)=2√πN(s)\n0,/angbracketleftN2/angbracketright(s)=3\n2/parenleftBig\nN(s)\n0/parenrightBig2\n,(8)\n/angbracketleftN/angbracketright(t)=Γ/parenleftbig1\n3/parenrightbig\n181/3N(t)\n0,/angbracketleftN2/angbracketright(t)=24/3π\n35/6Γ/parenleftbig1\n3/parenrightbig/parenleftBig\nN(t)\n0/parenrightBig2\n.(9)\nA comparison with the classical model [24] is provided in\nSec. IV of the SM [25].\nWhile the quasi-particle system is polarized, its spin\nquantization axis is not ﬁxed but diﬀuses with time, along\nwith the number of the polarized particles. This pro-\nduces a measurable spin noise that can be utilized for\nthe experimental veriﬁcation of the polarization, using\nthe setup sketched in Fig. 2 as explained below. The\nspin noise for a certain spin component can be estimated\nin terms of the noise of the number of particles SN,\nSspin/similarequal1\n3SN/similarequal/angbracketleft/angbracketleftN2/angbracketright/angbracketrighttf. Here,tfis a characteristic\ntime scale for the ﬂuctuations, that is estimated as tf/similarequal\nN0(N0/AV); it yields tf/similarequalτsandtf/similarequal(A¯Γt/V)−1/3\nin the SR and TA regimes, respectively. We have eval-\nuated numerically the particle number zero-frequency\nnoise in these two regimes to ﬁnd SN(0) = 0.5/angbracketleft/angbracketleftN2/angbracketright/angbracketrightτs\nandSN(0) = 0.6/angbracketleft/angbracketleftN2/angbracketright/angbracketright(A¯Γ2\nt/V)−1/3where the variances\n/angbracketleft/angbracketleftN2/angbracketright/angbracketrightin the regimes are given by Eqs. (8), (9).\nA ﬂux noise of substantial amplitude SΦ/similarequal\n10−12Φ2\n0/Hz at low frequencies is routinely mea-\nsured in superconducting quantum interference devices\n(SQUIDs), here Φ 0is the ﬂux quantum. This noise lim-\nits the performance of superconducting qubits, that mo-\ntivated its thorough investigation [32–34]. Nowadays its\norigin is commonly attributed to the slow dynamics of lo-\ncalized spins at the surface of a superconductor [35–37].\nWe note that the spins of non-equilibrium quasi-particles\nmay also contribute to this noise. In fact, the polariza-\ntion mechanism predicted in this article make these spins\nvery eﬀective noise sources: Nquasi-particle spins com-\nbined in a polarized cluster produce the same noise as4\nJJqp. trap\nSQUID loopNormal\nlead\nFIG. 2: Experimental setup for the detection of the polarized\nstate. The quasi-particles are injected from a normal lead\ninto a trap embedded in the arm of a SQUID loop. The\nﬂuctuations of their common spin produce in the loop the\nﬂux noise to be measured. The polarization is seen as an\nenhanced noise ∝N2\n0.\nN2localized spins, provided the time scale of their dy-\nnamics is the same. In distinction from localized spins,\nthe quasi-particles can be brought to the superconductor\nin a controllable way, for instance, by injection through\na normal lead separated from the superconductor by a\ntunnel barrier [38].\nThis leads us to the suggestion of a concrete experi-\nmental setup to observe the predicted polarized state. As\ndepicted in Fig. 2, one makes a quasi-particle trap em-\nbedded in the arm of a superconducting loop by reducing\nthe superconducting gap locally, and injects the quasi-\nparticles into the trap from a normal lead that is biased\nat a voltage that slightly exceeds the reduced gap. The\nﬂux noise is monitored at diﬀerent injection rates cor-\nresponding to diﬀerent quasi-particle numbers N0. As-\nsuming a width of 100 nm for the SQUID arm, one spin\ninduces a ﬂux/similarequal10−7Φ0through the SQUID loop [32].\nFor the following estimations, we assume that triplet an-\nnihilation dominates, N0= 104, and the trap volume is\n(100 nm)3. At these conditions, tf/similarequal0.5×10−4s, and\nthe ﬂuctuations of the polarized state produce the noise\nSΦ/similarequal10−12Φ2\n0/Hz, that exceeds the commonly observed\nlevel. If the particles were not polarized, the ﬂux noise\nwould be four orders of magnitude lower. An advan-\ntage of the setup is that the number of quasi-particles\ninduced, as well as the ﬂuctuations of this number, can\nbe monitored through the high-frequency inductance and\ninductance noise of the superconducting sample [39].\nIn this work, we assume that a possible external mag-\nnetic ﬁeld does not polarize the quasiparticle spins. This\nis valid provided the corresponding Zeeman energy EZ/lessmuch\nδ/epsilon1. On the level of the master equation, the polarizing\neﬀect of the magnetic ﬁeld can be taken into account by\nassigning an anisotropy to the spin relaxation, but we\nhave not investigated this.\nIn conclusion, we propose a novel scenario for spon-\ntaneous spin polarization of a ﬁnite system under out-of-equilibrium conditions. We predict that, owing to\nthe spin selectivity of recombination, the excess quasi-\nparticles in a superconductor may spontaneously polarize\nin clusters. The underlying mechanism diﬀers from that\nconsidered in Ref. [41] for homogeneous quasi-particle\nstates. For parameters of Al, such a polarized cluster may\ncontain 104quasi-particles and spread over microns. We\nshow that the polarization can be detected as an excess\nﬂux noise.\nWe acknowledge valuable discussions with A. Bespalov\nin the early stages of this work. 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Gray and H. W. Willemsen, J. Low Temp. Phys.\n31, 911 (1978).\n[39] P. J. de Visser, J. J. A. Baselmans, J. Bueno, N. Llom-\nbart, and T. M. Klapwijk, Nature Commun. 5, 3130\n(2014).\n[40] M. Houzet, K. Serniak, G. Catelani, M. H. Devoret, and\nL. I. Glazman, Phys. Rev. Lett. 123, 107704 (2019).\n[41] A. G. Aronov and B. Z. Spivak, J. Low Temp. Phys. 40,\n911 (1980).1\nSupplemental Materials for “Dynamical spin polarization of excess quasi-particles in\nsuperconductors ”\nI. CLEBSCH-GORDAN COEFFICIENTS\nUpon addition of a quasi-particle to a cluster, the probabilities to realize a state with a total spin S/primeand spin\nprojection S/prime\nzare determined by the corresponding Clebsch-Gordan coeﬃcients for combining the spin Sof the\nexisting cluster with the spin 1 /2 of the incoming quasi-particle [See, e.g., L. D. Landau and E. M. Lifshitz, Quantum\nMechanics: Non-Relativistic Theory , Vol. 3 (3rd ed., Pergamon Press, 1977)]. The states we consider, labelled by N\nandm, correspond to S=Sz= (N−m)/2. Thus, the relevant Clebsch-Gordan coeﬃcients are the following:\n/angbracketleftbiggN−m\n2,N−m\n2;1\n2,1\n2/vextendsingle/vextendsingle/vextendsingleN−m+ 1\n2,N−m+ 1\n2/angbracketrightbigg\n= 1 (S1)\n/angbracketleftbiggN−m\n2,N−m\n2;1\n2,−1\n2/vextendsingle/vextendsingle/vextendsingleN−m+ 1\n2,N−m−1\n2/angbracketrightbigg\n=/radicalbigg\n1\nN−m+ 1(S2)\n/angbracketleftbiggN−m\n2,N−m\n2;1\n2,−1\n2/vextendsingle/vextendsingle/vextendsingleN−m−1\n2,N−m−1\n2/angbracketrightbigg\n=/radicalbigg\nN−m\nN−m+ 1(S3)\nThe ﬁrst two lines correspond to processes with ∆ m= 0 whereas the third line describes a process with m→m+ 1.\nII. MASTER EQUATION AND FOKKER-PLANCK EQUATION\nThe master equation that describes the evolution of the probability PN,m, taking into account all processes listed\nin the main text, is\nd\ndtPN,m =AV\n2/bracketleftbigg/parenleftbigg\n1 +1\nN−m/parenrightbigg\nPN−1,m+/parenleftbigg\n1−1\nN−m/parenrightbigg\nPN−1,m−1−2PN,m/bracketrightbigg\n(S4)\n+¯Γ\nV[(N−m+ 1)(m+ 1)PN+2,m+1−(N−m)mPN,m]\n+¯Γt\n2V[(m+ 2)(m+ 1)PN+2,m+2+ 2(N−m+ 1)(m+ 1)PN+2,m+1+ (N−m+ 2)(N−m+ 1)PN+2,m\n−N(N−1)PN,m] +1\nτsf[(m+ 1)PN,m+1+ (N−m+ 1)PN,m−1−NPN,m].\nAssuming that the number of particles exceeds the average density of the unpolarized system, we may write a\nsimpliﬁed master equation for the fully polarized system, m→0, where singlet annihilation is the dominant process.\nIn that case, injection of a quasi-particle with opposite spin plus singlet annihilation yields a process N+ 1→Nwith\nrateAV/2. Similarly spin ﬂip plus singlet annihilation yields a process N+ 2→Nwith rate 1/τs. In particular,\nd\ndtPN=AV\n2/bracketleftbigg/parenleftbigg\n1 +1\nN/parenrightbigg\nPN−1+/parenleftbigg\n1−1\nN+ 2/parenrightbigg\nPN+1−2PN/bracketrightbigg\n(S5)\n+¯Γt\n2V[(N+ 2)(N+ 1)PN+2−N(N−1)PN] +1\nτs[(N+ 2)PN+2−NPN].\nIn the continuum limit, the probability P(N) obeys the equation\n∂P\n∂t=AV\n2∂\n∂N/bracketleftbigg\nN2∂\n∂N/parenleftbiggP\nN2/parenrightbigg\n+/parenleftbigg2¯Γt\nAV2N2+4\nAVτ sN/parenrightbigg\nP/bracketrightbigg\n. (S6)\nThe stationary solution is Eq. (7) of the main text.arXiv:1912.10661v1  [cond-mat.supr-con]  23 Dec 20192\nIII. THE CROSS-OVER TO AN UNPOLARIZED STATE\nWhile, in principle, our model is only valid at m/lessmuchN, we may still use it to better characterize the cross-over to\nan unpolarized state. The full stationary solution of Eqs. (3) and (4) of the main text can be written as\nAV\nN−m= 2¯Γ\nVm−¯Γt\nVN2\nN−m(S7)\n=/parenleftbigg¯Γt\nVN+2\nτs/parenrightbigg\n(N−2m). (S8)\nIf the polarization p= 1−2m/N is not too close to 1, namely 1 −p/greatermuchN−1\nc, the second term in Eq. (S7) may be\nneglected. Using dimensionless units a=A/A c,v=V/Vc, andn=N/N c, we then obtain\nn=1−p\np−2v, a = 2(1−p2)/parenleftbigg1−p\n2vp−1/parenrightbigg2\n. (S9)\nThe second equation yields the polarization as a function of aandv. Substitution of the result into the ﬁrst equation\nthen allows one to obtain the number of particles as function of aandv. Equations (5) and (6) of the main text are\nreproduced at 1−p/lessmuch1. The polarization as well as the number of particles in the ( a,v)-plane is shown in Fig. S1.\nThe polarization smoothly decreases as aorvare increased, see Fig. S1a). While nincreases with aorv, see Fig. S1b),\nthe ratioN/N unpol .decreases, see Fig. S1c). A large enhancement of the quasi-particle number, N/N unpol ./greatermuch1, can\nbe seen at small aandv.\na)\n b)\n c)\nFIG. S1: Cross-over between the polarized and unpolarized state as described by Eqs. (S9). a) The polarization as a function of\nvanda. b) The number of quasi-particles as a function of vanda. c) Detail of the enhancement of the number of quasi-particles\ncompared to the unpolarized case, N/(c0V).\nWithin this description, the cross-over between the unpolarized regime and the SR dominated polarized regime\nappears smoother than sketched in Fig. 1 of the main text. Namely, according to Eqs. (S9), a ﬁnite polarization p\nbuilds up at vanishing injection, a→0, for a suﬃcently small volume, v<v 0withv0= (1−p)/(2p). As the injection\nincreases, the volume corresponding to the same polarization decreases,\na≈8p21 +p\n1−p(v0−v)2atv0−v/lessmuchv0. (S10)\nIV. ISING SPIN MODEL\nWhile the 1 /(N−m) corrections in the ﬁrst line of Eq. (S4) were essential to obtain the mean ﬁeld result, they\nare much less important for the ﬂuctuations. To conﬁrm this, we study classical (or Ising) spins as considered in [A.\nBespalov et al. , Phys. Rev. Lett. 117, 117002 (2016)] for comparison. As the spin quantization axis is now “pinned”,\none loses the 1 /N-asymmetry that favors the spontaneous polarization in the quantum (or Heisenberg) case. We\nnote, however, that, if a polarized cluster is generated due to ﬂuctuations, spin relaxation or triplet annihilation are\nnecessary for it to relax, as the singlet recombination process does not change the total spin.\nThe master equation in that case is very similar to Eq. (S4), however, without the 1 /(N−m) terms in the ﬁrst\nline. Note that, by contrast to the quantum case, where the condition m/lessmuchNwas used to estimate the number3\nof available states, here no approximations are necessary: the equation holds for all m. Assuming that the number\nof particles exceeds the average density of the unpolarized system, we may again write a simpliﬁed equation for the\npolarized system, m→0 (orm→N), adapting Eq. (S5) (i.e., omitting the terms ∼1/(N±1)). In the continuum\nlimit, the stationary solution now obeys the equation\n0 =∂\n∂N/bracketleftbigg∂\n∂NPcl+/parenleftbigg2¯Γt\nAV2N2+4\nAVτ sN/parenrightbigg\nPcl/bracketrightbigg\n, (S11)\nyielding\nPcl(N) =Cclexp/bracketleftbigg\n−2¯Γt\n3AV2N3−2\nAVτ sN2/bracketrightbigg\n. (S12)\nIf spin relaxation dominates, one ﬁnds\n/angbracketleftN/angbracketright(s)\ncl=/radicalbigg\nAVτ s\n2π. (S13)\nIf triplet annihilation dominates, one ﬁnds\n/angbracketleftN/angbracketright(t)\ncl=61/3√π\nΓ(1/6)/parenleftbiggAV2\n¯Γt/parenrightbigg1/3\n. (S14)\nSurprisingly the average number of particles /angbracketleftN/angbracketrightcldiﬀers from the quantum case only by a prefactor. The same\nholds true for the ﬂuctuations. By contrast, the mean ﬁeld solution is now given by Ncl\n0=Nunpol ./lessmuch/angbracketleftN/angbracketrightcland\nmcl\n0=Nunpol ./2."    },    {        "title": "2110.00045v1.Spin_waves_in_doped_graphene__a_time_dependent_spin_density_functional_approach_to_collective_excitations_in_paramagnetic_two_dimensional_Dirac_fermion_gases.pdf",        "content": "Spin waves in doped graphene: a time-dependent spin-density-functional approach to\ncollective excitations in paramagnetic two-dimensional Dirac fermion gases\nMatthew J. Anderson,1Florent Perez,2and Carsten A. Ullrich1\n1Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n2Institut des Nanosciences de Paris, CNRS/Universit\u0013 e Paris VI, Paris 75005, France\n(Dated: October 4, 2021)\nIn spin-polarized itinerant electron systems, collective spin-wave modes arise from dynamical\nexchange and correlation (xc) e\u000bects. We here consider spin waves in doped paramagnetic graphene\nwith adjustable Zeeman-type band splitting. The spin waves are described using time-dependent\nspin-density-functional response theory, treating dynamical xc e\u000bects within the Slater and Singwi-\nTosi-Land-Sj olander approximations. We obtain spin-wave dispersions and spin sti\u000bnesses as a\nfunction of doping and spin polarization, and discuss prospects for their experimental observation.\nI. INTRODUCTION\nGraphene is a material with many fascinating struc-\ntural and electronic properties [1]. Most notably, it has\na Dirac cone feature in its energy bands that leads to the\nelectrons behaving as massless particles. In its pristine\nform, graphene is a semimetal; it can be made metallic\nthrough doping or gating, which then leads to a wealth of\napplications, for instance in plasmonics [2{5]. Together\nwith many other new 2D materials, graphene also shows\npromise for spintronics applications [6].\nPlasmons are collective charge-density excitations,\nwhich can be characterized as the collective response\nof an electron gas to an induced electrostatic perturba-\ntion. Typically, plasmon mode frequencies and disper-\nsions are calculated using the random-phase approxima-\ntion (RPA); such calculations were done for graphene\nearly on [7{11]. The left panels of Fig. 1 give a\nschematic illustration of plasmons in doped graphene,\nshowing inter- and intraband single-particle excitations\nand the plasmon dispersion. The latter is very similar\nto the plasmon dispersion in a two-dimensional electron\ngas (2DEG) [12], with a characteristicpqbehavior for\nsmall wavevectors q. However, in graphene there are also\ninterband excitations from the lower to the upper cone,\nand the associated interband single-particle continuum\na\u000bects the plasmon dispersion for larger q. Such inter-\nband excitations are absent in the 2DEG model.\nIn this paper we study a type of collective excitation\nthat has so far not attracted much attention in graphene,\nnamely, spin waves. As illustrated on the right side of\nFig. 1, we consider doped, magnetized graphene in which\nthe spin-up and spin-down bands are split by an e\u000bective\nZeeman energy Z\u0003. The upper right-hand panel shows\ninter- and intraband spin-\rip excitations, and the lower\nright-hand panel shows the dispersion of a collective spin-\n\rip mode or spin wave.\nThe corresponding spin waves in magnetic 2DEGs have\nbeen well studied theoretically and experimentally [13{\n26]. On the other hand, apart from a recent study based\non Fermi-liquid theory [27], spin waves in doped graphene\nhave not been investigated to our knowledge. A di\u000ber-\nent type of collective excitation, known as magnetoplas-\n(q)\n02kF\nq02F\n0|kF kF|\nq0L\nZ*FIG. 1. Top left: spin-conserving single-particle transitions\nof doped, nonmagnetic graphene near a Dirac point. Bottom\nleft: associated plasmon dispersion and intra- and interband\nsingle-particle continua. Top right: spin-\rip transitions of\ndoped, magnetized graphene near a Dirac point. The bands\nare split by the Zeeman energy Z\u0003. Bottom right: associated\nsingle-particle spin-\rip continuum and spin-wave dispersion,\nwhere!Ldenotes the Larmor frequency.\nmon, has been more widely studied in graphene, includ-\ning edges, nanoribbons, and other graphene nanostruc-\ntures [28{36]. Magnetoplasmons occur in the presence\nof Landau level quantization induced by perpendicular\nmagnetic \felds. Here, by contrast, we will consider situ-\nations where the spin splitting can be thought of as being\ninduced by in-plane magnetic \felds, hence there are no\nLandau levels.\nTraditional band theory, based on density-functional\ntheory (DFT), has been extremely successful in describ-\ning materials with Dirac-like topological features [37].\nThus, in principle, we could calculate the graphene band\nstructure using, for instance, the local spin-density ap-arXiv:2110.00045v1  [cond-mat.mes-hall]  30 Sep 20212\nproximation (LSDA), and then obtain the spin-wave\ndispersions using linear-response theory based on time-\ndependent density-functional theory (TDDFT) [38], sim-\nilar to the standard way of calculating magnons in mag-\nnetic materials from \frst principles [39{41].\nOn the other hand, electrons in graphene and other\ntopological materials close to the Dirac points are well\ndescribed by simple tight-binding model Hamiltonians\n[1, 42], which de\fnes the model system of a 2D Dirac\nfermion gas; the purpose of this paper is to study spin\nwaves within this model system. This simpli\fes the task\nenormously, since a fully-\redged band-structure calcu-\nlation is not needed, and the electronic single-particle\nstates are known analytically.\nHowever, when it comes to the calculation of spin\nwaves, the Dirac fermion model presents us with an in-\nteresting challenge. The formation of spin waves in itin-\nerant electron systems is due to electronic many-body\ne\u000bects beyond the RPA. Many-body e\u000bects in graphene\nhave been studied in the literature, see e.g. [43, 44]. In\nthe language of TDDFT, these are dynamical exchange-\ncorrelation (xc) e\u000bects, which have to be approximated as\nfunctionals of the (spin) density. Most standard approxi-\nmations in DFT, such as the LSDA or gradient-corrected\nfunctionals [45], are based on the homogeneous electron\ngas (or the 2DEG [46, 47]); but a homogeneous elec-\ntron gas is not an appropriate reference system for Dirac\nfermions. Thus, the standard DFT functionals are not\napplicable.\nA way out of this dilemma is to use xc functionals\nwhich are not tied to any reference system, such as the\nso-called \\orbital functionals\" of (TD)DFT [48]. In this\npaper we will use two orbital-dependent approximations,\nnamely, the local exchange functional of Slater [49], and\nthe Singwi-Tosi-Land-Sj olander (STLS) approach to in-\nclude correlation [12, 50]. These functionals were recently\nused to study the structure and dynamics in Hubbard\nsystems with noncollinear magnetism [51, 52]. Here, we\nwill use them to analyze the spin-wave dispersion and\nspin sti\u000bness of doped magnetized graphene as a function\nof doping concentration and degree of spin polarization.\nThis paper is organized as follows: Section II presents\nthe necessary formal background for describing collective\nexcitations within spin-TDDFT, namely, linear-response\ntheory for noncollinear spins and the de\fnitions of the\nSlater and STLS approximations. In Section III we de\fne\nour model for Zeeman-split Dirac fermions in graphene,\nand show how to calculate spin waves using the Slater and\nSTLS approximations. Section IV then presents results\nfor spin-wave dispersions and spin sti\u000bnesses for various\nparameters, and discusses prospects for experimental ob-\nservation. Conclusions are given in Section V. Further\ninformation regarding the derivation of the noninteract-\ning response function, a discussion of the magnetic \felds\nrequired to produce spin-split bands, and additional nu-\nmerical details are given in the Appendix. Atomic units\n(e=m= \u0016h= 4\u0019\u000f0= 1) are used throughout.II. COLLECTIVE EXCITATIONS WITH\nSPIN-TDDFT\nA. Linear response formalism\nThe excitations of interacting electronic systems are\nencoded in the many-body response function [12]. Here,\nwe are speci\fcally concerned with spin waves, which are\ncollective spin-\rip modes; thus, a spin-dependent linear-\nresponse formalism is required, which will be based on\nTDDFT for noncollinear spins. In this framework, the\nbasic variable is the spin-density-matrix,\nn(r) =\u0012\nn\"\"(r)n\"#(r)\nn#\"(r)n##(r)\u0013\n: (1)\nAlternatively, the theory can be formulated in terms of\nthe particle density n(r) =n\"\"(r) +n##(r) and the mag-\nnetization vector m(r) = trf\u001bn(r)g, where \u001bis the vec-\ntor of Pauli matrices.\nLet us now consider the response of the system to a\nfrequency-dependent perturbation \u000ev(r;!), which has a\nsimilar matrix form as the spin-density-matrix (1), and\ncouples to the charge and spin degrees of freedom. The\nlinear spin-density-matrix response is given by\n\u000en(r;!) =Z\ndr0\u001f(r;r0;!)\u000ev(r0;!); (2)\nwhere\u001fis the many-body spin-density-matrix response\ntensor. In TDDFT, Eq. (2) is rewritten as\n\u000en(r;!) =Z\ndr0\u001f(0)(r;r0;!)\u000eve\u000b(r0;!); (3)\nwhere\u001f(0)is the response tensor of the noninteract-\ning Kohn-Sham system, and the e\u000bective perturbation\nis de\fned as the sum of the physical perturbation plus\na Hartree and exchange-correlation (Hxc) contribution,\n\u000eve\u000b=\u000ev+\u000evHxc. The latter is given by\n\u000evHxc(r;!) =Z\ndr0fHxc(r;r0;!)\u000en(r0;!): (4)\nThe Hartree part of the Hxc kernel is diagonal in the spin\nindices,fH\n\u001b\u001b0;\u001c\u001c0(r;r0;!) =\u000e\u001b\u001b0\u000e\u001c\u001c0=jr\u0000r0j. The remain-\nder, the xc kernel fxc(r;r0;!), needs to be approximated.\nTo obtain the excitation energies of the physical sys-\ntems, we need an explicit expression for the interacting\nspin-density matrix response tensor \u001f. Comparing the\ntwo response equations, Eqs. (2) and (3), leads to\n\u001f=\u0010\n1\u0000\u001f(0)fHxc\u0011\u00001\n\u001f(0): (5)\nThe excitations are at those frequencies where \u001fis not\ninvertible. This can happen in two ways: when \u001f(0)is\nnot invertible, or when 1\u0000\u001f(0)fHxcis not invertible. The\nformer yields the single-particle excitation spectrum, the\nlatter the collective excitations.3\nThus, all we need is the noninteracting response ten-\nsor\u001f(0)and an approximation for fxc. Here, we con-\nsider the special case where the ground state is such\nthat the system is uniformly magnetized, i.e., the spins\nare collinear. In that case, \u001f(0)only has four nonvan-ishing components: \u001f(0)\n\"\";\"\";\u001f(0)\n##;##;\u001f(0)\n\"#;\"#;\u001f(0)\n#\";#\". The xc\ntensor, on the other hand, has the nonvanishing ele-\nmentsfxc\n\"\";\"\";fxc\n##;##;fxc\n\"\";##;fxc\n##;\"\";fxc\n\"#;\"#;fxc\n#\";#\". With this,\nwe \fnd that 1\u0000\u001f(0)fHxccan be represented in the fol-\nlowing 4\u00024 matrix form:\n1\u0000\u001f(0)fHxc=0\nBBBB@1\u0000(w+fxc\n\"\";\"\")\u001f(0)\n\"\";\"\"0 0 \u0000(w+fxc\n\"\";##)\u001f(0)\n\"\";\"\"\n0 1\u0000fxc\n\"#;\"#\u001f(0)\n\"#;\"#0 0\n0 0 1 \u0000fxc\n#\";#\"\u001f(0)\n#\";#\"0\n\u0000(w+fxc\n##;\"\")\u001f(0)\n##;##0 0 1 \u0000(w+fxc\n##;##)\u001f(0)\n##;##1\nCCCCA; (6)\nwherewrepresents the Coulomb interaction. This ma-\ntrix is noninvertible if its determinant is zero. We can\nrearrange the matrix in block diagonal form, so that the\ndeterminant factors into a product of the determinants\nof two 2\u00022 matrices:\ndetj1\u0000\u001f(0)fHxcj= detjMLjdetjMTj; (7)\nwhere the longitudinal block is\nML= \n1\u0000(w+fxc\n\"\";\"\")\u001f(0)\n\"\";\"\"\u0000(w+fxc\n\"\";##)\u001f(0)\n\"\";\"\"\n\u0000(w+fxc\n##;\"\")\u001f(0)\n##;##1\u0000(w+fxc\n##;##)\u001f(0)\n##;##!\n(8)\nand the transverse block is\nMT= \n1\u0000fxc\n\"#;\"#\u001f(0)\n\"#;\"#0\n0 1\u0000fxc\n#\";#\"\u001f(0)\n#\";#\"!\n: (9)\nHere, longitudinal and transverse refers to the spin quan-\ntization axis. Thus, the condition det jMLj= 0 yields the\nlongitudinal (or spin-conserving) collective excitations,\nand detjMTj= 0 yields the transverse (or spin-\rip) col-\nlective excitations. The former are the usual plasmon\nmode and a longitudinal spin excitation, and the latter\nare the spin waves. In the following we will make the\nadiabatic approximation for the xc kernel, i.e., we ignore\nthe frequency dependence of the fxc's.\nB. Construction of dispersion relations\nWe see that both longitudinal unpolarized charge\n(plasmon) and transverse spin (spin-wave) collective ex-\ncitations satisfy equations of the schematic form\n1\u0000f(q)\u001f(q;!) = 0; (10)\nwherefand\u001fare placeholders for the corresponding\nfunctions speci\fc to the underlying system.\nIn order to obtain the dispersion relation, !(q), that\nsatis\fes this condition, one must be able to invert the\nfunction\u001f(q;!) for!. It is possible to approximate the\ninverse of\u001fto arbitrary degree using a technique calledseries reversion [53]. The procedure consists of obtaining\na series approximation of \u001fin!to arbitrary degree, ob-\ntain the inverse of the series using series reversion, and\n\fnally evaluate the inverse series at 1 =f(q).\nWe expand the response in !around!(q= 0)\u0011!0:\nXn[q;!0](!) =nX\nl=0\u001f(0;l)(q;!0)\nl!(!\u0000!0)l; (11)\nwhereXn[q;!0](!) is the truncated series approxima-\ntion of\u001f(q;!), and we de\fne a shorthand for the partial\nderivatives of a function of the form g(x;y):\ng(m;n)(a;b) =\u0012@m\n@xm@n\n@yng(x;y)\u0013\nx=a;y=b: (12)\nWe then use Xin place of\u001fin Eq. (10),\n\u001f(q;!)\u0019Xn[q;!0](!) =1\nf(q); (13)\nand inversion of this yields\n!(q)\u0019Xinv\nn[q;!0]\u00121\nf(q)\u0013\n: (14)\nThe details of this inverse series are left to the speci\fcs\nof the system.\nC. The Slater and STLS approximations\nTo calculate spin-wave excitations, the transverse xc\nkernelsfxc\n\"#;\"#;fxc\n#\";#\"are needed. As discussed in the In-\ntroduction, we shall work with two orbital-dependent ap-\nproximations: Slater and STLS. Both xc kernels are in-\ndependent of the frequency !.\nThe Slater exchange kernel [12, 54] was generalized for\nnoncollinear spin dynamics in Ref. [51]. In the limit\nwhere the ground state has collinear spin, the Slater ex-\nchange kernel has the following elements:\nfx;S\n\u001b\u001b;\u001b\u001b (r;r0) =\u0000j\r\u001b\u001b(r;r)j2\nn\u001b\u001b(r)n\u001b\u001b(r0)jr\u0000r0j(15)\nfx;S\n\u001b\u0016\u001b;\u001b\u0016\u001b(r;r0) =\u00004\r\u001b\u001b(r;r0)\r\u0016\u001b\u0016\u001b(r0;r0)\nn(r)n(r0)jr\u0000r0j; (16)4\nwhere\r\u001b\u001b0(r;r0) is the spin-resolved reduced one-body\ndensity matrix of the Kohn-Sham system.\nFor an unpolarized 2DEG, the Fourier transform of\nthe Slater exchange kernel was worked out in Ref. [55].\nThe generalization to the spin-polarized 2DEG, with\nspin densities n\u001band total density n=n\"+n#and\ncorresponding Fermi wavevectors kF\u001b=p2\u0019n\u001band\nkF=p\n2\u0019n, is as follows:\nfx;S\n\u001b\u001b;\u001b\u001b (q) =\u00008\u0019\nk2\nF\u001bZ1\n0dx\nx2J0(qx)J2\n1(kF\u001bx) (17)\nfx;S\n\u001b\u0016\u001b;\u001b\u0016\u001b(q) =\u00008\u0019kF\u001bkF\u0016\u001b\nk4\nFZ1\n0dx\nx2J0(qx)J1(kF\u001bx)J1(kF\u0016\u001bx)\n(18)\nwhereJ0andJ1denote Bessel functions of the \frst kind.\nThe corresponding expressions for graphene will be dis-\ncussed below in Sec. III D.\nThe STLS xc kernel [12, 50] was generalized to sys-\ntems with noncollinear spin in Ref. [51]. More precisely,\nthe noncollinear formulation of Ref. [51] involved a sim-\npli\fed, scalar form of the original STLS scheme, termed\nsSTLS. Again, let us consider the special case where the\nground state is collinear. The elements of the xc tensor\nfxc;sSTLS(r;r0) are then given by\nfxc;sSTLS\n\u001b\u001b0;\u000b\u000b0(r;r0) =1\njr\u0000r0jR\u001b\u001b0;\u000b\u000b0(r;r0) (19)\n\u0002[S\u001b\u001b0;\u000b\u000b0(r;r0)\u0000\u000e\u001b\u000b\u000e(r\u0000r0)n\u000b0\u001b0(r)];\nwhereR\u001b\u001b0;\u000b\u000b0are the elements of the following matrix:\nR=0\nBBBBB@1\nn\"(r)n\"(r0)2\nn(r)n\"(r0)2\nn(r)n\"(r0)1\nn#(r)n\"(r0)\n2\nn\"(r)n(r0)4\nn(r)n(r0)4\nn(r)n(r0)2\nn#(r)n(r0)\n2\nn\"(r)n(r0)4\nn(r)n(r0)4\nn(r)n(r0)2\nn#(r)n(r0)\n1\nn\"(r)n#(r0)2\nn(r)n#(r0)2\nn(r)n#(r0)1\nn#(r)n#(r0)1\nCCCCCA:\n(20)\nS\u001b\u001b0;\u000b\u000b0(r;r0) are the elements of the generalized static\nstructure factor:\nS(r;r0) =\u00001\n\u0019Z1\n0=\u001f(r;r0;!)d!: (21)\nThe response tensor \u001f, in turn, follows from Eq. (5)\nevaluated with the sSTLS kernel (19), which closes the\nself-consistency loop. If in the \frst step of the iteration\nEq. (21) is initialized with \u001f(0), then Eq. (19) yields the\nSlater exchange kernel. Correlation enters in the subse-\nquent iteration steps.\nConsider again the homogeneous 2D case and carry out\na Fourier transformation of Eq. (19); speci\fcally, for the\ntransverse xc kernel in sSTLS approximation we obtain\nfxc;sSTLS\n\u001b\u0016\u001b;\u001b\u0016\u001b(q) =4\nn2X\nq0vq0[S\u001b\u0016\u001b;\u001b\u0016\u001b(q\u0000q0)\u0000n\u0016\u001b\u0016\u001b];(22)\nwherevq= 2\u0019=q. We also introduce an alternative form\nof the xc kernel, which directly generalizes the originalSTLS approach [50]:\nfxc;STLS\n\u001b\u0016\u001b;\u001b\u0016\u001b(q) =4\nn2X\nq0q\u0001q0\nq2vq0[S\u001b\u0016\u001b;\u001b\u0016\u001b(q\u0000q0)\u0000n\u0016\u001b\u0016\u001b]:\n(23)\nThe di\u000berence between the two schemes is that the scalar\nsSTLS is based on the e\u000bective potential whereas the\nfull STLS is based on the e\u000bective force [12]. Express-\ning the full STLS kernel in real space and for inhomoge-\nneous systems causes some technical complications, since\nforces formally couple to currents rather than densities\n[56]. However, in the homogeneous case the transition\nfrom Eq. (22) to (23) is straightforward.\nFor graphene, the construction of the (s)STLS xc ker-\nnels involves some subtleties, which we will discuss below.\nIII. COLLECTIVE EXCITATIONS IN\nGRAPHENE\nA. Model: Dirac fermions with Zeeman splitting\nThe tight-binding model is commonly used to describe\nthe band structure of graphene [1]. We consider a gener-\nalization in which the spin-up and spin-down bands are\nsplit by a Zeeman energy Z\u0003. Thus, we consider a Hamil-\ntonian of the form\n^H=\u0000tX\nhl;mi;\u001b(^cy\nl\u001b^cm\u001b+ H.c.) +Z\u0003\n2X\nj;\u001bs\u001b^cy\nj\u001b^cj\u001b;(24)\nwheret(\u00192:8 eV) is the nearest-neighbor hopping en-\nergy, ^cy\nl\u001b(^cl\u001b) is the creation (annihilation) operator for\nan electron with spin \u001bat thelth site, and s\u001b= +1 and\n\u00001 for\u001b=\"and#, respectively. The sum over hl;miis\nrestricted to nearest neighboring sites.\nIn our model, Z\u0003is treated as an adjustable parameter.\nNotice that Z\u0003denotes the Zeeman energy renormalized\nby electronic many-body e\u000bects, which in general dif-\nfers from the bare Zeeman energy Z[21]. To determine\nZ\u0003from \frst principles would require a self-consistent\ncalculation of the band structure in the presence of an\nexternally applied uniform in-plane magnetic \feld, or a\nmagnetic \feld induced by proximity to a ferromagnetic\nsubstrate; how this could be realized will be further dis-\ncussed in Sec. IV B. In Appendix B we calculate the ef-\nfective magnetic \feld strengths required to produce given\nvalues ofZ\u0003in graphene.\nWe are particularly interested in the electronic states\nclose to the Dirac points. Instead of working with the\neigenstates of the full tight-binding Hamiltonian (24), one\ncan consider the eigenstates of the reduced Hamiltonian\n^HK, valid around the Kpoint of the graphene Brillouin\nzone [1, 57]:\n^HK=\r\u0010\n^\u001b(b)\nxkx+ ^\u001b(b)\nyky\u0011\n+Z\u0003\n2^\u001b(s)\nz: (25)5\nFIG. 2. Spin-split Dirac cones of doped graphene. The Zee-\nman splitting gives rise to di\u000berent Fermi surfaces for the\nmajority and minority spins.\nHere,\r= 3at=2, whereais the C-C bond length in\ngraphene, ^\u001bx;y;zare Pauli matrices operating on the band\n(b) or spin (s) degrees of freedom, and k= (kx;ky) is a\nwave vector measured with respect to the Dirac point K.\nThe energy eigenvalues of ^HKare\n\"kb\u001b=b\rjkj+s\u001bZ\u0003\n2; (26)\nwhereb=\u00061 is the band index. The spin-split Dirac\ncones are illustrated in Fig. 2. The associated eigenstates\nare\n K\nkb\u001b(r) =eik\u0001r\np\n2\u0012\ne\u0000i\u001ek\nb\u0013\n\ns\u001b; (27)\ninvolving the product of a two-component pseudospinor\n(since there are two sites within a unit cell) with the\ntwo-component (up/down) spinor s\u001b. The eigenstates\n K0\nkb\u001b(r) around the K0point are obtained by replacing b\nwith\u0000bin Eq. (27).\nB. Noninteracting response function\nAs discussed in Sec. II A, the noninteracting response\ntensor\u001f(0)is the fundamental object needed to calculate\ncollective excitations. The generic de\fnition is\n\u001f(0)(r;r0;!) =X\njl(fl\u0000fj) j(r) y\nl(r) y\nj(r0) l(r0)\n!\u0000(\"j\u0000\"l) +i\u0011;(28)\nwhere the  jare single-particle spinor wave functions\nlabeled by a set of quantum numbers j,\"jare the associ-ated single-particle energies, fjare occupation numbers\n(here, either 0 or 1), and \u0011is a positive in\fnitesimal.\nThe noninteracting response function of graphene\n(within the Dirac fermion model) is obtained by sub-\nstituting the single-particle energies (26) and eigenstates\n(27) into Eq. (28). The spin-independent form of the\ngraphene response function is well known from the lit-\nerature [7, 9, 57]; here, we generalize it to the spin-\ndependent case. Details of the derivation are given in\nAppendix A. Furthermore, instead of real frequencies !\nwe evaluate the response function for complex frequencies\nz, which has certain technical advantages, as discussed in\nAppendix C. The result for the non-spin-dependent re-\nsponse function at Z\u0003= 0,\u001f(0)=\u001f(0)\n\"\";\"\"+\u001f(0)\n##;##, is as\nfollows:\n\u001f(0)(q;z)\ngsgv=\u0000kF\n2\u0019\r\u0000q\n16\rr\n1\u0000\u0010\nz\n\rq\u00112\n+\u00061X\n\u000bq\n16\u0019\rG\u0012\u000bz\n\rq;\u000bz+ 2\rkF\n\rq\u0013\n;(29)\nwheregsandgvare the spin and valley degeneracies,\nrespectively, kF=p\n4\u0019n=gsgvis the Fermi wave vec-\ntor associated with the 2D electron density of the upper\nbandsn, and the function G(a;x) is de\fned as\nG(a;x) =x(x2\u00001)\u0000p\n1\u0000x2arcsinx\n(ax\u00001)q\n(1\u0000x2)(1\u0000a2)\n(1\u0000ax)2: (30)\nEq. (29) reduces to the real frequency form from previous\nwork by taking lim \u0011!0+(z=!+i\u0011). It is important to\nnote that Eq. (30) cannot be further reduced since the\nproper branch cuts must be preserved.\nThe transverse spin-dependent response functions at\n\fniteZ\u0003,\u001f(0)\n\"#;\"#and\u001f(0)\n#\";#\", are given by\n\u001f(0)\n\u001b\u0016\u001b;\u001b\u0016\u001b(q;z)\ngv=\u0000kF\u001b+kF\u0016\u001b\n4\u0019\r\u0000q\n16\rr\n1\u0000\u0010\nz+\"\u001b\u0016\u001b\n\rq\u00112\n+q\n16\u0019\r\u0014\nG\u0012z+\"\u001b\u0016\u001b\n\rq;z+\"\u001b\u0016\u001b+ 2\rkF\u001b\n\rq\u0013\n\u0000G\u0012z+\"\u001b\u0016\u001b\n\rq;z+\"\u001b\u0016\u001b\u00002\rkF\u0016\u001b\n\rq\u0013\u0015\n;(31)\nwhere \u0016\u001b=#if\u001b=\"and vice versa, kF\u001b=p2\u0019n\u001bis the\nFermi wavevector associated with the spin density of the\nupper band n\u001b, and\"\u001b\u0016\u001b=\"kb\u001b\u0000\"kb\u0016\u001b=s\u001b\u0000s\u0016\u001b\n2Z\u0003is the\nsingle-particle spin-\rip energy, which is independent of b\nandkfor the simple Zeeman splitting considered here.\nThe renormalized Zeeman energy can be expressed as\nZ\u0003=\rjkF\"\u0000kF#j=\rp\u0019n\f\f\fp\n1 +\u0010\u0000p\n1\u0000\u0010\f\f\f;(32)\nwhere\u0010= (n\"\u0000n#)=nis the spin polarization of the con-\nduction electrons. Figure 3 shows Z\u0003within the density-\npolarization parameter space. The range of doping den-\nsitiesn(1011\u00001013cm\u00002) is chosen such that the Dirac6\nFIG. 3. The renormalized Zeeman energy, Z\u0003, as a function\nof doping concentration nand spin polarization \u0010.\nmodel is still valid, i.e., the Fermi level does not reach\nthose parts of the conduction band where the band dis-\npersion deviates signi\fcantly from linearity. We \fnd that\nZ\u0003can reach values of a few hundreds of meV for strong\ndoping and high degrees of spin polarization.\nC. Mode dispersions and spin sti\u000bness\n1. Plasmons\nLet us \frst consider the spin-unpolarized case. The\ngraphene plasmon dispersion energy goes to zero as q\napproaches zero, see Fig. 1. This is problematic for\nthe inverse series procedure because the response func-\ntion has a singularity in the q-!plane at (0 ;0). The\nlimit of the response function depends on the direction\nas (0;0) is approached. For collective excitations, it is\nimportant to calculate the dispersion in the dynamical\nlong-wavelength limit (DLWL) [12], i.e. !\u001dvFq, where\nvFis the Fermi velocity. It is useful to introduce the\nparameter\u0017=!=\rq , which de\fnes the slope of a line\npassing through the origin and thus the direction of the\nlimit. In order to obtain the low- qbehavior and still be\nin the DLWL, we expand the response function in \u0017near\nin\fnity. This is equivalent to expanding in 1 =\u0017near 0.\nThe \frst few terms of the series expansion of the re-\nsponse function of Eq. (29) in \u0017are\n\u001f(0)(q;\rq\u0017 ) =kF\n\u0019\r\u00172+kF\n2\u0019\r\u00174+O\u00121\n\u00176\u0013\n: (33)\nThe corresponding inverse series is\n\u0017(y) =!(y)\n\rq\u0019r\n\rkF\n\u0019\u00121\n\rpy+\u0019\n4kFpy\u0013\n+O\u0010\ny3=2\u0011\n;\n(34)wherey= 1=fHxc(q). Thus, the plasmon dispersion re-\nlation becomes\n!pl(q) =s\n\rkF=\u0019\nfxc(q) +2\u0019\nq\u0012\n2\u0019+q\u0012\nfxc(q) +\r\u0019\n4kF\u0013\u0013\n+O\u00121\nfHxc(q)\u00133=2\n: (35)\nIn principle, it is straightforward to obtain higher order\nterms by including more terms in Eq. (33). However, it is\nbest to stop at the 4th order terms because of the DLWL.\nThe series diverges quickly for higher order terms.\n2. Spin waves\nThe spin polarization \u0010of the conduction electrons can\nbe positive or negative. Let us consider the case where\n\u0010 >0: this implies n\">n#and therefore, from Eq. (26),\nthe upper (lower) of the two spin-split conduction bands\nhas spin\u001b=#(\u0016\u001b=\"). Hence, (s\u001b\u0000s\u0016\u001b)=2 =\u00001. The spin\nwaves are obtained from Eq. (9), but only the condition\n1\u0000fxc\n#\";#\"\u001f(0)\n#\";#\"= 0 is needed. The case \u0010 <0 works in\nan analogous fashion, except that \u001band \u0016\u001bare reversed.\nThe spin waves in graphene have a \fnite dispersion\nenergy asqgoes to 0, see Fig. 1. The series can be cal-\nculated directly with !. The low-qspin-wave dispersion\nrelation becomes:\n!sw(q) =\"\u001b\u0016\u001b\u0012\n\u00001 +fxc\n0\n2\u0019\r(kF\u001b+kF\u0016\u001b)\u0013\n+qk2\nF\u0010(s\u001b\u0000s\u0016\u001b)fxc\n00\n2\u0019\n+q2\n2(s\u001b\u0000s\u0016\u001b)\n2S+O(q3); (36)\nwhere\nS=\r(p1 +s\u001b\u0010+p1 +s\u0016\u001b\u0010)\n2kF\u0010+\u0019\r2\nk2\nF\u0010fxc\n0\n+(s\u001b\u0000s\u0016\u001b)\n2fxc\n0\n4\u0019\u0012\nln\f\f\f\ffxc\n0\u00004\u0019\rp1 +s\u0016\u001b\u0010\nkF\u0010(s\u001b\u0000s\u0016\u001b)\f\f\f\f\n\u0000ln\f\f\f\ffxc\n0+4\u0019\rp1 +s\u001b\u0010\nkF\u0010(s\u001b\u0000s\u0016\u001b)\f\f\f\f\u0013\n+k2\nF\u0010fxc\n000\n2\u0019:(37)\nHere, we use the abbreviations fxc\n0,fxc\n00andfxc\n000for\ntheq= 0 limit of fxc\n\u001b\u0016\u001b;\u001b\u0016\u001b(q) and its \frst and second\nderivatives with respect to q, respectively. Notice that\nfor the Slater and STLS approximation we consider here,\nwe havefxc\n00= 0 and the linear term in the spin-wave\ndispersion (36) vanishes.\nThe generic form of spin-wave dispersions in itinerant\nparamagnetic electron liquids for small qis as follows:\n!sw(q) =!L+Ssw\n2q2+O\u0000\nq4\u0001\n: (38)\nHere,!Lis the Larmor frequency, which indicates a col-\nlective precessional motion of all spins about the mag-\nnetic \feld direction. For the case of graphene we \fnd7\nx,S\n,() fqσσ σσ(0)\n,()qσσ σσS(0)\n,(, )qσσ σσχω\n()hqzeroth order sSTLS :\nxc,STLS\n,() fqσσ σσ ,()qσσ σσS,(, )qσσ σσχωself-consistent full STLS:\nFIG. 4. Modi\fed STLS approach for Dirac fermions. To ob-\ntain the STLS xc kernel, an integration cuto\u000b h(q) is needed,\nwhich follows from the requirement that the zeroth iteration\nof the sSTLS scheme yields the Slater exchange kernel.\n!L=Z\u0003(1 +fxc\n0=2\r2), which is smaller than Z\u0003since\nfxc\n0<0. From electronic many-body theory one would\nhave expected that !L=Z(Larmor's theorem), where\nZis the bare Zeeman energy, i.e., all many-body ef-\nfects cancel out exactly in the Larmor precessional state\n[21]. However, Larmor's theorem does not apply here\nsince the band structure is obtained from a tight-binding\nDirac fermion Hamiltonian and not from \frst principles;\nin other words, Z\u0003is given but Zremains unknown.\nOne should therefore refer to !Lmore appropriately as\npseudo-Larmor frequency.\nThe spin-wave sti\u000bness, Ssw, determines the curva-\nture of the spin-wave dispersion for small q; it can have\npositive or negative values depending on the parame-\nters characterizing the electron liquid. Here, we have\nSsw= (s\u001b\u0000s\u0016\u001b)S=2.\nD. Slater and STLS kernels for Dirac fermions\nThe Slater approximation for Dirac fermions uses the\nsame expressions as for the 2DEG, Eqs. (15) and (16).\nWe use the graphene eigenstates (27) to construct the\ndensity matrix:\n\r\u001b\u001b(r;r0) =occX\nbk y\nbk\u001b(r0) bk\u001b(r) = 2occX\nbkeik\u0001(r\u0000r0);(39)\nwhere the factor 2 accounts for the valley degeneracy.\nWithin the Dirac model, the so de\fned density matrix\nnominally involves a diverging integral over an in\fnite\nlower band. To avoid this problem, we impose a \fnite\ncuto\u000b to the lower band at a wavevector kv. The nat-\nural choice for this cuto\u000b is that which reproduces the\nundoped density of graphene, nv= 1:91\u00021015cm\u00002:\nkv=p\u0019nv= 0:41a\u00001\n0: (40)Since\r\u001b\u001b(r;r0) only depends on the coordinate di\u000ber-\nence, we can make the substitution r\u0000r0=\u001a, and we\nalso de\fne an occupation function fb(k) which depends\non the band index b:\n\r\u001b\u001b(\u001a) =2\n(2\u0019)2X\nbZ1\n0kdkfb(k)Z2\u0019\n0d\u0012eik\u001acos(\u0012)\n=1\n\u0019X\nbZ1\n0kdkfb(k)J0(k\u001a)\n=1\n\u0019\u001a[kF\u001bJ1(kF\u001b\u001a) +kvJ1(kv\u001a)]: (41)\nWith this, the transverse Slater kernel for Dirac fermions\nbecomes:\nfx;S\n\u001b\u0016\u001b;\u001b\u0016\u001b(\u001a) =\u00004\n\u00192n2\u001a3[kF\u001bJ1(kF\u001b\u001a) +kvJ1(kv\u001a)]\n\u0002[kF\u0016\u001bJ1(kF\u0016\u001b\u001a) +kvJ1(kv\u001a)]: (42)\nFourier transform of this yields\nfx;S\n\u001b\u0016\u001b;\u001b\u0016\u001b(q) =\u00008\n\u0019n2Z1\n0d\u001a\n\u001a2[kF\"J1(kF\"\u001a) +kvJ1(kv\u001a)]\n\u0002[kF#J1(kF#\u001a) +kvJ1(kv\u001a)]J0(q\u001a): (43)\nThe Slater kernel is typically dominated by the valence\nband contribution because of the larger number of parti-\ncles compared to the conduction band.\nLet us now discuss how to implement the STLS scheme\nfor Dirac fermions. As for the Slater approximation, a\ncuto\u000b for the lower Dirac cone is necessary; otherwise,\nthe static structure factor S(q) diverges as q!1 (this\nhappens because the structure factor is proportional to\nthe density in the high- qlimit). We introduce the same\ncuto\u000b as above, Eq. (40), which ensures that the static\nstructure factor remains \fnite and bounded for all q.\nThe next problem arises from the shape of the struc-\nture factor itself. The calculation of the xc kernel con-\nverges only when S(q)\u0000nis asymptotically smaller than\n1=q. However, the tail of the structure factor goes as\nnv=2 +2k3\nF\u0000k3\nv\n6\u0019q, which approaches the wrong value for\nthe density as 1 =q. This produces an unavoidable sin-\ngularity in the integrand of the xc kernel. As a remedy,\nwe alter the integration limits in Eq. (22) such that the\nsSTLS xc kernel remains \fnite for all q. We \fx this limit\nby enforcing that the zeroth sSTLS iteration reproduces\nthe Slater kernel [12, 51]:\n4\nn2h(q)Z\n0q0dq0\n(2\u0019)22\u0019Z\n0d\u0012vq0\u0010\nS(0)\n\u001b\u0016\u001b;\u001b\u0016\u001b(q\u0000q0)\u0000n\u0016\u001b\u0016\u001b\u0011\n=fx;S\n\u001b\u0016\u001b;\u001b\u0016\u001b(q)\n(44)\nThe integration limit h(q) can be determined numerically\nusing standard root \fnding algorithms. We then use the\nsame integration limit for the non-scalar, full STLS ker-\nnel, Eq. (23). Our modi\fed STLS scheme is schemati-\ncally illustrated in Fig. 4. In the following, all spin-wave\nresults are obtained with the so de\fned full STLS kernel.8\nFIG. 5. Slater and STLS transverse-spin local \feld factors\ngxc\n#\";#\"(q) forn= 1:89\u00021013cm\u00002and\u0010= 0:82. The region\nin which spin-waves can exist is left of the dashed grey line\ncorresponding to the wavevector Z\u0003=\r=jkF\"\u0000kF#j.\n0 1\nq/|k k|\n01/Z*\n=0.4\n0.0 0.20.81.01011cm2\n1012cm2\n1013cm2\nFIG. 6. Spin-wave dispersions for various doping densities\nand polarization \u0010= 0:4. The dispersions are scaled by the\nrenormalized Zeeman energy, Eq. (32). The grey region is\nthe spin-\rip continuum. The response function has a \fnite\nimaginary part in this region and thus the spin-wave damps\naway.\nFigure 5 shows the Slater and STLS spin-\rip local-\n\feld factors, de\fned via fxc\n#\";#\"(q) =\u0000vqgxc\n#\";#\"(q), for\nn= 1:89\u00021013cm\u00002and\u0010= 0:82. The xc kernels\nare dominated by the scale set by the valence electron\ndensitynv. For the spin-wave dispersions, only the re-\ngion of small qvalues is relevant, in which the local-\feld\nfactors have a linear behavior, as indicated in the \fgure\nby the vertical dashed lines. It can be seen that Slater\nhas a larger slope than STLS, which directly a\u000bects the\nspin-wave dispersions, as we will see below.IV. RESULTS AND DISCUSSION\nA. Spin wave characteristics\nFigure 6 shows spin-wave dispersions, calculated using\nSTLS, for\u0010= 0:4 and three doping densities: n= 1011,\n1012, and 1013cm\u00002. For smaller densities, the dispersion\ncurves lie closer to the boundary of the spin-\rip contin-\nuum;Z\u0003\u0000!Lincreases with n. The inset to the \fgure\nshows a close-up of the spin-wave dispersions for small\nq: this illustrates how, for smaller n, the spin waves are\nmore and more squeezed into a narrow corner below the\nspin-\rip continuum, which causes the spin-wave sti\u000bness\nSswto increase.\nTo summarize the characteristic behavior of the spin-\n\rip waves, Fig. 7 shows 1 \u0000!L=Z\u0003andSswas a func-\ntion ofnand\u0010, calculated using Slater (left panels) and\nSTLS (right panels). The quantity 1 \u0000!L=Z\u0003represents\ntheq= 0 o\u000bset of the spin wave with respect to Z\u0003,\ni.e., the position of the Larmor mode with respect to the\nspin-\rip continuum. Large values of 1 \u0000!L=Z\u0003indicate\nthat the Larmor mode is well separated from the con-\ntinuum, which increases its lifetime and the chance of it\nbeing experimentally observable (see the discussion be-\nlow). The red line in the top panels of Fig. 7 indicates\nthatZ\u0003\u0000!0= 0:5 meV, which is comparable to typical\nlinewidths of spin waves found in 2DEGs [17].\nComparing Slater and STLS, we \fnd that the STLS\nspin waves tend to lie signi\fcantly closer to the contin-\nuum than Slater. This is because exchange is negative\nand correlation gives a positive correction. We also see\nthis from the slopes in Fig. 6.\nThe associated spin-wave sti\u000bnesses Ssware shown in\nthe lower panels of Fig. 7. The sti\u000bnesses in Slater and\nSTLS are very similar. We \fnd that Sswdiverges as\nnand\u0010approach zero. This is because the window in\nwhich the spin-wave can exist is shrinking: 0 < ! <\nZ\u0003\u0000\rq. The curvature must get larger in order to \ft in\nthis window. At some point this window shrinks to the\npoint of physical irrelevance, which implies that the spin\nwave merges with the continuum and ceases to exist as\na well-de\fned collective mode; however, it will still show\nup as a resonance feature that can be distinguished from\nthe broad background of the continuum. We also mention\nthat it is, in principle, possible to observe positive values\nofSsw; however, these would be for much larger values\nofn, where the Dirac model is no longer applicable.\nB. Prospects for experimental observation\nFor the experimental observation of spin waves in\ngraphene, doping concentrations of order n\u00181011to\n1013cm\u00002and signi\fcant spin polarizations \u0010are needed.\nIn Appendix B we show that for free-standing graphene\nthis would require the applications of in-plane magnetic\n\felds of tens to hundreds of Tesla, which is clearly not\npractical. Instead, suitable values of nand\u0010should be9\nFIG. 7. Top left: The pseudo-Larmor frequency, !L, of spin-waves calculated with the Slater approximation and scaled by\nthe renormalized Zeeman energy, Z\u0003. The red line indicates where Z\u0003\u0000!L= 0:5meV. Bottom left: The spin-wave sti\u000bness\ncalculated with the Slater approximation. The sti\u000bness diverges as it approaches the origin. Right: The same as the left but\ncalculated with the STLS scheme.\nattainable by depositing graphene on a magnetic sub-\nstrate. For instance, experimental and theoretical stud-\nies of graphene on Ni(111) have shown that interfacial\nhybridization of graphene atoms with the top interface\natoms of the magnetic layer causes a spin polarization\nin the graphene layer [58{60]. Similarly, Wang et al.\n[61] demonstrated proximity-induced ferromagnetism in\ngraphene/YIG structures. Wei et al. [62] observed\nstrong interfacial exchange \felds (in excess of 14 T) in\ngraphene/EuS structures, with the potential to reach\nhundreds of Tesla; device properties may be further im-\nproved by encapsulation of the graphene sheet in hexag-\nonal boron nitride [63].\nAssuming, then, that the necessary conditions (doping\nand spin polarization) can be achieved in graphene, the\nnext question is how to create and detect spin waves. For\n2DEG systems in semiconductor quantum wells, spin-\rip\nexcitations and spin waves have been studied using in-\nelastic light scattering (also known as electronic Raman\nscattering) [13{19, 26]. For this technique to work, the\npresence of spin-orbit coupling in the material is essen-\ntial; clearly, this rules out pristine graphene, where thespin-orbit coupling is very small [1]. Proximity-induced\nRashba-type spin-orbit coupling in graphene has been\nwell documented in the literature [6, 64, 65]. However,\nfor light-induced spin dynamics, L\u0001S-type spin-orbit cou-\npling in deeper valence bands is needed to enable spin\nmixing of interband electron-hole pairs in the 1 eV en-\nergy range. Whether these conditions can be achieved by\nproximity is an open question. Alternatively, one could\nexcite the magnetization dynamics in the ferromagnetic\nproximity layer and in this way trigger the spin dynamics\nin graphene. However, the resulting hybrid spin modes\nare expected to be more complex than the pure spin\nwaves considered here, requiring a theoretical description\nbeyond the model considered in this paper.\nAn alternative approach could be to use microscopy.\nPlasmons in graphene have been studied using near-\feld\nmicroscopy [66{68]. Spin-sensitive scanning probes such\nas spin-polarized scanning tunneling microscopy (SP-\nSTM) [69] have been used to probe spin structure and\ndynamics of ferromagnets at the atomic scale, including\nmagnon excitations [70{72]. There have been STM stud-\nies of the electronic and magnetic properties of quantum10\nHall states in graphene [73{75], and it seems conceivable\nthat similar techniques could be used for spin waves.\nV. CONCLUSION\nIn this paper we have presented a detailed study of\nspin waves in doped graphene with in-plane spin polar-\nization, using linear-response TDDFT. From a (TD)DFT\nperspective, many-body e\u000bects in graphene pose an inter-\nesting challenge, since Dirac fermions do not lend them-\nselves to a treatment using approximate density function-\nals derived from the homogeneous electron gas. Thus, we\nplaced some emphasis on the development and implemen-\ntation of orbital-based functionals, and showed that the\nSlater and STLS approximations can be successfully used\nfor the charge and spin dynamics in doped graphene.\nWe calculated spin-wave dispersions and spin sti\u000b-\nnesses for a wide range of doping concentrations and\nspin polarizations, and identi\fed regions where the spin\nwaves are well separated from the spin-\rip continuum,\nwhich means that they should be su\u000eciently long-lived\nto be observable. Creating and detecting spin waves in\ngraphene is without doubt a signi\fcant practical chal-\nlenge, and we discussed various experimental techniques\nthat appear promising.\nOur calculations are based on the ideal model of free-\nstanding graphene with a given Zeeman splitting. In\npractice, achieving a spin-polarized Dirac fermion gas\nmost likely involves interaction with a magnetic sub-\nstrate, which will also introduce spin-orbit coupling. To\naccount for these e\u000bects, our model can be generalized to\ninclude Rashba-type spin-orbit coupling; if the Rashba\nterms are not too strong, this will preserve the essential\nfeatures of the spin waves (as is the case in the 2DEG [17{\n21]). On the other hand, if the spin waves in graphene\nare coupled with spin excitations in the magnetic sub-\nstrate, more complex hybrid modes may occur. This will\nbe the subject of future studies.\nACKNOWLEDGMENTS\nThis work was supported by DOE Grant No. DE-\nSC0019109.\nAppendix A: Derivation of the noninteracting\nresponse functions\nStarting from Eq. (28) we get the de\fnition of the non-\ninteracting response function for a Dirac model. We \frst\nconsider the nonmagnetic case with Z\u0003= 0, where the\nsingle-particle energies are \"kb=b\rjkj, and the occupa-\ntion factors become fkb=\u0012(\"F\u0000\"kb). The labels jand\nlof the single-particle states are replaced with j!(b;k)\nandl!(b0;k0), whereb;b0=\u00061 are the band indices.The summation over kimplies the substitution\nX\nk!1Z\n0kdk\n(2\u0019)22\u0019Z\n0d\u001ek: (A1)\nSetting k0=k+q, we introduce the orbital overlap func-\ntion\nF\f(k;q) =1\n2(1 +\fcos(\u001ek0\u0000\u001ek)); (A2)\nwhere\ncos(\u001ek0\u0000\u001ek) =k+qcos\u001ekp\nk2+q2+ 2kqcos\u001ek\n=k0\u0000qcos\u001ek0p\nk02+q2\u00002k0qcos\u001ek0:(A3)\nThe response function then becomes\n\u001f(0)(q;!) =gsgvX\nbb0k(fkb\u0000fk0b0)Fbb0\n+(k;q)\n!+\"b(k)\u0000\"b0(k0) +i\u0011;(A4)\nwheregsandgvare the spin and valley degeneracies and\n\u0011is a positive in\fnitesimal required to preserve causality.\nNext, we introduce the complex frequency zand separate\nthe response function based on the b=\u00061 terms:\n\u001f(0)(q;z) =\u001f+(q;z) +\u001f\u0000(q;z); (A5)\nwhere\n\u001f+(q;z)\ngsgv=X\n\u000b\fkF\f(k;q)\n\u000bz+\"kb\u0000\"k0\f(A6)\nand\n\u001f\u0000(q;z)\ngsgv=X\n\u000bkF\u0000(k;q)\n\u000bz+\"k\u0000\u0000\"k0+\n=\u0000q\n16\rr\n1\u0000\u0010\nz\n\rq\u00112; (A7)\nwhere\u000b=\u00061 comes from separating the fkb\u0000fk0b0terms\nand transforming the integration limits of the k0integrals\nand\f=bb0=\u00061 is the (intra)interband transition. The\n\u001f\u0000term is a direct continuation from Ref [7].\nNext, we perform the \fsum in\u001f+to eliminate di\u000ecult\nterms:\n\u001f+(q;z)\ngsgv=X\n\u000bk1\n2\rk0\nB@1\u0000\u0010\n\u000b~z+2k\nq\u00112\n1\u0000\u000b~z\nq(\u000b~z+2k\nq) +2k\nqcos\u001ek\u000011\nCA;\n(A8)\nwhere ~z=z=\r. The angular integral evaluates to:\n2\u0019Z\n0d\u001e[1\u0000(a+b)2]\n1\u0000a(a+b) +bcos\u001e=1\u0000(a+b)2\n(1\u0000a(a+b))\n\u00022\u0019q\n1\u0000b2\n(1\u0000a(a+b))2(A9)11\nand therefore\n\u001f+(q;z)\ngsgv=\u0000kF\n2\u0019\r+X\n\u000b1\n4\u0019\rkFZ\n0dk1\u0000\u0010\n\u000b~z+2k\nq\u00112\n\u0010\n1\u0000\u000b~z\nq\u0010\n\u000b~z+2k\nq\u0011\u0011\n\u00021r\n1\u0000(2k\nq)2\n(1\u0000\u000b~z\nq(\u000b~z+2k\nq))2: (A10)\nThe radial integral evaluates to:\nZdx(1\u0000x2)\n(1\u0000ax)q\n1\u0000(x\u0000a)2\n(1\u0000ax)2=x(x2\u00001)\u0000p\n1\u0000x2arcsinx\n2(ax\u00001)q\n(1\u0000x2)(1\u0000a2)\n(1\u0000ax)2\n=1\n2G(a;x): (A11)\nThis \fnally gives\n\u001f+(q;z)\ngsgv=\u0000kF\n2\u0019\r+X\n\u000bq\n16\u0019\rG\u0012\u000b~z\nq;x\u0013\f\f\f\fx=\u000b~z+2kF\nq\nx=\u000b~z\nq\n=\u0000kF\n2\u0019\r+X\n\u000bq\n16\u0019\r\u0012\nG\u0012\u000b~z\nq;\u000b~z+ 2kF\nq\u0013\n\u0000G\u0012\u000b~z\nq;\u000b~z\nq\u0013\u0013\n=\u0000kF\n2\u0019\r+X\n\u000bq\n16\u0019\rG\u0012\u000b~z\nq;\u000b~z+ 2kF\nq\u0013\n(A12)\nThe spin-resolved response functions at \fnite Z\u0003follow\nin a similar way, except that the occupation factors are\nnow spin dependent and that the frequency is shifted by\n\"\u001b\u0016\u001b. We substitute j!(b;\u001b;k) andl!(b0;\u001b0;k0), use\nthe single-particle energies \"kb\u001b=b\rjkj+s\u001bZ\u0003\n2and the\noccupation factors fkb\u001b=\u0012(\"F\u0000\"kb\u001b), and we de\fne\nz\u001b\u001c=!+\"\u001b\u0016\u001b+i\u0011: (A13)\nThe response function then becomes:\n\u001f(0)\n\u001b\u0016\u001b;\u001b\u0016\u001b(q;!)\ngv=X\nkbb0(fkb\u001b\u0000fk0b0\u0016\u001b)\n\u0002Fbb0(k;q)\n!+\"kb\u001b\u0000\"k0b0\u0016\u001b+i\u0011:(A14)\nTo do the k-integration we then follow the same proce-\ndure as in the non-spin-polarized case above, taking care\nto note the di\u000berent spin occupation factors; in this way,\nwe arrive at Eq. (31).\nAppendix B: Magnetic \feld estimates\nThe e\u000bective Zeeman energy can be written as\nZ\u0003=g\u0016B(Bext+Bxc) =g\u0016BBe\u000b; (B1)\n0.0 0.2 0.4 0.6 0.8 1.0\n101110121013n[cm2]\n10 T50 T100 T250 T500 T1000 TFIG. 8. Contour lines of constant magnetic \feld in the\nn\u0000\u0010parameter space, illustrating that rather large magnetic\n\felds are required to generate signi\fcant spin polarization in\ngraphene via the Zeeman e\u000bect.\nwhere\u0016Bis the Bohr magneton and the e\u000bective mag-\nnetic \feldBe\u000bis the sum of the externally applied mag-\nnetic \feldBextand an additional magnetic \feld Bxcdue\nto exchange-correlation many-body e\u000bects [21]. Using\nthe experimental g-factor of graphene, g= 1:952 [76],\nwe can calculate the Be\u000bthat produces a given value of\nZ\u0003. Using Eq. (32) we can then relate Be\u000bto the spin\npolarization \u0010and doping concentration n.\nThis is illustrated in Fig. 8, which shows lines of con-\nstantBe\u000bin then\u0000\u0010parameter space. Clearly, a high\ndegree of spin polarization in strongly doped graphene\nwould require very large \feld strengths. Notice that Bxc\nis not available in our tight-binding model. Therefore,\nwe cannot obtain the external magnetic \feld Bextthat\nproduces\u0010for a given n; however, Be\u000bprovides a rea-\nsonable estimate for Bextsince xc e\u000bects can be expected\nto be comparatively small.\nAppendix C: Some numerical details\n1. Nonuniform q-grid\nIt is important that our choice of grid spacing for qbe\nsensitive to all of the relevant scales in the model, de-\ntermined by the three characteristic wavevectors kF,kv,\nandjkF\"\u0000kF#j. It is also important for the q-grid to ex-\ntend all the way to in\fnity to account for the integration\nlimits in Eqs. (22) and (23). We satisfy these conditions\nthrough the repeated use of the following transformation:\nt=\u0010\n1 +q\nk\u0011\u00001\n; (C1)\nwherekis one of the aforementioned wavevectors, and\nq=k2[0;1) maps tot2(0;1]. We then create a uni-\nformly spaced t-grid and transform back to a nonuniform\nq-grid. This results in about half of the qpoints lying12\nbelowkand the remaining points having a successively\nlarger spacing. Finally, we repeat this procedure for each\nwavevector and merge all of the grids together. The in-\ntegration along qis then carried out using integration\nroutines appropriate for nonuniform grids.\n2. Frequency integration\nIt is numerically convenient to use an alternate de\fni-\ntion of the structure factor. One can use a special con-\nstruction of the Cauchy integral theorem to show that\nthe integral in Eq. (21) can be transformed into\nS(r;r0) =\u00001\n\u0019Z1\n0<\u001f(r;r0;iu)du: (C2)This expression for the structure factor, involving inte-\ngration along the imaginary frequency axis, is numeri-\ncally much better behaved than Eq. (21). The reason\nis that=\u001f(r;r0;!) has minute details along the !-axis,\nwhereas<\u001f(r;r0;iu) is quite smooth away from the real\nfrequency axis. This transformation is the reason why\nin Sec. 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Lett.\n119, 066802 (2017)."    },    {        "title": "1107.4265v1.Theory_of_a_c__spin_current_noise_and_spin_conductance_through_a_quantum_dot_in_the_Kondo_regime_I__The_equilibrium_case.pdf",        "content": "arXiv:1107.4265v1  [cond-mat.mes-hall]  21 Jul 2011Theory of a.c. spin current noise and spin conductance throu gh a quantum dot\nin the Kondo regime I: The equilibrium case\nC. P. Moca,1,2I. Weymann,3and G. Zarand1,4\n1Department of Theoretical Physics, Institute of Physics,\nBudapest University of Technology and Economics, H-1521 Bu dapest, Hungary\n2Department of Physics, University of Oradea, 410087, Orade a, Romania\n3Department of Physics, Adam Mickiewicz University, 61-614 Pozna´ n, Poland\n4Dahlem Center for Complex Quantum Systems and Fachbereich P hysik,\nFreie Universit¨ at Berlin, 14195 Berlin, Germany\n(Dated: June 25, 2018)\nWe analyze the equilibrium frequency-dependentspin curre nt noise and spin conductance through\na quantum dot in the local moment regime. Spin current correl ations behave markedly diﬀerently\nfrom charge correlations. Equilibrium spin correlations a re characterized by two universal scal-\ning functions in the absence of an external ﬁeld: one of them i s related to charge correlations,\nwhile the other one describes cross-spin correlations. We c haracterize these functions using a com-\nbination of perturbative and non-perturbative methods. We ﬁnd that at low temperatures spin\ncross-correlations are suppressed at frequencies below th e Kondo scale, TK, and a dynamical spin\naccumulation resonance is found at the Kondo energy, ω∼TK. At higher temperatures, T > T K,\nsurprising low-frequency anomalies related to overall spi n conservation appear in the spin noise and\nspin conductance, and the Korringa rate is shown to play a dis tinguished role. The transient spin\ncurrent response also displays universal and singular prop erties.\nPACS numbers: 72.25.-b, 73.63.Kv, 72.15.Qm, 72.70.+m\nI. INTRODUCTION\nNanoelectronic devices are likely to provide our future\ntechnology and serve as basic tools for storing informa-\ntion, quantum computation1,2or spin manipulation.3,4\nDue to recent developments in fabrication, it is now pos-\nsible to produce and measure spin accumulation in meso-\nscopic circuits, or ﬁlter the generated spin currents.5–8\nOne of the next most prominent goals of spintronics is to\ngo further towards the microscopic regime,9–11and try\nto measure and manipulate single spins in quantum dots\nusing spin biased circuits. Understanding the structure\nof spin current noise and response through quantum dots\nis therefore of primary importance. Moreover, the inter-\nplay of strong interactions and the impact of quantum\nﬂuctuations of the spin on spin transport are also im-\nportant questions of fundamental interest on their own,\nand quantum dots, being the simplest strongly interact-\ning systems, also play a prominent role in this regard:\ntheyallowforthesystematicandcontrolledexperimental\nand theoretical study of strongly interacting states. Al-\nthoughnoteasytomeasure,12–17dynamicalnoisespectra\nand responsefunctions wouldallowtogain aclearinsight\nto the structure of interactions.18\nIn spite of its obvious importance, however, surpris-\ningly little is known about the dynamical spin current\nresponse and the spin current noise spectrum of a quan-\ntum dot. Spin current shot noise in the sequential tun-\nneling regime has been theoretically studied in Ref. [19],\nand later in the co-tunneling regime by Kindermann.20\nHowever, these calculations focussed almost exclusively\non d.c. properties, and have avoided the strong coupling\n(Kondo) regime,21which is much more diﬃcult to reachand understand theoretically.\nIn the present paper we make an attempt to character-\nize the equilibrium spin noise spectrum and the dynami-\ncal spin response of a quantum dot. Here we focus on the\nlocal moment regime,22where charge ﬂuctuations can be\nneglected, and the dot can simply be described in terms\nof a local spin operator, S(S= 1/2), coupled to the left\nand right lead electrons through an exchangecoupling, j.\nWe shall study how time dependent spin polarized cur-\nrents can be injected through the dot at various temper-\natures, and how these spin polarized currentsﬂuctuate in\ntime. To obtain a coherent and clearpicture, we combine\nvarious numerical and analytical methods such as nu-\nmerical renormalization group (NRG), perturbative and\nrenormalization group calculations, and a master equa-\ntion approach.\nThough some of the results presented here are also\nvalid in non-equilibrium, in this paper we focus exclu-\nsively on the equilibrium case, and leave the detailed pre-\nsentation of the rather technical non-equilibrium quan-\ntum Langevin calculation to a subsequent publication.23\nNevertheless, even in this equilibrium case, the spin noise\nandthespinconductancedisplayanextremelyrichstruc-\nture; in addition to the temperature, T, and the Kondo\ntemperature, TK, belowwhichthedot spingetsscreened,\nnew time scales emerge. In the regime, T≫TK, e.g., we\nﬁnd that the Korringa rate (the characteristic spin decay\nrate),\nEK≈πT\nln2(T/TK), (1)\nplays a distinguished role. Furthermore, incorporating\nthe eﬀect of external spin relaxation processes also turns2\nJRn'(t')=?\nn n’ VLn(t)\nJRn'(t')\nn’ \nn JLn(t)spin filter \nspin filter spin filter \nFIG. 1: (Color online) Sketch of the setups to measure\nthe spin-dependent left-right conductance, Gnn′\nLR(ω), and the\nnoise,Snn′\nLR(ω). The arrows indicate the direction of the spin\npolarization in each lead. In the upper setup the spin ﬁlter\non the right side detects the spin-resolved current at time t′,\nJn′\nR(t′), induced by applying a spin-dependent voltage to the\nleft lead at time t,Vn\nL(t). The lower setup allows for inject-\ning spins and measuring the spin-resolved currents in both\nelectrodes.\nout to be important, and introduces an additional new\nrate, 1/τs.\nIn our analysis, we focus on two crucial quantities. On\nthe one hand, we study the dynamical spin conductance\nGnn′\nrr′(ω). Thisquantitycharacterizeshowacurrent Jn\nr(t)\nofcarrierswith spins polarizedin direction nis generated\nin the left ( r=L) or right ( r=R) lead by a time\ndependent chemical potential shift, δµn′\nr′(t) =eVn′\nr′(t),\nactinginlead r′oncarrierspolarizedalong n′(seeFig.1),\n∝an}bracketle{tJn\nr(t)∝an}bracketri}ht=/integraldisplay\ndt′Gnn′\nrr′(t−t′)Vn′\nr′(t′).(2)\nFurthermore, we also investigate time dependent corre-\nlations of the currents Jn\nr(t),\nSnn′\nrr′(t−t′)≡1\n2∝an}bracketle{t{Jn\nr(t),Jn′\nr′(t′)}∝an}bracketri}ht,(3)\nand determine the corresponding noise spectra. Of\ncourse, in the equilibrium case studied here, Gnn′\nrr′and\nSnn′\nrr′are not independent, but are related by the ﬂuctu-\nation dissipation theorem [see Eq. (26)].\nOne of our main results is, that - in the absence of ex-\nternal spin relaxation and external magnetic ﬁeld - both\nthe noise and the conductance take on simple, universal\nforms, and apart from some geometry dependent prefac-\ntors, are characterized by just two universal functions.\nThe left-right ( r=L,r′=R) conductance, e.g., for\nn=σˆzandn′=σ′ˆzreads\nGσσ′\nLR(ω) =e2\nhsin2(φ) [δσσ′˜g(ω,T)+σσ′g(ω,T)],(4)\nwhere only the prefactor depends on the speciﬁc geome-\ntry of the dot, characterized by the angle φ(see Sec. II\n−ℜe g(ω)∝ ℜe G↑↓LR(ω)\nT≪TK\n∼1/ln2(ω/TK)\nT≫TK\n∼ω2\nω∼TK\nFIG. 2: (Color online) Sketchof the properties of the real pa rt\nof the universal function, −g(ω,T)∼G↑↓\nLR(ω), characterizing\nthe cross-spin conductance through the dot.\nand Ref. [24]), but the functions gand ˜gare univer-\nsal functions of ω/TKandT/TK. The prefactor,e2\nhin\nEq. (4), denotes the universal conductance quantum.49\nThe function ˜ gdescribeschargeconductance through the\ndot, whilegdetermines the cross-spin ( σ=↑,σ′=↓) con-\nductance,G↑↓\nrr′(ω). Our main goal is to study the proper-\nties ofgand those of the corresponding noise component\nin detail. The characteristic features of gare shown in\nFig. 2, which gives a concise summary of our most im-\nportant results. As shown in Fig. 2, gvanishes in the\nlimit,ω→0, and|g|develops a dip at ω < T K, and a\nbroad resonance at ω∼TKfor temperatures T≪TK.\nThe vanishing of the d.c. conductance is a consequence\nof the fact that spin can only be transferred from the\nspin-up channel to the spin-down channel by ﬂipping the\ndot spin,S. Thus the amount of spin transfer is limited,\nand no d.c. spin-cross conductance is possible in the ab-\nsence of external spin relaxation. Temporarily, however,\none can transfer spin between these two channels at the\nexpense of accumulating spin on the dot. This amounts\nin the appearance of the broad resonance.\nThe above-mentioned dip in |ℜe g|also survives at\ntemperatures, T≫TK: There, by the simple argu-\nment above, the cross-spin conductance also goes to zero\nℜe GLR(ω)/2\nT EK ωT≫TKℜe G↑↑LR(ω)\nℜe GLR(ω)/6\nFIG. 3: (Color online) Real part of the conductance G↑↑\nLR(ω)\nthrough the dot for T≫TK. The anomaly below the Ko-\nrringa relaxation rate, EK, is a consequence of correlations\nbetween consecutive spin-ﬂip processes, and is related to t he\ndip in|ℜe g(ω)|(see Fig. 2).3∝angbracketleftJ↓\nR(t)∝angbracketright\ntT=0\n∼1\nln2(t)\n∼h/TK∼e−t∆∼e2/h δV↑\nL\nFIG. 4: (Color online) Real time response of the current\n/angbracketleftJ↓\nR/angbracketright(t), as generated by a sudden change of amplitude, δV↑\nL.\nasω→0, however, this happens only below the Ko-\nrringa rate, ω < E K[see Eq. (1)], where correlations\nbetween consecutive spin-ﬂip events become important.\nAs a consequence of this, the spin-up – spin-up conduc-\ntance,G↑↑\nLR(ω), develops a peakfor frequencies ω <E K.\nThe relative size of the peak is universal (see Fig. 3), and\nis determined by the ratio of spin-ﬂip in the high tem-\nperature regime vs spin-diagonal scattering processes.\nBy just analyzing the analytical structure of the con-\nductance,Gσσ′\nrr′(ω), we can also make rather general con-\nclusions on the structure of transient response to a sud-\nden potential change at time t= 0,Vσ\nr(t) =δVσ\nrΘ(t),\nwhere Θ(t) denotes the step function. The linear re-\nsponse is universal, and it depends only on tTKand\nT/TK. Figure 4 sketches the structure of the zero-\ntemperature response, ∝an}bracketle{tJ↓\nR(t)∝an}bracketri}htgenerated by a sudden\nchange ofV↑\nL. The response is found to be logarithmi-\ncally singular at time t= 0, and at T= 0 temperature,\none observesat a time scale t≈h/TKan induced current\nbump of amplitude ∼(e2/h)δV↑\nL, followed by an expo-\nnential decay of the current response. The exponential\ndecay we ﬁnd is somewhat counter-intuitive: the long-\ntime behavior is typically associated with Fermi liquid\nproperties,25and therefore one could naively expect an\nalgebraic decay. However, the exponential decay found\nfollows precisely from Fermi liquid theory, which implies\ncertain analytical properties for the response functions.\nAs mentioned before, the properties of the equilibrium\nnoise spectrum are related to those of the conductance.26\nThus the noise has a structure similar to Eq. (4), and\ncan also be described in terms of just two universal func-\ntions, ˜s(ω) ands(ω), describing charge and cross-spin\ncorrelations, respectively. The left-right noise compo-\nnentsSσσ′\nLR(ω), e.g., can be expressed as\nSσσ′\nLR(ω) =−e2\nhTKsin2(φ) [δσσ′˜s(ω,T)+σσ′s(ω,T)],\n(5)\nwith the two universal functions ˜ sandsdisplaying\nmarkedly diﬀerent behavior. At T→0, e.g., ˜s(ω)∼\n|ω|/TK, while spin cross-correlations scale as s(ω)∼\n|ω|3/T3\nK.−S↑↑\nLR(ω)\nEKT≫TK\nT|S↑↑\nLR(ω)|,|S↑↓\nLR(ω)|\nω−S↑↓\nLR(ω)\nFIG. 5: (Color online) Structure of the spin noise component s\nS↑↑\nLR(ω) andS↑↓\nLR(ω) forT≫TK.\nThis diﬀerence between sand ˜scarries over to ﬁnite\ntemperatures, where ˜ sremains ﬁnite in the d.c. limit,\nwhilethespin-dependentcomponent softhenoisealways\nscales to zero as ω→0. In particular, at temperatures\nT≫TKthe charge component of the noise is similar to\nwhat is found for an ordinary tunnel junction,27,28\n˜s(ω,T≫TK)≈3π2\n16j2(ω,T)ω\nTKcoth/parenleftBigω\n2T/parenrightBig\n,(6)\nand remains proportional to the temperature, T, for\nω→0. Here the only nontrivial physics is carried by\nthe prefactor j2(ω,T)≈1/ln2(max{|ω|,T}/TK), which\naccounts for the Kondo-renormalized amplitude of the\ncurrent, mediated by the exchange processes. Similar to\ng, however, the spin-dependent part shas a non-trivial\nstructure, and exhibits adip at frequenciesbelowthe Ko-\nrringa rate, ω<E K, where correlations between consec-\nutive spin-ﬂip processesdrivethe cross-spinnoise tozero.\nThis amounts in somewhat unexpected low frequency\nanomalies in the noise spectra Sσσ′\nrr′(ω), as sketched in\nFig. 5 for the particular case of the left-right noise com-\nponents.\nThe results discussed so far hold in the case, where\nspin relaxation on the dot is induced exclusively by the\nexchange coupling, j, to the lead electrons. Then the\nd.c. cross-spin conductance and the shot noise vanish\ndue to spin conservation. External spin relaxation, how-\never, slightly modiﬁes the picture above, and in its pres-\nence we ﬁnd a small but ﬁnite spin shot noise and d.c.\nspin conductance. The precise dependence of the noise\non the spin relaxation rate, 1 /τs, is rather complex, and\nis analyzed in Sec. VI in detail. Experimentally, how-\never, the most relevantregimeseemsto be the one, where\nh/τs≪T,EK. There we ﬁnd a residual cross-spin noise\nS↑↓\nLR(0)∼\n\n−e2\nhh\nτs, ifT≫TK,\n−e2\nhT/parenleftBig\nh\nτsTK/parenrightBig2\n,ifT≪TK.(7)\nFor typical quantum dots, this residual noise and the\ncorrespondingconductance arethus verysmall compared4\nto all other features, and therefore inclusion of 1 /τsonly\nslightly modiﬁes the picture obtained within the Kondo\nmodel.\nThe paper is organized as follows. In Sec. II we de-\nscribe the system’s Hamiltonian and deﬁne the spin cur-\nrent operators. We then derive compact expressions for\nthe spin noise and spin conductance in terms of universal\nfunctions using the Kubo formula. The zero temperature\nbehavior is discussed in Sec. III, where we ﬁrst show\nthe analytical results in the perturbative and Fermi liq-\nuid regimes, and then present NRG results for universal\nfunctions obtained in the absence/presence of external\nmagnetic ﬁeld. As shown in Section III.B, application of\nan external magnetic ﬁeld leads to a more complex be-\nhavior and has somewhat similar eﬀect as the external\nspin relaxation for spin currents polarized perpendicular\nto the applied ﬁeld.\nSection IV is devoted to devoted to the study of the\nrole of a ﬁnite temperature. In the perturbative regime,\nto determine the ω-dependence of universal functions we\nuse a combination of master equation and perturbation\ntheory, while in the strong coupling regime we resort to\nthe Fermi liquid arguments. Transient response of the\ndot to a sudden switch of a time-dependent spin-resolved\nvoltage is discussed in Sec. VI, while in Sec. VII we\nconsider the eﬀect of external spin relaxation on the dy-\nnamics of spin correlations. Finally, the conclusions are\ngiven in Sec. VII.\nII. THEORETICAL FRAMEWORK\nA. Hamiltonian\nThroughout this paper, we focus our attention on the\nlocal moment regime of a quantum dot, where the dot is\noccupied by a single electron and it can be described in\nterms of a spin S, coupled to the leads. The Hamiltonian\nof the system consists of a part describing electrons in\nthe leads,Hleads, and an interacting part, Hint, which\naccounts for the exchange coupling between the dot and\nthe leads,H=Hleads+Hint. As usual, electrons in\nthe leads are assumed to be non-interacting, and in the\nabsence of a charge or spin bias, they are described by\nthe Hamiltonian,\nHleads=/summationdisplay\nr,σ/integraldisplayD\n−Dεc†\nrσ(ε)crσ(ε)dε, (8)\nwithc†\nrσ(ε) being the creation operator for a spin- σelec-\ntron with energy εin the left ( r=L) or right ( r=R)\nelectrode and 2 Dbeing the bandwidth. The energy ε\nis measured from the chemical potential, and the opera-\ntorscrσ(ε) satisfy the usual anticommutation relations:\n{c†\nrσ(ε),cr′σ′(ε′)}=δrr′δσσ′δ(ε−ε′).50\nThe interaction part, Hint, can be most easily con-\nstructedin termsoftheﬁelds, ψrσ=/integraltextD\n−Dcrσ(ε)dε, whichdestroyelectronsofspin σin the leadr. In terms ofthese\nﬁelds, the Kondo interaction is given by\nHint=/summationdisplay\nr,r′=L,R/summationdisplay\nσ,σ′j\n2vrvr′Sψ†\nrσσσσ′ψr′σ′,(9)\nwithja dimensionless coupling related to the Kondo\ntemperature, TK≈De−1/j, andσthe Pauli spin ma-\ntrices. Here the strength of the hybridization of the dot\nlevel with the leads is incorporated in the exchange cou-\npling,j, while its left-right asymmetry appears through\nthe dimensionless parameters, vL/R. These satisfy the\nrelationv2\nL+v2\nR= 1, and are parametrized by an an-\ngleφasvL= cos(φ/2),vR= sin(φ/2).51Using this\nparametrization the maximal, T= 0 temperature con-\nductance is given by, G0= (2e2/h) sin2(φ), wheree2/h\nis the conductance quantum. In the equilibrium case,\nstudied here, the problem can be further simpliﬁed by\nnoticing that only the combination Ψ ≡vLψL+vRψR\noccurs in Eq. (9). Therefore, rewriting the Hamiltonian\nin terms of the ’even’ (Ψ) and ’odd’ ( ˜Ψ) ﬁelds,\nΨσ≡cos/parenleftbiggφ\n2/parenrightbigg\nψLσ+sin/parenleftbiggφ\n2/parenrightbigg\nψRσ\n˜Ψσ≡sin/parenleftbiggφ\n2/parenrightbigg\nψLσ−cos/parenleftbiggφ\n2/parenrightbigg\nψRσ,(10)\nthe interaction part simpliﬁes to\nHint=j\n2S(Ψ†σΨ), (11)\nand the ’odd’ ﬁeld completely decouples from the dot.\nB. Spin current, spin conductance, and spin noise\nTo calculate the spin noise and the spin conductance,\nwe ﬁrst need to deﬁne the spin current operators. Let\nus ﬁrst focus on the case, where only the z-component is\nmeasured, and deﬁne the operators,\nQσ\nr=e/integraldisplayD\n−Dc†\nrσ(ε)crσ(ε)dε, (12)\nmeasuring the total charge of the electrons in lead rand\nspin component, σ, whereeis the electron charge. The\ncorresponding current operator can then be deﬁned as\nJσ\nr≡dQσ\nr\ndt. (13)\nNotice that with this deﬁnition a positive current implies\na charge ﬂow towards the leads. Using the equation of\nmotion we obtain\nJσ\nr=iej\n2/summationdisplay\nr′,σ′vrvr′S/parenleftbig\nψ†\nrσσσσ′ψr′σ′−h.c/parenrightbig\n.(14)\nThe charge current is then simply expressed as the sum\nof spin currents, Jr=/summationtext\nσJσ\nr.5\nAt this point, it is useful to also express the current\noperators in terms of the even and odd ﬁelds, Ψ and ˜Ψ.\nIntroducing the so-called composite fermion operators,29\nFσ≡(SσΨ)σ, we can write the current operator as a\nsum of two components,\nJσ\nr=Iσ\nr+˜Iσ\nr,\nIσ\nr=ej γri(Ψ†\nσFσ−F†\nσΨσ),\n˜Iσ\nr=ej˜γri(˜Ψ†\nσFσ−F†\nσ˜Ψσ), (15)\nwith the prefactorsdeﬁned as γL/R= (1±cos(φ))/4, and\n˜γL/R=±sin(φ)/4.\nIt is instructive to analyzethe propertiesof the current\noperators, Eq. (15). The components Iσ\nrsatisfy\nI↑\nr+I↓\nr≡0, (16)\nimplying that the components Iσ\nrdo not contribute to\nthe charge current, Jr=/summationtext\nσ˜Iσ\nr. Furthermore, since the\ncomponents ˜Iσ\nr, satisfy current conservation at the op-\nerator level,\n˜Iσ\nL+˜Iσ\nR≡0, (17)\nthere is no charge accumulation on the dot, and JL(t)+\nJR(t) = 0 is satisﬁed at the operator level at any time t.\nThe current components, Iσ\nr, on the other hand, do not\nsatisfy current conservation, Iσ\nL+Iσ\nR∝ne}ationslash= 0, implying that\nspin accumulation is possible on the dot. These proper-\nties follow naturally in the local moment (Kondo) limit,\nwherechargeﬂuctuationsarecompletelysuppressed, and\nonly spin ﬂuctuations are allowed on the dot.\nC. Conductance and noise\nWe shall now analyze the structure of the current re-\nsponse of the dot generated by an external perturbation,\nδH=Vσ\nr(t)Qσ\nr(t), which shifts the potential of carriers\nwith spinσin leadr. In linear response, the current is\ndetermined by the conductance Gσσ′\nrr′in Eq. (2), given by\nthe Kubo formula,\nGσσ′\nrr′(t,t′) =−iΘ(t−t′)∝an}bracketle{t[Jσ\nr(t),Qσ′\nr′(t′)]∝an}bracketri}ht.(18)\nThis expressionstill contains the chargeoperators, which\nare non-local operators. To eliminate them, we diﬀer-\nentiate with respect to time t′and use the equation of\nmotion. After Fourier transformation we then obtain the\nfollowing form of the Kubo formula,\niωGσσ′\nrr′(ω) = (Jσ\nr;Jσ′\nr′)ω−(Jσ\nr;Jσ′\nr′)ω=0,(19)\nwhere we introduced a compact notation for the Fourier\ntransform of the retarded correlation function of any two\noperators,\nGR\nAB(ω)→(A;B)ω. (20)The second term in Eq. (19) originates from the disconti-\nnuity ofGσσ′\nrr′(t) att= 0, which can be shown to be real\nand just equal to ( Jσ\nr;Jσ′\nr′)ω=0.\nLet us now focus on the SU(2) symmetrical case. To\nsimplify the expression of Gσσ′\nrr′, we can make use of the\ndecomposition of Iσ\nr, Eqs. (15), and exploit the property\n(16) as well as the fact that all cross-correlations of Iσ\nr\nand˜Iσ′\nr′vanish. Taking furthermore the spin symmetry\ninto account we ﬁnd that Gσσ′\nrr′reduces to the following\nform,\nGσσ′\nrr′(ω) =e2\nh[δσσ′˜arr′˜g(ω,T)+σσ′arr′g(ω,T)].\n(21)\nHere the information on the geometry of the dot is ex-\nclusively carried by the matrices ˜a(φ) anda(φ),\n˜a=/parenleftbigg\n−sin2φsin2φ\nsin2φ−sin2φ/parenrightbigg\n,a=/parenleftbigg\n4cos2φ\n2sin2φ\nsin2φ4sin2φ\n2/parenrightbigg\n,\nwhile the functions gand ˜gare completely independent\nof the dot geometry. They are deﬁned in terms of the\n’reduced current operators’,\nIσ≡i(Ψ†\nσFσ−h.c.) (22)\n˜Iσ≡i(˜Ψ†\nσFσ−h.c.) (23)\nand are given by the following expressions,\ng(ω) = +πj2\n8iω[(I↑;I↑)ω−(I↑;I↑)ω=0],\n˜g(ω) =−πj2\n8iω/bracketleftBig\n(˜I↑;˜I↑)ω−(˜I↑;˜I↑)ω=0/bracketrightBig\n.(24)\nHerewe havechosena normalizationsuch that forthe or-\ndinary conductance through the dot we recover the well-\nknown expression,\nGLR(ω,T) = 2e2\nhsin2(φ) ˜g(ω,T),(25)\nwith ˜g(ω,T→0) = 1 in the unitary limit.\nImportantly, both g(ω,T) and ˜g(ω,T) are dimension-\nless functions, and, apart from some universal prefac-\ntors, they both determine physically measurable quanti-\nties. Bythebasicprinciplesoftherenormalizationgroup,\nthis immediately implies that in the scaling limit, j→0,\nD→∞, andTKﬁnite, they must both reduce to func-\ntions of the ratios, ω/TKandT/TK. This can, of course,\nbe checked explicitly by performing perturbation theory\nin the exchange coupling, j.\nIn the equilibrium case, to which we restricted our-\nselves here, the symmetrized current noise [Eq. (3)]\nis simply related to the retarded Green’s functions,\n(Jσ\nr;Jσ′\nr′)ω, and thus to the conductance by the\nﬂuctuation-dissipation theorem,\nSσσ′\nrr′(ω) =−ωcoth/parenleftBigω\n2T/parenrightBig\nℜeGσσ′\nrr′(ω).(26)6\nUsing Eq. (21), we then immediately ﬁnd\nSσσ′\nrr′(ω) =−e2\nhTK[δσσ′˜arr′˜s(ω,T)+σσ′arr′s(ω,T)],\nwheresand ˜sare two real dimensionless universal func-\ntions characterizing the symmetrized equilibrium noise,\n/parenleftbigg\ns(ω,T)\n˜s(ω,T)/parenrightbigg\n=/parenleftbigg\nℜeg(ω,T)\nℜe˜g(ω,T)/parenrightbiggω\nTKcoth/parenleftBigω\n2T/parenrightBig\n.(27)\nThe structures of the universal functions g, ˜g,s, and ˜s\nshall be thoroughly discussed in the following Sections.\nD. Non-collinearity\nThe discussion above can readily be generalized to the\ncase, where the polarizationsof the injected and detected\ncurrents arearbitrary, but the system still exhibits SU(2)\nsymmetry. In this case we ﬁrst deﬁne the charge of car-\nriers polarized along nas\nQn\nr≡e/summationdisplay\nσ,σ′/integraldisplayD\n−Dc†\nrσ(ε)1\n2(1+nσ)σσ′crσ′(ε)dε.(28)\nThe corresponding current operators, Jn\nr=d\ndtQn\nrcan be\nexpressed as\nJn\nr=In\nr+˜In\nr,\nthe even and odd current components being deﬁned as\nIn\nr=ej γri(Ψ†PnF−F†PnΨ),\n˜In\nr=ej˜γri(˜Ψ†PnF−F†Pn˜Ψ),(29)\nwith the projector P ngiven by\nPn= (1+nσ)/2. (30)\nApart from the above changes of deﬁnition, the calcu-\nlations presented in Subsection IIC trivially generalize\nto this case, and the ﬁnal expression of the conductance\nassumes the following simple form,\nGnn′\nrr′(ω) =e2\nh/bracketleftBig\n˜arr′1+nn′\n2˜g(ω,T)\n+arr′nn′g(ω,T)/bracketrightBig\n, (31)\nwith the functions gand ˜gdeﬁned by Eqs. (24). Through\nthe ﬂuctuation-dissipation theorem, we then obtain the\nfollowing expression for the noise,\nSnn′\nrr′(ω) =−e2\nhTK/bracketleftBig\n˜arr′1+nn′\n2˜s(ω,T)\n+arr′nn′s(ω,T)/bracketrightBig\n, (32)\nwith the functions sand ˜sdeﬁned in Subsection IIC.\nWe emphasize again that the above expressions hold\nonly in the presence of spin rotation invariance, and in\nan external magnetic ﬁeld the conductance cannot be\nparametrizedin terms ofjust two universalfunctions (see\nSec. IIIB).III. THE T= 0TEMPERATURE LIMIT\nBefore discussing the more complex case of ﬁnite tem-\nperature, let us focus on the much simpler T= 0 case.\nThere we can compute the universal scaling functions g\nands”numerically exactly” using the machinery of nu-\nmerical renormalization group (NRG),30,31and can un-\nderstand all their important features relatively easily by\ncombining perturbative renormalization group methods\nwith Fermi liquid arguments.\nA. Analytical considerations\nPerturbation theory. The simplest one can do to com-\npute the noise is to evaluate the current-current correla-\ntion functions order by order in j. The 0-th order con-\ntribution to Sσσ′\nLRis found to be\nS↑↑\nLR=1\n2S↑↓\nLR=−e2\nhsin2φ|ω|π2j2\n16,(33)\nfrom which we can extract gand ˜gusing Eqs. (21) and\n(26),\n˜g=3π2j2\n16+...;g=−π2j2\n8+... . (34)\nEvaluating higher order terms to Sσσ′\nLR, one obtains loga-\nrithmic corrections to gand ˜g. The so-called leading log-\narithmiccorrectionscanbe summedup byaperturbative\nrenormalization group procedure,32,33which amounts in\nsimply replacing j→j(ω) = 1/ln(ω/TK), and gives\n˜g≈3π2\n161\nln2(|ω|/TK);g≈−π2\n81\nln2(|ω|/TK),(35)\nfor large frequencies, |ω|≫TK. By the ﬂuctuation dis-\nsipation theorem, Eq. (27), we then ﬁnd for |ω|≫TK\n˜s≈3π2\n16|ω|/TK\nln2(|ω|/TK);s≈−π2\n8|ω|/TK\nln2(|ω|/TK).(36)\nFermi liquid regime. The above expressions approx-\nimate the scaling functions only at large frequencies,\n|ω|≫TK. In the opposite limit, |ω|≪TK, perturbation\ntheory injbreaks down. However, the relevant processes\ncan be captured by a simple Fermi liquid approach.25In\nthis very small frequency limit both gand ˜gare analyt-\nical. AtT= 0 temperature the charge conductance is\nunitary in the limit, ω→0, and correspondingly,\n˜g= 1+O(ω2),˜s(ω/TK) =|ω|\nTK+... . (37)\nIn contrast to ˜ g, however, G↑↓\nrr′(ω→0) and thus\ng(ω→0) must vanish. This follows from the simple\nobservation that it is impossible to generate a steady\nspin-down current in any of the leads by injecting only7\nspin-up electrons. This simply follows from Fermi liquid\ntheory.25According to this latter, at T= 0 temperature\nthe impurity spin is completely screened by the Kondo\neﬀect, and electronsinjected at the Fermi energyareonly\nsubject to elastic scattering. As a consequence, a spin-up\nelectron injected right at the Fermi energy conserves its\nspin. Electrons (quasiparticles), however, interact with\neach-other through a local interaction at the impurity\nsite. For electrons injected with energy ω, this resid-\nual electron-electron interaction generates spin-ﬂip pro-\ncesses, and leads to a ﬁnite spin cross-conductance. By\nsimple phase space arguments, the amplitude of these\nlatter processes scales with the square of the energy of\nthe incoming electron, ∼ω2, and therefore g(ω)∼ω2.\nWe thus conclude that\ng=−αω2\nT2\nK+... , s =−α|ω|3\nT3\nK+... , (38)\nwithαa universal parameter.\nB. NRG results\nIn Section IIB we related the functions g, and ˜gto the\ncorrelation functions of the local operators, Iσand˜Iσ,\nrespectively [see Eq. (24)] . At T= 0 temperature, such\nlocal correlation functions can be accurately computed\nby NRG calculations,30,31which, as we show below, are\nindeedinfull agreementwith ourpreviousanalyticalcon-\nsiderations.\n1. No external ﬁeld, B= 0\nLet us start by sketching how one can perform the\nNRG calculations in the case where we have no external\nmagneticﬁeld, B. Tocompute ˜ g, we exploit the fact that\ncorrelation functions of the operators ˜Ψ†\nσFσandF†\nσ˜Ψσ\ncan be factorized onto correlation functions of F†\nσand\n˜Ψ†\nσ. The latter being trivial, after some algebra we then\nobtain for the real part of ˜ g,\nℜe˜g(ω,T) = (39)\nπ2j2\n4/integraldisplaydω′\n2ω̺F(ω′,T) [f(ω′−ω)−f(ω′+ω)],\nwithf(ω) the Fermi function and ̺F(ω,T)≡\n−1\nπℑm(F↑;F†\n↑)ωthe spectral function of the compos-\nite fermion operator. The prefactor π2j2/4 in Eq. (39)\ncan be eliminated by observing that ˜ g(0,0) = 1,24and\ntherefore\nπ2j2\n4̺F(0,0) = 1. (40)\nUsing this relation, we can also express the real part of\nthe scaling function gas\nℜeg(ω,T) =−1\n2ω̺I↑I↑(ω,T)\n̺F(0,0)(41)10-210010200.20.40.60.81−ℜe g,ℜe˜g\n−ℜe g\nω/TKℜe˜g\n10-2100102 10-810-610-410-2100\n∼/parenleftbiggω\nTK/parenrightbigg˜s∼/parenleftbiggω\nTK/parenrightbigg3\n−s\nω/TK−s,˜s\nFIG. 6: (Color online) The zero temperature universal func-\ntionsg, ˜gfor the a.c.-conductance (upper panel) and for the\nspin noise sand ˜s(lower panel) computed by NRG.\nwith̺I↑I↑(ω,T) =−1\nπℑm(I↑;I†\n↑)ωthe spectral func-\ntion of the operator, Iσ≡i(Ψ†\nσFσ−F†\nσΨσ). Computing\nthus the local spectral functions, ̺Fand̺I↑I↑we can\ndetermine the real parts of the functions, gand ˜g. From\nthese, we can then directly determine the universal spin\nand charge current noise functions sand ˜s.\nThe real parts of gand ˜gand the corresponding scal-\ning functions, sand ˜s, as obtained by NRG52are shown\nin Fig. 6. The results presented were obtained by the\nBudapest Flexible-DMNRG code,34,35and clearly dis-\nplay all features discussed in the previous subsection. All\nfunctions were plotted as a function of ω/TK, withTK\ndeﬁned as a half-width of the function ℜe˜g. This lat-\nter is essentially the a.c. charge conductance through\nthe dot, studied in Ref. [36], and displays the expected\nKondo resonance, which appears as a peak at frequencies\nω<TK.\nThe behavior of the universal function, −ℜeg∼G↑↓\nLR,\nis somewhat more surprising. It exhibits a maximum at\na frequency, ω∼TK, in agreement with the results of\nour analytical considerations, Eqs. (35) and (38). This\nmaximum correspondsto a temporaryspin accumulation\non the quantum dot at the “resonance” frequency, ω∼\nTK.\nThe lower panel of Fig. 6 shows the universal functions\nfor the spin noise on a logarithmic scale. The function ˜ s\nshows a clear linear behavior at frequencies ω<TK. The\ndeviations at larger frequencies from the linear behavior8\n10-310-210-110010110210300.10.20.30.4\nPSfrag replacements\nω/TK−ℜe gzBz/TK= 0\nBz/TK= 0.1\nBz/TK= 0.2\nBz/TK= 0.5\nBz/TK= 1\nBz/TK= 2\nBz/TK= 5\nBz/TK= 10\n10-310-210-110010110210300.20.40.60.81\nPSfrag replacements\nω/TKℜe˜g\nBz/TK= 0\nBz/TK= 0.1\nBz/TK= 0.2\nBz/TK= 0.5\nBz/TK= 1\nBz/TK= 2\nBz/TK= 5\nBz/TK= 10\n10-310-210-110010110210310-1210-1010-810-610-410-2100\nPSfrag replacements\nω/TK−sz\nBz/TK= 0\nBz/TK= 0.1\nBz/TK= 0.2\nBz/TK= 0.5\nBz/TK= 1\nBz/TK= 2\nBz/TK= 5\nBz/TK= 1010-310-210-110010110210310-410-310-210-1100101102\nPSfrag replacements\nω/TK˜s\nBz/TK= 0\nBz/TK= 0.1\nBz/TK= 0.2\nBz/TK= 0.5\nBz/TK= 1\nBz/TK= 2\nBz/TK= 5\nBz/TK= 10\nFIG. 7: (Color online) Real parts of the universal functions gzand ˜g, and the corresponding noise scaling functions, szand ˜s,\nas obtained by performing zero-temperature NRG calculatio ns with a magnetic ﬁeld Bzalong the zth direction.\nare due to logarithmic corrections. At high frequencies\n−sfollows the behavior of ˜ s, and is linear apart from the\naforementioned logarithmic corrections. For ω < T K,\nhowever, it deviates strongly, and scales to zero as ∼ω3.\n2. Eﬀect of Zeeman ﬁeld, B/negationslash= 0\nThe presence of a Zeeman ﬁeld, HB=−B·S, breaks\nthe spin SU(2) symmetry. Therefore, for general spin\npolarizations, nandn′, the spin current noise and the\nspin conductance cannot be characterized by just two\nuniversal functions, as before. Using group theoretical\narguments and exploiting the electron-hole symmetry of\nthe Kondo model we can show that 5 universal functions\nneed be used to characterize the complete nandn′de-\npendenceof Gnn′\nrr′. Moreover,thesefunctionswillnotjust\nbe functions of ω/TKandT/TK, but they also depend\non the ratio, B/TK.\nHere we do not venture to characterize all these func-\ntions. Rather, we focus our attention to the special case,\nwheren||n′. Without loss of generality, we can then\nassume that nandn′are both parallel to the zaxis,\nn=σˆzandn′=σ′ˆz. In this special case, one can\nshow that, as a consequence of electron-hole symmetry\nand rotationalinvariancearoundthe axis ofthe magneticﬁeld, the contribution of the current ˜Iσ\nrdoes not depend\non the direction of the magnetic ﬁeld, and that the total\nconductance has the structure,\nGσσ′\nrr′(ω,θ) =e2\nh/braceleftBig\nδσσ′˜arr′˜g(ω,B) (42)\n+σσ′arr′/bracketleftbig\ncos2(θ)gz(ω,B)+sin2(θ)g⊥(ω,B)/bracketrightbig/bracerightBig\n,\nwhereθis the angle between the magnetic ﬁeld and n,\nand two new scaling functions replace the originalscaling\nfunction,g(ω,T). Notice that all these scaling functions\nnow depend on the size of the magnetic ﬁeld, too. Appli-\ncation of the ﬂuctuation dissipation theorem yields then\na similar scaling form for the noise, with corresponding\nscaling functions, ˜ s,sz, ands⊥,\nSσσ′\nrr′(ω,θ) =−e2\nhTK/braceleftBig\nδσσ′˜arr′˜s(ω,B) (43)\n+σσ′arr′/bracketleftbig\ncos2(θ)sz(ω,B)+sin2(θ)s⊥(ω,B)/bracketrightbig/bracerightBig\n.\nTo determine the scaling functions, ˜ gandgz, and the\ncorresponding noise scaling functions, we performed the\nzero-temperature NRG calculations for the case, where\nthe magnetic ﬁeld is parallel to the zaxis,Bz=B(i.e.,\nθ= 0). The results are summarized in Fig. 7. Applica-\ntionofamagneticﬁeldgraduallyremovesthespin degen-\neracy of the dot, and leads to a splitting and suppression9\n10-310-210-110010110210300.10.20.30.4\nPSfrag replacements\nω/TK−ℜe g⊥Bx/TK= 0\nBx/TK= 0.05\nBx/TK= 0.1\nBx/TK= 0.2\nBx/TK= 0.5\nBx/TK= 1\nBx/TK= 2\nBx/TK= 5\n10-310-210-1100101102103 10-510-410-310-210-1100101102\nPSfrag replacements\nω/TK−s⊥\nBz/TK= 0\nBz/TK= 0.1\nBz/TK= 0.2\nBz/TK= 0.5\nBz/TK= 1\nBz/TK= 2\nBz/TK= 5\nBz/TK= 10\nFIG. 8: (Color online) Zero temperature NRG results for the\nreal part of the universal function g⊥, and the corresponding\nnoise scaling function s⊥for several values of magnetic ﬁeld\npointing along the xdirection.\nof the Kondo resonancefor B≫TK. The splitting of the\nresonance can readily be observed in the scaling function\n˜g(i.e., the charge conductance through the dot). At very\nlarge ﬁelds, B≫TK, the conductance has a logarithmic\ntail forω≫B, but is strongly reduced at frequencies\nω<B, where spin-ﬂip processes are forbidden.\nSimilartog, the scalingfunction gzcontinuesto vanish\natω= 0 for any magnetic ﬁeld. This property follows\nfrom the overall conservation of the total spin compo-\nnent parallel to the external magnetic ﬁeld. The only\neﬀect of a magnetic ﬁeld is thus to increase the region of\nsuppressed conductance.\nNext, to determine the scaling function, g⊥, we per-\nformed calculations with a ﬁeld Bxapplied along the x\ndirection (θ=π/2). We also checked that ˜ gis the same\nas before, and does not depend on the direction of the\napplied ﬁeld. The function g⊥, however, is markedly dif-\nferent from gz. Of course, for B= 0 they are both equal\ntog. However, any ﬁnite magnetic ﬁeld results in a ﬁ-\nnite d.c. conductance, g⊥(ω= 0)∝ne}ationslash= 0. The reason of\nthis is that in this case we measure a spin component\nwhich is not conserved for any ﬁnite magnetic ﬁeld. In a\nsense, the perpendicular magnetic ﬁeld in this case playsa role similar to the external spin relaxation, discussed\nin Section VI, and allows the impurity spin to ﬂip back\nand forth, independently of the conduction electrons. As\na consequence, the ”pseudogap” feature in g⊥is gradu-\nally ﬁlled up with increasing Bx, reaching maximum for\nBx∼TK.\nIV. THE FINITE TEMPERATURE CASE\nAlthough the NRG calculations presented in the previ-\nous section could, in principle be extended to any ﬁnite\ntemperature, a serious technical problem appears. As\nwe mentioned in the introduction, the cross-spin conduc-\ntance,G↑↓\nLRand thusg, must vanish at any ﬁnite tem-\nperatureTin theω→0 limit. However, currently used\nﬁnite temperature NRG broadening schemes all produce\nlinear (∝ω) spectral weight in ̺I↑I↑(ω,T), and there-\nfore by Eq. (41) lead to a ﬁnite and thus unphysical d.c.\ncross-spin conductance.\nWedonotknowofanywaytogetaroundthis problem.\nTherefore,atﬁnite T,wehadtorelyonanalyticalresults,\nand combine them with the T= 0 temperature results\nto obtain a coherent picture. At very high temperatures,\nT≫TK, we can make use of perturbative approaches.\nHowever, as explained below, even in this regime simple-\nminded perturbation theory is insuﬃcient, and we need\nto combine it with a master equation approach to ob-\ntain the complete ωdependence of the spin conductance\nand noise. Combining these perturbative results with\nscaling and Fermi liquid arguments, we are then able to\nunderstand the complete frequency and temperature de-\npendence of the scaling functions, s, ˜s,gand ˜g.\nA. Perturbation theory and master equation\napproach\nAt temperatures T≫TKcorrections to the leading\norder perturbative results are small, and much of the\nnoise spectrum can be understood based upon perturba-\ntive results. However, to understand the limitations of\nperturbation theory, we ﬁrst need to understand the im-\nportanttimescalesinthishightemperaturelimitandthe\nway they inﬂuence spin transport. At T≫TK, trans-\nport through the dot occurs through individual exchange\nprocesses, whereby just one electron tunnels from one\nside of the dot to the other side of it. The typical time\nbetween such events is given by the ”Korringa time”,\nτK, which we deﬁne as the inverse of the Korringa rate,\nτK∼h/EK. To the leading order in j, it is given as,\nτK∼h/j2T. Tunneling events are, however, not instan-\ntaneous in the sense that they are dressed by the in-\nternal dynamics of the electron-hole excitations, created\nthroughout the tunneling process. Correspondingly, the\n”duration” of a tunneling process is given by the thermal\ntime,τT≡h/T≪τK. At very short times below the\nthermal time, t<h/T≡τT(or at frequencies ω >T),10\ncurrent-current correlations reﬂect just the internal and\ncoherent dynamics of such a spin-ﬂip event, as well cap-\ntured by the usual connected second order contribution\nto the current-current correlation function.\nAs just stated, the usual bubble diagramonly accounts\nfor the structure of a singletunneling event. At times\nt>τK, however, several independent incoherent tunnel-\ning processes take place. These processes are correlated,\nsince a spin-ﬂip process that changes the dot spin from\nup into down (⇑→⇓) must necessarily be followed by an\nopposite process when the dot spin ﬂips from down to up\n(⇓→⇑). These correlations turn out to be important for\nspin transport, and are obviously not captured by sim-\nple perturbation theory. Fortunately, for times t > τT\n(ω < T), the internal dynamics of a spin-ﬂip event can\nbe ignored, and we can make use of a master equation\nmethod as a complementary approach. There tunneling\nprocesses are taken to be instantaneous, just character-\nized by some rates, but correlations between individual\nspin-ﬂipeventsareproperlyaccountedforthroughaclas-\nsical rate equation.\nFrom these simple arguments we thus conclude that\nthe master equation approach must be valid for frequen-\nciesω < T, while simple-minded perturbation theory\nworks for frequencies EK<ω. SinceEK<T, the range\nof validity of these two approaches overlaps, as also con-\nﬁrmed by the actual calculations presented below and by\nthe results of Ref. [20].\n1. Perturbation theory\nAs explained before, for T≫TKand timest < τK\n(frequencies ω>E K), perturbationtheory(PT)isthusa\ngoodapproximation,thoughit failsatlongertimeswhere\nalready several spin-ﬂip events occur, and the correla-\ntions between these spin-ﬂip events cannot be neglected.\nSimplest 0-th order perturbation theory yields, e.g., for\nthe left-right components of the symmetrized frequency-\ndependent spin noise\nS↑↓\nLR(ω) =−e2\nhsin2φπ2j2\n8ωcoth/parenleftBigω\n2T/parenrightBig\n+... ,\nS↑↑\nLR(ω) =−e2\nhsin2φπ2j2\n16ωcoth/parenleftBigω\n2T/parenrightBig\n+... ,\nwith the dots referring to higher order corrections in j.\nThe corresponding universal functions then read for ω>\nEK,\nsPT(ω) =−2\n3˜sPT(ω) =−π2j2\n8ω\nTKcoth/parenleftBigω\n2T/parenrightBig\n+... ,\ngPT(ω) =−2\n3˜gPT(ω) =−π2j2\n8+... . (44)Higher order terms give logarithmiccorrections, and lead\nto a renormalization of jin these expressions.\n2. Master equation approach\nLet us now focus on the ”classical” frequency regime,\nω<T. Here we can use a simple master equation (ME)\napproach,37where we assume that, at any instance, the\nspin on the dot is either in a spin-up state S=⇑or in a\nspin-down state S=⇓, with corresponding probabilities,\nPS(t). Conduction through the dot and spin relaxation\nare generated by scattering events, q, generated by the\nexchange interaction, Eq. (9). These scattering events\nare taken to be instantaneous, and consist of the scatter-\ning of a spin σelectron from lead rto a spinσ′state in\nleadr′while changing the dot spin, S→S′,\nq↔{r′,σ′,S′←r,σ,S}.\nThey occur with a rate, γ(q) =γS′←S\nr′σ′←rσ∝j2v2\nrv2\nr′, fol-\nlowing Fermi’s golden rule.\nThe dynamics of the dot spin is described by a simple\nmaster equation,\nd\ndt/parenleftbigg\nP⇑\nP⇓/parenrightbigg\n=/parenleftbigg\n−Γ Γ\nΓ−Γ/parenrightbigg/parenleftbigg\nP⇑\nP⇓/parenrightbigg\n,(45)\nwith the relaxation rate Γ given as Γ ≡/summationtext\nr,r′,σ,σ′γ⇑←⇓\nrσ←r′σ′. From Eq. (45) it follows that\nspin polarization on the dot relaxes exponentially,\n∝an}bracketle{tSz∝an}bracketri}ht∼e−2Γt. Thus the rate Γ is related to the Korringa\nspin relaxation rate as, EK,0≡2 Γ, which for the simple\nequilibrium case considered here takes on the usual\nexpression,\nEK,0= 2 Γ =πj2T . (46)\nHere the label ”0” indicates that this is just the leading\norder expression of the Korringa rate, and higher order\nterms in perturbation theory renormalize it [see Eq. (1)].\nTo compute current-current correlations, we ﬁrst no-\ntice that a scattering event qat some time τinduces\ncurrent pulses in the leads, Jσ\nr(t) = ∆Qσ\nr(q)δ(t−τ),\nwith ∆Qσ\nr(q)∈{±e,0}the amountofchargetransferred.\nSimilarly, a series of events, {τn,qn}generates a current,\nJσ\nr(t) =/summationdisplay\nn∆Qσ\nr(qn)δ(t−τn). (47)\nAs a consequence, the stationary current-current corre-\nlation function can be simply expressed as11\n∝an}bracketle{tJσ\nr(t)Jσ′\nr′(0)∝an}bracketri}ht=δ(t)/summationdisplay\nqP(q) ∆Qσ\nr(q)∆Qσ′\nr′(q)+/summationdisplay\nq,q′P(q,t;q′,0) ∆Qσ\nr(q) ∆Qσ′\nr′(q′). (48)\nHere the ﬁrst term describes the auto-correlation of indi-\nvidual scattering events, while the second term describes\ncorrelationsbetweendistincttunnelingevents. Theprob-\nabilityP(q) in Eq. (48) denotes the stationary rate for a\ngiven type of event, q={r′,σ′,S′←r,σ,S}, and can\nbe expressed as\nP(q) =γ(q)PS, (49)\nwithPSthe stationary probability distribution of the\ndot spin. The quantity P(q,t;q′,0) denotes the joint\nprobability rate of a scattering event qat timetand an\neventq′at timet= 0. For the events q={r2,σ2,S2←\nr1,σ1,S1}andq′={r′\n2,σ′\n2,S′\n2←r′\n1,σ′\n1,S′\n1}it can be\nexpressed as\nP(q,t;q′,0) =γ(q)PS1←S′\n2(t)γ(q′)PS′\n1,(50)\nwherePS1←S′\n2(t) denotesthe conditionalprobabilitythat\nthe dot spin evolves from state S′\n2toS1during time\nt. The functionPS1←S′\n2(t) is determined by the mas-\nter equation, Eq. (45), and it is obviously this and only\nthis quantity that generates time-dependent correlations\nbetween consecutive scattering events. Thus spin cur-\nrent correlationsin this perturbativemaster equation ap-\nproach are directly related to the time evolution of the\ndot spin.\nHaving set up this general framework, the detailed cal-\nculation of the classical noise spectrum is somewhat te-\ndious, but straightforward. Therefore, instead of pre-\nsenting further details on it, let us just continue with the\ndiscussion of the ﬁnal results. Within the master equa-\ntion approach, the left-right components ofthe spin noise\nread\nS↑↓\nLR(ω<T)≈ −e2\nhπ\n4EK,0ω2sin2φ\nω2+E2\nK,0, (51)\nS↑↑\nLR(ω<T)≈ −e2\nhπ\n8EK,0/parenleftbig\nω2+3E2\nK,0/parenrightbig\nsin2φ\nω2+E2\nK,0.\nFrom these equations we extract the following approxi-\nmations for the universal scaling functions,\nsME(ω) =−π2\n4j2T\nTKω2\nω2+E2\nK,0,(52)\n˜sME(ω) =3π2\n8j2T\nTK. (53)\nRemarkably, for T≫ω≫EK,0, these results precisely\ncoincide with the perturbative results, Eq. (44). This is\nindeed also clearly visible in Fig. 9, where we compare\nperturbation theory results for Sσσ′\nLR(ω) with results of0.01 0.1 1 1000.10.20.30.40.5\nPSfrag replacements\nω/T−Sσσ′\nLR/bracketleftBigTe2/h/bracketrightBig\nME PTEK/TS↑↓\nLR\nS↑↑\nLR\n|g| ∼ω2\nT21\nln2T\nTK\nT= 0, T≪TK\n|g|,˜g∼1\nln2/parenleftBig\nω\nTK/parenrightBig\n0.01 0.1 1 1000.10.20.30.40.5\nPSfrag replacements\nω/T−Sσσ′\nLR/bracketleftBigTe2/h/bracketrightBig\nMEPTEK/TS↑↓\nLR\nS↑↑\nLR\n|g| ∼ω2\nT21\nln2T\nTK\nT= 0, T≪TK\n|g|,˜g∼1\nln2/parenleftBig\nω\nTK/parenrightBig\nFIG. 9: (Color online) Upper panel: Frequency-dependent\nspin current cross-correlations ( S↑↓\nLR) and auto-correlations\n(S↑↑\nLR)intheabsenceofspinrelaxation, ascalculatedusingthe\nmaster equation approach (ME, solid line) and second-order\nperturbation theory (PT, dashed line). In the calculations we\nassumed j= 0.2. Lower panel: Eﬀect of a ﬁnite (rather large)\nexternal spin relaxation rate, 1 /τs≡EK,0/2.\nthe master equation approach. We remark that one can\nbridge these two approaches through a systematic but\nmuch more formal and diﬃcult quantum Langevin ap-\nproach, already brieﬂy sketched in Ref. [38], and to be\ndiscussed in a subsequent publication, Ref. [23].\nThe ﬂuctuation dissipation theorem, Eq. (27), in this\nclassical regime, ω<T, amounts in the following expres-\nsions,\nℜegME(ω)≈TK\n2TsME(ω) =−π2\n8j2ω2\nω2+E2\nK,0,(54)\nℜe˜gME(ω)≈TK\n2T˜sME(ω) =3π2\n16j2. (55)\nNotice that, in contrast to sandg, the functions ˜ sand\n˜gare completely featureless in this frequency range. On12\nthe other hand, in agreement with our earlier statement,\nS↑↓\nLR(ω),ℜe G↑↓\nLR(ω),sandℜe gall exhibit a dip be-\nlowEK, and scale to zero as ω→0. There is a sim-\nple heuristic picture behind this fact: a spin-ﬂip process\n⇑→⇓pumps spin-up electrons into the leads. However,\nit must necessarily be followed by a reverse spin-ﬂip pro-\ncess,⇓→⇑, where, on average, exactly the same amount\nof spin is pumped back as it has been pumped in before.\nThese processes exactly balance each other in the long\ntime limit, and lead to a vanishing cross-spin conduc-\ntance in equilibrium. We remark that if, however, there\nare external spin relaxation processes, then the spin may\nﬂip backspontaneouslybefore pumping backthe injected\nspin through the reversespin-ﬂip process. In this case, as\nwe shall see in Sec. VI, the conductance G↑↓\nLR(ω) remains\nﬁnite even in the ω→0 limit (see also Fig. 9).\nA rather curious consequence of the dip in sis that,\nwhileS↑↓\nLR(ω) develops a dipbelow the Korringarate, the\nnoise component S↑↑\nLR(ω) exhibits a peakof equal size,\nwhich precisely cancels the dip of −S↑↓\nLR(ω) in the charge\nnoise. This peak in S↑↑\nLR(ω) orthe similarpeak in S↑↑\nLL(ω)\nmay be more conveniently detected experimentally than\ncross-spin correlations.\nB. Beyond perturbation theory\n1. Logarithmic corrections\nIn the previous subsection we discussed only the\nleading order perturbative and master equation re-\nsults. Performing, however, perturbation theory in\njgives rise to logarithmic corrections. As long as\nmax{T,|ω|}≫TK, these corrections can be summed up\nusing renormalization group methods,32,33and amount\nin the replacement of jby it’s renormalized value, j→\n1/ln(max{T,|ω|}/TK). Apart from this substitution,\nhowever,theresultsofSubsectionsIVA1andIVA2con-\ntinue to be valid as long as T≫TK. Forω > T, e.g.,\nwe just recover the T= 0 temperature results, Eqs. (35)\nand (36), while in the opposite limit, ω<T, we obtain\ns(ω)≈ −π2\n41\nln2(T/TK)T\nTKω2\nω2+E2\nK(T),\n˜s(ω)≈3π2\n81\nln2(T/TK)T\nTK, (56)\nwithEK=EK(T) =πT/ln2(T/TK) the renormalized\nKorringa rate of Eq. (1). Similarly, for the scaling func-\ntionsgand ˜gwe obtain in this regime,\nℜeg(ω)≈TK\n2Ts(ω) =−π2\n81\nln2(T/TK)ω2\nω2+E2\nK(T),\nℜe˜g(ω)≈TK\n2T˜s(ω) =3π2\n161\nln2(T/TK). (57)\nFig. 10 gives a concise summary of these results.|s| ∼ω2\nTTKln2T\nTK\nω T EKT≫TK\n|s|,˜s∼T\nTK1\nln2T\nTK|s|,˜s∼ω\nTK1\nln2ω\nTK˜s,|s|\nωℜe˜g,|ℜe g|\nT|g|,˜g∼1\nln2ω\nTK\nEK|g|,˜g∼1\nln2T\nTK\n|g| ∼ω2\nT2ln2T\nTKT≫TK\nFIG. 10: (Color online) Sketch of the universal scaling func -\ntionss(continuous line), and ˜ s(dashed line), and the real\nparts ofg(continuous line) and ˜ g(ω/TK,T/TK) (dashed line)\nforT≫TK. The charge conductance and noise functions,\n˜gand ˜sshow no particular feature, while the spin scaling\nfunctions exhibit an anomaly below the Korringa rate, EK.\n2. Fermi liquid regime, T≪TK\nIn the Fermi liquid regime,25T≪TK, perturbation\ntheory injbreaks down. However, we can derive the\nbehavior of the scaling functions by two simple observa-\ntions. We ﬁrst observe that in this Fermi liquid regime,\nj→ ∞,25,30and therefore the only remaining energy\nscales areTandTK. Our second observation is that\nat the Fermi liquid ﬁxed point, the residual electron-\nelectron interactions are irrelevant,25and therefore phys-\nical quantities are analytical functions of ω. In partic-\nular, the asymptotic forms, Eqs. (37) and (38) remain\nvalid up to the higher order corrections even at ﬁnite\ntemperatures,\nℜeg(ω) =−αω2\nT2\nK+O(ω4,T2ω2),(58)\nℜe˜g(ω) = 1+O(ω2,T2). (59)\nThe scaling functions of the noise, sand ˜scan then yet\nagain be read out of the ﬂuctuation dissipation theorem,\nEq. (27), yielding\ns(ω≪TK)≈ −αω3\nT3\nKcoth/parenleftBigω\n2T/parenrightBig\n,(60)\n˜s(ω≪TK)≈ω\nTKcoth/parenleftBigω\n2T/parenrightBig\n. (61)13\nT≪TK\nlnω\nTK|s| ∼|ω|3\nTK|s|,˜s∼ω\nTK1\nln2(ω/TK)\n|s| ∼|ω|2T\nT2\nK˜s∼ω\nTK\nlnT\nTK0˜s∼T\nTKln(|s|), ln(˜s)\nT≪TK\nlnω\nTKlnT\nTK 0|ℜe g|,ℜe˜g\n˜g≈1\n˜g,|g| ∼1\nln2ω\nTK\n|g| ∼ω2\nT2\nK\nFIG. 11: (Color online) Sketch of the universal scaling func -\ntionss(continuousline) and ˜ s(dashed line) and the real parts\nofg(continuous line) and ˜ g(dashed line) for T≪TK.\nThese equations reduce to the T= 0 expressions in the\nω≫Tlimit, and are valid as long as ω < T K. Notice\nthat in the ω→0 limit, the charge noise component\n˜sscales to a constant, ∝T, while the spin component\nsscales quadratically to zero, just as in the T≫TK\nregime. Of course, for ω > T Kthe scaling functions\nmust also reduce to their T= 0 temperature expres-\nsions, Eqs. (35) and (36). The overall behavior of these\nfunctions for T≪TKis sketched in Fig. 11\nV. TRANSIENT RESPONSE\nLet us now turn to the discussion of real time tran-\nsient response, i.e., the time dependent current response\nin the left lead when a spin-dependent voltage of the\nformV↑\nR(t) =δV↑\nRθ(t) is applied to the right electrode\n(see Fig. 1). Within linear response theory, the average\ncurrent pulse is just given by\n∝an}bracketle{tJσ\nL(t)∝an}bracketri}ht=i/integraldisplay∞\n−∞Gσ↑\nLR(ω)1\nω−iδe−iωtdωδV↑\nR,(62)\nwithi/(ω−iδ) the Fourier transform of the θfunction.\nSurprisingly, just using Eq. (62) and the analytical\nproperties of the functions Gσ↑\nLR(ω), we are able to\nmake rather strong statements on the transient response,\n∝an}bracketle{tJσ\nL(t)∝an}bracketri}ht. Letusstartbybrieﬂysummarizingtheseanalyt-\nical properties. First ofall, being retarded responsefunc-\ntions,Gσσ′\nLR(ω), are analytical on the upper half plane.\nMoreover, as assured by Fermi liquid theory, they areBranch cut- i∆∞\nC0\nCImω\nReω −∞\nBranch cutPole\nReωImω\n- i ∆\nC- iδ\nFIG. 12: (Color online). Pole structure of the integrand in\nEq. (62), for the spin-up – spin-down (upper panel) and spin-\nup – spin-up (lower panel) channels. In the spin- ↑↓channel,\nthe pole at ω= 0 is canceled by the ω2dependence of g(ω,T),\nwhile in the case of spin- ↑↑this pole survives and gives aﬁnite\nresponse as t→ ∞.\nalso analytical in an extended region around ω= 0 at\nany temperature. However, from perturbation theory we\nknow that at very large frequencies, ω≫TK,T, they\nhave logarithmic tails, |Gσσ′\nLR(ω)| ∼1/ln2(ω/TK) and\nthus tend to zero even in the universal scaling limit,\nD→ ∞,TKﬁnite. Their asymptotic behavior and\ntheir symmetries [ ℜe Gσσ′\nrr′(ω) =ℜe Gσσ′\nrr′(−ω) while\nℑmGσσ′\nrr′(ω) =−ℑmGσσ′\nrr′(−ω)] imply the presence of a\ncutalong the negative imaginary axis with an endpoint,\n−i∆, with ∆∝max{T,TK}(see Fig. 12).53Further-\nmore, as already discussed, G↑↓\nrr′(ω= 0) = 0, while the\ncomponents G↑↑\nrr′(ω= 0) =Grr′(ω= 0)/2 remain ﬁnite.\nLet us now discuss the properties of the response,\nEq. (62). First, we notice that due to the asymptotic\n1/ln2(ω/TK) fall-oﬀ of Gσσ′\nLR(ω) and the analyticity on\nthe upper half-plane, the integral contours in Eq. (62)\ncan be closed upwards for any time t≤0. Therefore,\n∝an}bracketle{tJσ\nL(t)∝an}bracketri}ht= 0 fort≤0, i.e., it respects causality. The14\nresponse being zero even at t= 0 is not entirely trivial:\nin the master equation approach, e.g., G↑↑\nrr′(ω) remains\nﬁnite in the ω→∞limit, and one obtains an unphysi-\ncal jump at t= 0. We notice that the statement on the\nt= 0 response being zero is equivalent to the Kramers-\nKronig relation. Though the response ∝an}bracketle{tJσ\nL(t)∝an}bracketri}htvanishes\nat timet= 0 and is continuous for times t≥0, theslope\nof the response,d\ndt∝an}bracketle{tJσ\nL(t)∝an}bracketri}ht|t=0is, however, inﬁnite, since\nthe integral/integraltext∞\n−∞dωG↑↑\nrr′(ω) logarithmically diverges.\nFor timest >0, the contours must be closed down-\nwards, as shown in Fig. 12. In the spin-up – spin-down\nchannel,G↑↓\nLR(ω= 0) = 0, and therefore the pole at −iδ\ndoes not give any contribution. The contribution of the\ncut to the spin up-down response can be written as\n∝an}bracketle{tJ↓\nL(t)∝an}bracketri}ht\nδV↑\nR=−/integraldisplay\nCG↑↓\nLR(z)1\nze−iztdz (63)\n=e2\nhsin2φe−∆t/integraldisplay∞\n0δg(∆+y)e−ytdy\ny,\nwithδg(x) = 2ℑm g(−ix+δ) the cut of the universal\nconductance function, g(ω). Clearly, the contribution of\nthe cut falls oﬀ as ∼e−∆tfor long times. At T= 0 tem-\nperature we have ∆ ∝TK, and furthermore δgmust be\nalso a universal function, δg(∆+y) =δg(y/TK). There-\nfore, the response is a universal function of tTK. We\ncan also tell the short time asymptotics of the response.\nMaking use of the fact that the response is continuous at\nt= 0, we obtain for t≪1/∆ the expression,\n∝an}bracketle{tJ↓\nL(t)∝an}bracketri}ht∼e2\nhsin2φ/integraldisplay∞\n0δg(∆+y)(e−yt−1)dy\ny.(64)\nSince the cut scales for large energies as ∼1/ln3(y/TK),\nwe get,\n∝an}bracketle{tJ↓\nL(t≪1/∆)∝an}bracketri}ht∼δV↑\nR\nln21\ntTK. (65)\nRemarkably, this result does not depend on the temper-\nature, since it is determined only by the high frequency\npart of the conductance. Furthermore, since the length\nof the current pulse is determined by the exponential\nprefactor,∼e−∆t, we can also read out of Eq. (65) its\nheight: for T≪TKone has ∆∼TK, and the height of\nthe pulse is∝an}bracketle{tJ↓\nL(t)∝an}bracketri}ht∼δV↑\nR. ForT≫TK, on the other\nhand, we have ∆ ∼T, and the height of the current\npulse, is∼δV↑\nR/ln2T\nTK.\nFigure 13 summarizes all the characteristic features of\nthe current response ∝an}bracketle{tJ↓\nL(t)∝an}bracketri}ht, discussed above. The total\nchargepumped into spin-down channel ofleft lead is sim-\nply given by the integral of the transient response, and\nis approximately\n∆Q↓\nL∼e2\nhsin2φ/braceleftBigg\n1\nTln2(T/TK),ifT≪TK,\n1\nTK, ifT≫TK.(66)\n∝angbracketleftJ↓\nL(t)∝angbracketright/δV↑\nR[e2\nh]\nt∼h/TT≫TK ∼e−t∆T= 0O(1)\n1\nln2t TK\n∼h/TKO/parenleftbigg\n1/ln2T\nTK/parenrightbigg∝angbracketleftJ↑\nL(t)∝angbracketright/δV↑\nR[e2\nh]\ntT= 0\nT≫TK1\nln2t TK∼1/ln2T\nTK\n∼h/T ∼h/TKO(1)\nFIG. 13: (Color online). Transient current response in the\nspin-down (upper panel) and spin-up (lower panel) channels ,\nupon a constant bias applied at t = 0 in the spin-up channel.\nRemarkably, the coeﬃcients appearing in this expression\nare just the high-temperature and low temperature ex-\npressions of the spin susceptibility.39\nThe analysis of the response of the spin-up carriers fol-\nlows very similar lines. The only major diﬀerence is that\ninthiscasethepoleat −iδgivesaﬁnitetimeindependent\ncontribution, and leads to an asymptotic response,\n∝an}bracketle{tJ↑\nL(t→∞)∝an}bracketri}ht=G↑↑\nLR(ω= 0)δV↑\nR.(67)\nOtherwise, our discussions on the universal form of the\nresponse, and its short time 1 /ln2(t) singularity carry\nover to this case, too. Instead of giving further details\non∝an}bracketle{tJ↑\nL(t→∞)∝an}bracketri}ht, we just summarized its properties in\nFig. 13.\nVI. SPIN RELAXATION EFFECTS\nAll results presented so far were obtained under the\nassumption that spin relaxation is generated by the ex-\nchange coupling j, and there are no external sources of\nspin relaxation. In reality, however, external spin relax-\nation channels are always present. In quantum dots, the\ndominant channel of (external) spin relaxation is usually\ndue to hyperﬁne interaction between the conﬁned elec-\ntron and nuclear spins in the host material, leading typ-\nically to a dephasing time of the order of τs∼10 ns or\nlonger in the absence of magnetic ﬁeld.40,41These hyper-\nﬁne relaxation processes are thus characterized by an en-\nergyscaleh/τs∼1−10 mK, typically much smaller than15\n∼1\nτs1\nln2/parenleftbigg\n1\nτsTK/parenrightbiggT≪TK, ω= 0\n1\nτsS↑↓\nLR∼T/parenleftBigg\n1\nτsTK/parenrightBigg2\nT T K\nEK1\nτsT≫TK, ω= 0\n∼T\nln2/parenleftbigg\n1\nτsTK/parenrightbigg∼T\nln2/parenleftbigg\nT\nTK/parenrightbiggS↑↓\nLR\n∼1\nτs+/parenleftbigg1\nτs/parenrightbigg21\nTln2/parenleftbiggT\nTK/parenrightbigg\nT\nFIG. 14: (Color online) Shot noise S↑↓\nLR(ω= 0,T) as a func-\ntion of the spin relaxation rate1\nτsin the Fermi liquid regime\n(T≪TK) (upper panel) and in the perturbative regime\n(T≫Tk) (lower panel).\nthe temperature. Coupling to piezoelectric phonons42or\nelectromagnetic ﬂuctuations43through spin-orbit inter-\naction or polaron dephasing processes due to coherent\nacoustic phonon generation are, in general, characterized\nby even longer dephasing times and smaller relaxation\nrates.44Therefore, for typical experimental parameters,\nwe would naively expect 1 /τsto be small compared to\nthe temperature, T. Nevertheless, a ﬁnite τsleads to\nqualitatively diﬀerent results, since its presence lifts the\nconstraint of spin conservation, and allows to have a ﬁ-\nnite d.c. spin cross-conductance, G↑↓\nLR(ω= 0)∝ne}ationslash= 0.\nTo investigate the eﬀect of a ﬁnite τs, let us consider\nthe perturbative ( T≫TK) and Fermi liquid ( T≪TK)\nregimes separately. In the regime T≫TK, we can read-\nily extend our master equation analysis to include 1 /τs\nand obtain,\nS↑↓\nLR(ω<T)≃−e2\nhπ\n4EK/parenleftBig\nω2+1\nτ2s+EK\nτs/parenrightBig\nsin2φ\nω2+EK+1\nτ2\ns.\nHere we incorporated already logarithmic corrections by\nreplacing the bare Korringa rate, EK,0by its renormal-\nized value, given by Eq. (1). Clearly, at large frequencies,\nω≫EK, external spin relaxation does not play a role.\nHowever, for ω < E K, it suppresses the dip in S↑↓\nLR(ω),\nand leads to a ﬁnite cross-spin ”shot noise”,\nS↑↓\nLR(ω= 0)≈ −e2\nhsin2φπ\n4EK1\nτs\nEK+1\nτs\n≈ −sin2φe2\nhπ\n4min{EK,1\nτs}.(68)In other words, in the most relevantcase, 1 /τs<EK, the\ncross-spin current noise is simply proportional to 1 /τs,\nas also found by Kindermann.20The full frequency spec-\ntrum of the noise in the perturbative regime ( T≫TK)\nis presented in Fig. 9, lower panel.\nIn the Fermi liquid regime, T≪TK, ﬁnite external\nspin relaxation also leads to a ﬁnite cross-spin noise. To\nestimate it, we ﬁrst notice that 1 /τsjust introduces a\nnew frequency scale, and therefore we expect\nG↑↓\nLR(ω→0)∼1\nτ2sT2\nK. (69)\nNotice that Tdoes not appear in this equation. Corre-\nspondingly, the noise behaves for T <T Kas\nS↑↓\nLR(ω→0)∼T\nτ2sT2\nK. (70)\nThe overalldependence ofthe cross-spinshot noisesig-\nnal on the spin relaxation rate is sketched in Fig. 14.\nThere we also display the physically not too relevant\nregime, 1/τs> T, where 1/τsbecomes the dominant\nenergy scale, and therefore we have j→1/ln(1/τsTK).\nVII. CONCLUSIONS\nIn the present paper we studied the equilibrium spin\ncurrent noise and spin conductance through a quantum\ndot in its Kondo (local moment) regime. We have shown\nthat in the absence of external ﬁelds, they are both char-\nacterizedbyapairofuniversalfunctions, anddetermined\nthe properties of these functions. We have shown that –\nin contrast to the charge conductance ( GLR) – the d.c.\nspin cross-conductance ( G↑↓\nLR) vanishes. Put in another\nway, there is no spin drag, and a spin-up current cannot\ngenerate a steady spin-down current, at least not within\nlinear response. At T= 0 temperature this obviously\nfollows from Fermi liquid properties, and is thus valid for\nanyinteracting system with no spin-orbit coupling and\nwith a Fermi liquid ground state. However, somewhat\nsurprisingly, though it is not true for any interacting sys-\ntem, for a quantum dot, this property also carries over\nfor ﬁnite temperatures. It is related to the simple struc-\nture of the Kondo (or the underlying Anderson) mod-\nels, where spin transfer between spin-up and spin-down\nstates can occur only through a single point, namely the\ndot state (or the dot spin in the Kondo model). There-\nfore,thespincurrentsgeneratedbyconsecutive ⇑→⇓and\n⇓→⇑ﬂips ofthe dot spin precisely cancel each-other,and\nno d.c. cross-spin currents appear. Correspondingly, the\ncross-spin shot noise, S↑↓\nLR(ω= 0,T) also vanishes at any\ntemperature, and the noise spectra, S↑↓\nLR(ω) andS↑↑\nLR(ω)\nboth exhibit related low frequency anomalies.\nExternal spin relaxation slightly changes the picture\nabove. It partly removes the correlations between con-\nsecutive spin-ﬂip processes, and makes G↑↓\nLR(ω= 0) and16\nFIG. 15: (Color online) Correspondence between a single\nquantum dot device and a spinless double dot device.\nS↑↓\nLR(ω= 0) ﬁnite. However, since the external spin-ﬂip\nrate, 1/τsis typically much smallerthan the other energy\nscales (T,TK,EK), it only leads to small changes in the\noverall behavior of the noise and conductance functions.\nAs we also demonstrated in detail, simple-minded per-\nturbation theory accounts only for the structure of in-\ndividual coherent processes, and fails badly to capture\nthese correlations between consecutive processes, which\nhappen to dominate the spin response at small frequen-\ncies. Therefore, one must be very careful when calcu-\nlating spin transport properties. Even in the pertur-\nbative regime, T≫TK, simple-minded perturbation\ntheory is valid only for frequencies above the Korringa\nrate,ω > E K. To capture the physics at frequencies\nω < E K, a supplementary master equation approach\n(valid forω < T) can be employed. Alternatively, one\ncan use a more systematic but also more technical quan-\ntum Langevin approach, which works for any frequency\nin the perturbative regime, T≫TK, but neglects loga-\nrithmic corrections (see Ref. [23,45]).\nAlthough ﬁnite-frequency noise measurements are nowavailable,12–17andspinpolarizedcurrentscanalsobe rel-\natively easily produced,6,46measuring the low-frequency\nanomalies of spin cross-correlations, S↑↓\nLR(ω), seems to be\na diﬃcult task. However, the predicted low frequency\nanomalies are also present in the spin polarized conduc-\ntance,G↑↑\nLR(ω), and noise, S↑↑\nLR(ω) (see Figs. 3 and 5).\nThese are experimentally much more easily accessible,\nsince carriers must be polarized in the same direction.\nAlternatively, one can measure these cross-correlations\nin thecharge sector , by using capacitively coupled dou-\nble dots (see Fig. 15).47,48In the spin polarized case, the\nHamiltonian of the double dot system maps to that of\nthe Anderson model with anisotropic hybridization pa-\nrameters. Measuring noise or conductance between leads\nattached to the upper or lower leads of the double dot\ndevice shown in Fig. 15 is thus equivalent to cross-spin\nmeasurements in the single quantum dot setup.\nAcknowledgments\nThis research has been supported by Hungarian Sci-\nentiﬁc Research Funds Nos. 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Rev.\nLett.98, 056801 (2007).\n49Although in this paper we use units of /planckover2pi1=kB= 1, in\ncertain formulas we shall restore and display the Planck\nconstant to clarify physical dimensions.\n50The density of states is incorporated in the ﬁelds, crσ(ε),\nto obtain the normalization in the main text.\n51The couplings vLandvRcan be taken to be real.\n52In NRG calculations we kept 1024 states at each iteration\nand assumed Λ = 1 .8 andj= 0.18.\n53We remark that while perturbation theory indeed repro-\nduces this cut, the master equation approach fails to do\nthat, and only produces a pole at −iEK."    },    {        "title": "0705.3174v1.Spin_dynamics_in_rolled_up_two_dimensional_electron_gases.pdf",        "content": "arXiv:0705.3174v1  [cond-mat.mes-hall]  22 May 2007Spin dynamics in rolled-up two dimensional electron gases\nMaxim Trushin and John Schliemann\nInstitute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany\n(Dated: May 2007)\nA curved two dimensional electron gas with spin-orbit inter actions due to the radial conﬁnement\nasymmetry is considered. At certain relation between the sp in-orbit coupling strength and curvature\nradius the tangential component of the electron spin become s a conserved quantity for anyspin-\nindependent scattering potential that leads to a number of i nteresting eﬀects such as persistent spin\nhelix and strong anisotropy of spin relaxation times. The eﬀ ect proposed can be utilized in the\nnon-ballistic spin-ﬁeld-eﬀect transistors.\nI. INTRODUCTION\nSpin-orbit coupling is one of the key ingredients for\nelectrical control and manipulation spins in semiconduc-\ntor nanostructures and therefore a major issue of both\nexperimental and theoretical studies in semiconductor\nspintronics. A paradigmatic example for a spintronics\ndevice is the spin ﬁeld-eﬀect transistor(SFET) proposed\nby Datta and Das over ﬁfteen years ago1. The origi-\nnal proposal envisaged a two-dimensional electron gas\n(2DEG) in a semiconductor quantum well with Rashba\nspin-orbit coupling2,3. This contribution to spin-orbit in-\nteraction stems from an asymmetry of the conﬁning po-\ntential in the growth direction and can be particularly\npronounced for material such as InAs. Most noteworthy,\nthe strength of the Rashba term can be tuned in experi-\nment via a gate voltage across the quantum well4,5,6,7,8.\nThis is in contrast to the Dresselhaus coupling, another\neﬀective contribution to spin-orbit interaction in 2DEGs\nresulting from the lack of inversion symmetry in zinc-\nblende III-V semiconductors9. In particular, for typical\nparameters of realistic materials it is in principle possi-\nble to tune the Rashbacoupling to be equalin magnitude\nto the Dresselhaus coupling10. In this situation an ad-\nditional conserved spin quantity arises which opens the\nperspective to possibly operate an SFET also in the dif-\nfusive regime11, apart from other interesting eﬀects such\nas persistent spin helix12and strong anisotropy of spin\nrelaxation times13. In the present paper we investigate\na similar interplay between the Rashba coupling and the\neﬀects of a ﬁnite curvature of a cylindrical 2DEG. Such\ncurved systems have been produced recently by several\nindependent groups14,15,16,17and studied regarding their\nmagnetotransport properties17,18,19,20. Our theoretical\nresults obtained here predict the existence of a conserved\nspin component for appropriately tuned system param-\neters, very analogously to the balancing of Rashba and\nDresselhaus coupling in a ﬂat 2DEG. Moreover, within\nthe framework of second quantization, this observation\ncanbeextended toafull su(2)algebraofconservedquan-\ntities, in full analogyto recent ﬁndings for a ﬂat 2DEG12.\nFinally, we also discuss our results with respect to the\nzitterbewegung of electrons due to spin-orbit coupling in\ntwo-dimensional semiconductor structures21.kt\nkzkt\nkzkzkt\nradial asymmetric\nconfinement2DEG\nU(r)(b) (a)\n(c) (d)y\nxz\nR\nFIG. 1: (a) The system under consideration: A rolled-up\n2DEG with spin-orbit coupling induced by the asymmetric\nradial conﬁnement U(r). (b) In general case, Fermi contours\nofthe rolled-up2DEG are anisotropic. Here, kzandktare the\nlongitudinal and tangential components of the electron mo-\nmenta respectively. The arrows show the spin orientation in\ntheeigenstates(7)–(8). (c)Intheplanarcase R≫/planckover2pi12/(2m∗α)\nthe Fermi contours represent just two concentric circles, i . e.\nthe dispersion law is isotropic. Here, the spin orientation de-\npends on the momentum. (d) At α=−/planckover2pi12/(2m∗R) the Fermi\ncontours are two circles as well. Here, the spin orientation\ndoes not depend on the momentum within a spin-split sub-\nband, i. e. the tangential component of the electron spin is\nconserved.\nII. RESULTS AND DISCUSSION\nLet us consider the Hamiltonian describing electrons\nin a rolled-up layer of radius Rdepicted in Fig. Ia. Fol-\nlowing Rashba2,3, we rely on the eﬀective mass model,\nand, hence, the Hamiltonian reads\nH=Hkin+HSO+V(z,ϕ)+U(R),(1)2\nwhereU(R) is the radial conﬁning potential U(r) atr=\nR,V(z,ϕ) is the arbitrary scalar potential describing,\nfor example, the inﬂuence of impurities or imperfections.\nThe spin-orbit coupling term has the form22\nHSO=α(σϕkz−σzqϕ/R), (2)\nwherekz=−i∂\n∂zandqϕ=−i∂\n∂ϕare the longitudi-\nnal and angular momentum operators respectively, σϕ=\n−σxsinϕ+σycosϕ,σzare the corresponding Pauli ma-\ntrices, and αis the spin-orbit coupling constant. The\nkinetic term reads\nHkin=/planckover2pi12k2\nz\n2m∗+ε0q2\nϕ, (3)\nwhereε0=/planckover2pi12/(2m∗R2) is the size conﬁnement energy,\nandm∗is the eﬀective electron mass.\nThe spin dynamics can be described utilizing the com-\nmutation relations between the spin projection operators\nsz=1\n2σz,sr=1\n2(σxcosϕ+σysinϕ),sϕ=1\n2(−σxsinϕ+\nσycosϕ) and the Hamiltonian (1). The corresponding\nequations read\ndsz\ndt=−α\n/planckover2pi1kz/parenleftbigg\n0 e−iϕ\neiϕ0/parenrightbigg\n, (4)\ndsr/dt=\ni\n/planckover2pi1/parenleftbigg\n−iαkz e−iϕ/parenleftbig\nε0+α\nR/parenrightbig/parenleftbig1\n2−qϕ/parenrightbig\neiϕ/parenleftbig\nε0+α\nR/parenrightbig/parenleftbig1\n2+qϕ/parenrightbig\niαkz/parenrightbigg\n,\n(5)\ndsϕ\ndt=ε0+α/R\n/planckover2pi1/parenleftbigg\n0 e−iϕ/parenleftbig1\n2−qϕ/parenrightbig\n−eiϕ/parenleftbig1\n2+qϕ/parenrightbig\n0/parenrightbigg\n.\n(6)\nNote, that the left-hand sides of Eqs. (4)–(6) are noth-\ning else than the correspondingspin precessionfrequency\noperators.\nLet us have a look at the special case α=−ε0R. Here\nEq. (II) becomes diagonal, whereas the right hand side\nof Eq. (6) vanishes. The latter means that the tangential\nspinsϕdoes not precess at all, i. e. sϕis the conserved\nquantity for arbitrary potentialV(z,ϕ). It is important\nto emphasize here, that in the planar case with Rashba\ncoupling none of all the possible spin projections is con-\nserved.\nThe eﬀect hasthe followinggeometricalinterpretation.\nOn the one hand the spin rotation angle in the 2DEG\nwithRashbaspin-orbitcouplingdependsexplicitlyonthe\nlengthLofthe electronpath, namely∆ θso= 2m∗αL//planckover2pi12.\nOn the other hand the spin rotation angle of an elec-\ntron moving adiabatically along the arc of radius R\nis ∆θg=L/R. Here, the index gmeans “geometri-\ncal”. Now one can see easily that in the special case\n1/R=−2m∗α//planckover2pi12the spin rotation angle of geometrical\norigin ∆θg=−2m∗αL//planckover2pi12completely compensates thespin rotation angle ∆ θsowhich is due to the spin-orbit\ncoupling alone.\nThe phenomena found here is similar to what is pro-\nposedbySchliemannetal.11fortheplanar2DEGinpres-\nence ofbothRashba and Dresselhaus interactions. The\ninterplay between them can lead to the conservation of\nthe spin quantity Σ =1√\n2(σx±σy), that might be uti-\nlized in non-ballistic SFETs. In contrast to Ref.11, for us\nit is enough that the spin-orbit coupling stems from the\nasymmetry of the conﬁnement U(r) only, and the bulk\nspin-orbit eﬀects are not necessary. Nevertheless, all the\nproposals regarding the non-ballistic SFET11are valid\nfor the device studied here as well.\nTo show that we consider the Hamiltonian (1) at\nV(z,ϕ)+U(R) = 0. Then, its eigenstates are\nψ+=/parenleftbigg\nisinγe−iϕ/2\ncosγeiϕ/2/parenrightbigg\nei(kzz+lϕϕ), (7)\nψ−=/parenleftbigg\ncosγe−iϕ/2\nisinγeiϕ/2/parenrightbigg\nei(kzz+lϕϕ), (8)\nwhere tan2 γ=−αkz/[(ε0+α/R)lϕ], and the spectrum\nreads\nE±=/planckover2pi12k2\nz\n2m∗+ε0l2\nϕ+ε0\n4+α\n2R±/radicalbigg\nα2k2z+/parenleftBig\nε0+α\nR/parenrightBig2\nl2ϕ.\n(9)\nNote, that the expectation values of sz,sϕcalculated for\ntheeigenstates(7)–(8) are, ingeneral,momentum depen-\ndent (see Fig. Ib,c). Therefore, the electron spin becomes\nrandomized due to the momentum scattering, and any\ngiven spin-polarization of the electron beam vanishes at\nthelengthsoftheorderoftheelectronmeanfreepath. At\nα=−ε0Rthe tangential spin-polarization remains un-\nchangedforanyspin-independentscattering(seeFig.Id).\nThus, assuming two spin-polarized contacts at the ends\nof the rolled-up 2DEG one can modulate the electric cur-\nrent viaRashba constant αasdiscussed in Ref.11in great\ndetails.\nAs an important property of the system studied here,\nthespinorsinEqs.(7),(8)dependexplicitlyonthespatial\ncoordinateϕ, i.e. spin and orbital degrees of freedom are\nentangled. This observation corresponds to the fact that\ntangential momentum operator qϕ/Rdoes not commute\nwiththespinoperators sϕandsr, diﬀerentlyfromthesit-\nuation in a planar 2DEG with spin orbit coupling of, e.g.\nRashba and Dresselhaus type. This property of rolled-up\n2DEGs has essentially geometrical origin since, generally\nspeaking, an electron spin moving along a path with ﬁ-\nnite curvature Rchanges its direction depending on the\nadiabaticity parameter 2 αm∗R//planckover2pi12, see Ref.23. However,\nthe expectation values of sϕ,sz, andsrwithin the eigen-\nstates (7),(8) are independent of the angle coordinate ϕ.\nAnother promising application of the eﬀect proposed\nis the observation of the persistent spin helix studied re-\ncently in Ref.12. In fact, the exact su(2) symmetry nec-\nessary for the persistent spin helix can be found not only3\nin ﬂat 2DEGs with both Rashba and Dresselhaus inter-\nactions but in rolled-up 2DEGs with Rashba interaction\nalone. Indeed, the exact su(2) symmetry is generated by\nthe following operators\nS+=/summationdisplay\nkz,lϕc†\nkz+kR,lϕ,−ckz−kR,lϕ,+,(10)\nS−=/summationdisplay\nkz,lϕc†\nkz−kR,lϕ,+ckz+kR,lϕ,−,(11)\nSz=/summationdisplay\nkz,lϕ/parenleftBig\nc†\nkz,lϕ,−ckz,lϕ,−−c†\nkz,lϕ,+ckz,lϕ,+/parenrightBig\n,(12)\nwhereckz,lϕ,sare the annihilation operators of the par-\nticles with the spin-index s∈ {±}, andkR=m∗α//planckover2pi12.\nThese operators and Hamiltonian written as\nH=/summationdisplay\nkz,lϕ,sEs(kz,lϕ)c†\nkz,lϕ,sckz,lϕ,s(13)\nobey the following commutation relations\n/bracketleftbig\nH,S±/bracketrightbig\n= 0,[H,Sz] = 0, (14)\n/bracketleftbig\nS+,S−/bracketrightbig\n= 2Sz,/bracketleftbig\nSz,S±/bracketrightbig\n=±S±.(15)\nThus, the operators S±andSzcommutewith the Hamil-\ntonianandformarepresentationofsu(2), andallﬁndings\nof Ref.12are valid for our system as well.\nLet us ﬁnally make some remarks regarding the zit-\nterbewegung of electrons in rolled-up 2DEGs. Just as in\nthe classic case of free relativistic electrons described by\nthe Dirac equation, this phenomenon is nothing else but\na beating between diﬀerent dispersion branches split in\nenergy21. To investigate the zitterbewegung of electrons\nin rolled-up 2DEGs we ﬁnd the components of the time\ndependent position operator in the Heisenberg picture\nwhich read\nzH(t) =z(0)+[z,R]+1\n2[[z,R],R]+1\n6[[[z,R],R],R]+...\n(16)\nϕH(t) =ϕ(0)+[ϕ,R]+1\n2[[ϕ,R],R]+1\n6[[[ϕ,R],R],R]+...\n(17)\nwhereR=−iHt//planckover2pi1. In the particular case ε0=−α/R\nneither of the position operator components contains os-\ncillating terms, and the zitterbewegung is absent, simi-\nlarly to the case of a ﬂat 2DEG with balanced Rashba\nand Dresselhaus spin-orbit coupling24.The key problem regarding the present proposal is the\nexperimental realization of the rolled-up 2DEGs fulﬁll-\ning the required relation between Randα. In the Table\nI, we present the values for Rashba constant which are\nnecessary for the realization of the non-ballistic SFET\nTABLE I: Critical Rashba constants α=−/planckover2pi12/(2m∗R) for\nsome rolled-up structures reported in the literature.\nQuantum well, Refs Rm∗/mα=−ε0R(eVm)\nAlGaAs/GaAs/AlGaAs178µm0.067 6·10−14\nAlGaAs/GaAs/AlGaAs184µm0.0671.2·10−13\nSiGe/Si/SiGe16,25270nm 0.19 6·10−13\nTABLE II: Critical curvature radii R=−/planckover2pi12/(2m∗α) accord-\ning to the Rashba parameters of some ﬂat 2DEGs reported in\nthe literature.\nQuantum well, Refs α(eVm) m∗/m|R|\nInAlAs/InGaAs/InAlAs47.2·10−120.0583nm\nInP/InGaAs/InP65.3·10−120.041150nm\nSiGe/Si/SiGe255.5·10−150.1933µm\nproposed. In the Table II, the curvature radius is cal-\nculated for a given α. Here, the Rashba constant is as-\nsumed to be the same asforthe planarcase. This is quite\nroughassumptionsince the spin-orbitinteractionscan be\nchanged because of the additional strain at the bending.\nTherefore, the Rashbaconstant should be remeasuredfor\nrolled-up 2DEGs even if its value for the planar case is\nalready known.\nIII. CONCLUSIONS\nIn conclusion, we have investigated the spin dynam-\nics in rolled-up 2DEGs with interactions of Rashba type\nusing the Hamiltonian which includes an arbitrary scat-\ntering potential as well. We have found, that at certain\nrelation between the Rashba constant and radius of cur-\nvature the tangential spin is conserved. This is the most\nstriking feature of the rolled-up 2DEG as compared with\nits planar analogue. Apart from its fundamental impor-\ntance, the eﬀect proposed can be utilized in non-ballistic\nSFETs. In addition, su(2) spin rotation symmetry and\nzitterbewegung have been investigated.\nWe acknowledge ﬁnancial support from Collaborative\nResearch Center 689.\n1S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).2E. Rashba, Sov. Phys. Solid State 2, 1109 (1960).4\n3Y. Bychkov and E. Rashba, JEPT Lett. 39, 78 (1984).\n4J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n5C.-M. Hu, J. Nitta, T. Akazaki, H. Takayanagi, P. Pfeﬀer,\nand W. Zawadzki, Phys. Rev. B 60, 7736 (1999).\n6G. Engels, J. Lange, T. Sch¨ apers, and H. L¨ uth, Phys. Rev.\nB55, 1958 (1997).\n7T. Matsuyama, R. K¨ ursten, C. Meissner, and U. Merkt,\nPhys. Rev. B 61, 15588 (2000).\n8D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).\n9G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n10S. Giglberger, L. E. Golub, V. V. Bel’kov, S. N. Danilov,\nD. Schuh, C. Gerl, F. Rohlﬁng, J. Stahl, W. Wegscheider,\nD. Weiss, et al., Phys. Rev. B 75, 35327 (2007).\n11J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett.\n90, 146801 (2003).\n12B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev.\nLett.97, 236601 (2006).\n13N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582\n(1999).\n14V. Y. Prinz, V. A. Seleznev, A. K. Gutakovsky, A. V.\nChehovskiy, V. V. Preobrazhenskii, M. A. Putyato, and\nT. A. Gavrilova, Physica E 6, 828 (2000).15O. G. Schmidt and K. Eberl, Nature 410, 168 (2001).\n16O. G. Schmidt and N. Y. Jin-Phillipp, Appl. Phys. Lett\n78, 3310 (2001).\n17S. Mendach, O. Schumacher, C. Heyn, S. Schn¨ ull,\nH. Welsch, and W. Hansen, Physica E 23, 274 (2004).\n18A. B. Vorob’ev, V. Y. Prinz, Y. S. Yukecheva, and A. I.\nToropov, Physica E 23, 171 (2004).\n19N. Shaji, H. Qin, R. H. Blick, L. J. Klein, C. Deneke, and\nO. G. Schmidt, Appl. Phys. Lett. 90, 42101 (2007).\n20K.-J. Friedland, R. Hey, H. Kostial, A. Riedel, and K. H.\nPloog, Phys. Rev. B 75, 45347 (2007).\n21J. Schliemann, D. Loss, and R. M. Westervelt, Phys. Rev.\nLett.94, 206801 (2005).\n22L. I. Mararill, D. A. Romanov, and A. V. Chaplik, JETP\n86, 771 (1998).\n23M. Trushin and A. Chudnovskiy, JETP Lett. 83, 318\n(2006).\n24J. Schliemann, D. Loss, and R. M. Westervelt, Phys. Rev.\nB73, 085323 (2006).\n25Z. Wilamowskii, W. Jantsch, H. Malissa, and U. R¨ ossler,\nPhys. Rev. B 66, 195315 (2002)."    },    {        "title": "0912.1676v1.Microscopic_Theory_of_Current_Spin_Interaction_in_Ferromagnets.pdf",        "content": "arXiv:0912.1676v1  [cond-mat.mes-hall]  9 Dec 2009October 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\nMICROSCOPIC THEORY OF CURRENT-SPIN INTERACTION\nIN FERROMAGNETS\nH. KOHNO∗, S. KAWABATA, T. NOGUCHI, S. UETA\nDepartment of Materials Engineering Science, Graduate Sch ool of Engineering Science,\nOsaka University, Toyonaka, Osaka 560-8531, Japan\n∗E-mail: kohno@mp.es.osaka-u.ac.jp\nJ. SHIBATA\nKanagawa Institute of Technology, Atsugi, Kanagawa, 243-0 292, Japan\nG. TATARA\nGraduate School of Science, Tokyo Metropolitan University , Hachioji, Tokyo 192-0397, Japan\nInterplay between magnetization dynamics and electric cur rent in a conducting ferromagnet is\ntheoretically studied based on a microscopic model calcula tion. First, the eﬀects of the current\non magnetization dynamics (spin torques) are studied with s pecial attention to the “dissipative”\ntorques arising from spin-relaxation processes of conduct ion electrons. Next, an analysis is given\nof the “spin motive force”, namely, a spin-dependent ‘volta ge’ generation due to magnetization\ndynamics, which is the reaction to spin torques. Finally, an attempt is presented of a uniﬁed\ndescription of these eﬀects.\nKeywords : Current-driven magnetization dynamics; domain wallmoti on; spin torque; spin-transfer\ntorque; spin relaxation; Gilbert damping; spin motive forc e; gauge ﬁeld; eﬀective action\n1. Introduction\nThe fact that electrons have spin degree of\nfreedom as well as electric charge enables\nus to control, in principle, magnetism by\nelectrical means, and vice versa, without re-\ncourse to the relativistic eﬀect of spin-orbit\ncoupling. This type of magnetoelectric cou-\npling has been actively studied over these\ntwo decades based on nanostructured ferro-\nmagnets, where the interplay of electric cur-\nrent and magnetization leads to giant/tunnel\nmagnetoresistance, current-induced magne-\ntization reversal, and so on.1,2\nMicroscopic origin of such phenomena is\nthes-dexchange interaction\nHsd=−M/integraldisplay\nd3xn(x)·ˆσ(x),(1)\nbetween the spin ˆσ(x) of conduction elec-\ntrons and magnetization n(x). For example,\nif an electronmovesthrougha magnetizationtexturen(x), its spin feels a time-dependent\n‘ﬁeld’Mnand is aﬀected. The electron, in\nturn, exertsareactiontorque3,4(spintorque)\ntsd=Mn(x)×/angb∇acketleftˆσ(x)/angb∇acket∇ight,(2)\non the magnetization, which enables us to\ncontrol the magnetization by current.\nIn this paper, we present our microsopic\nstudy on the spin torque and its reciprocal\neﬀect (spin motive force). The magnetiza-\ntion is treated as a classical object, whereas\nelectrons are treated quantum-mechanically.\n2. Spin torques\n2.1.Case of domain wall\nTo illustrate how an electric current ﬂowing\nin a ferromagnet aﬀects the magnetization\ndynamics, let us ﬁrst considera magnetic do-\nmain wall (DW) as an example.3,4,5,6\nFor a rigid DW, there are twodistinct ef-\nfects of the current. If a conduction electron\n1October 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\n2\n(b) momentum-transfer effect(a) spin-transfer effect\nFig. 1. Two eﬀects of electric current on a domain\nwall (DW) via the s-dexchange interaction. (a) Adi-\nabatically transmitted electron transfers spin angu-\nlar momentum to the DW, and exerts a torque (in\na narrow sense) on the DW. (b) Reﬂected electron\ntransfers linear momentum to the DW, and exerts a\nforce on the DW.\npasses through the DW adiabatically and its\nspin is ﬂipped after the passage (Fig.1(a)),\nthis change of electron spin should be com-\npensated by the change of magnetization ow-\ningtototalangularmomentumconservation,\nthereby driving the DW. This is the cele-\nbrated spin-transfer eﬀect. If, instead, an\nelectron is reﬂected by the DW, a linear mo-\nmentum is transferred to the DW and the\nelectron exerts a force on it (Fig.1(b)).\nThe latter process is nonadiabatic, and\nwill be negligible for a ‘thick’ DW as real-\nized in typical metallic magnets. However,\nif the electron system admits spin-relaxation\nprocesses, a new adiabatic torque (called β-\nterm, see below) arises which has the same\neﬀect (i.e., force) on a DW and crucially af-\nfects the dynamics of the DW.7,8,9,10\n2.2.Landau-Lifshitz-Gilbert\nequation under current\nFor general but slowly-varying (in space and\ntime) magnetization conﬁgurations, the dy-\nnamics is described by the Landau-Lifshitz-\nGilbert (LLG) equation,\n˙n=γ0Heﬀ×n+α0˙n×n+t′\nsd.(3)\nHere,n=n(r,t) is a unit-vector ﬁeld rep-\nresenting the d-spin direction, and the dot\nrepresents time derivative. The ﬁrst term, aprecessional torque around the eﬀective ﬁeld\nγ0Heﬀ, and the second term (Gilbert damp-\ning) comefromprocesseswithout conduction\nelectrons. The eﬀects ofconductionelectrons\nare contained in the third term, t′\nsd≡tsd\n×(a3//planckover2pi1S) (S: magnitude of dspin,a3: vol-\nume per d-spin), called spin torque.\nIn this paper, we focus on adiabatic spin\ntorques,11which are ﬁrst order in space/time\nderivative and are expressed as\nt′\nsd=−(v0\ns·∇)n−βsrn×(v0\ns·∇)n\n−αsr(n×˙n)−δS\nS˙n. (4)\nThe ﬁrst term on the right-hand side is the\ncelebrated spin-transfer torque,12where\nv0\ns=−a3\n2eSjs (5)\nis the (unrenormalized) “spin-transfer veloc-\nity”, with js=j↑−j↑being the spin-\ncurrent density. The second term, called ‘ β-\nterm’,8comes from spin-relaxation processes\nof electrons,7and acts as a force on a rigid\nDW. Here βsris a dimensionless constant.\nThe third term is the Gilbert damping, also\nresulting from spin relaxation of electrons.\nThe fourth term contributes as a “renor-\nmalization”ofspin;7it canbe combinedwith\nthe term on the left-hand side of eq.(3) to\nform (1+ δS/S)˙n= (Stot/S)˙n, where\nStot=S+δS, (6)\nis the total (“renormalized”) spin with δS\nbeing the contribution from conduction elec-\ntrons. Then, dividing both sides of the equa-\ntion byStot/S, we arrive at\n˙n=γHeﬀ×n−α(n×˙n)\n−(vs·∇)n−βn×(vs·∇)n,(7)\nwhereγ= (S/Stot)γ0,α= (S/Stot)(α0+\nαsr),β=βsr, and\nvs=S\nStotv0\ns=−a3\n2eStotjs,(8)\nis the “renormalized” spin-transfer velocity.\nNote that βis not renormalized by this pro-\ncedure.October 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\n3\nIn the parameterspace of the LLG equa-\ntion (7), the manifold of α=βprovides a\nveryspecialcaseforthedynamics. Forexam-\nple, any static solution n(r) in the absence\nof spin current can be used to construct a\nsolution n(r−vst) in the presence of spin\ncurrentvsifα=β. Since the controversy\non the current-drivendomain-wallmotion,6,9\nwhether the relation α=βholds generally\nor not has been a theoretical issue.\nThe relation α=βwas originally sug-\ngested in ref.9based on the assumption of\nGalilean invariance of the system. Although\none may argue that the Galilean invariance\nshould be valid for long-wavelength and low-\nfrequency dynamics in which the underlying\nlattice structure is irrelevant, the αandβ\ncome from spin-relaxation processes,7which\nare usually intimately related to the lattice,\ne.g., through the spin-orbit coupling. Also,\nforamany-electronsystemhavingFermisur-\nfaces, Galilean invariance is not an obvious\nproperty. Therefore, it is desired to carry\nout a fully microscopic calculation without\nintroducing any phenomenological assump-\ntions once a microscopic model is ﬁxed.\n2.3.Microscopic model\nFor conceptual simplicity, we take a local-\nizedpictureforferromagnetism,andconsider\nthe so-called s-dmodel consisting of local-\nizeddspins,S=Sn, and conducting selec-\ntrons (as we alreadyused in the previous sec-\ntions). Theyaremutuallycoupled viathes-d\nexchange interaction Hsd[eq.(1)] and obey,\nrespectively, the LLG equation (3) and the\nSchr¨ odinger equation\ni/planckover2pi1˙c=/bracketleftbigg\n−/planckover2pi12\n2m∇2−Mn·σ+Vimp/bracketrightbigg\nc.(9)\nThe impurity potential Vimpincludes poten-\ntial scattering as well as spin scattering\nVs\nimp=us/summationdisplay\njSj·σδ(r−R′\nj) (10)\ndue to quenched magnetic impurities Sj.\nThe latter has been introduced as a micro-scopic modeling of spin-relaxation processes.\nThe averaging over the impurity spin direc-\ntion is taken as Sα\ni= 0 and\nSα\niSβ\nj=1\n3S2\nimpδijδαβ.(11)\nTo obtain the torque tsd, we calcu-\nlate the s-electron spin density /angb∇acketleftˆσ⊥/angb∇acket∇ightne[see\neq.(2)]. ( ⊥means perpendicular component\nton.) Here the average /angb∇acketleft···/angb∇acket∇ightneis taken in\nthe following nonequilibrium states for elec-\ntrons depending on the type of the torque.\n(a) Nonequilibrium states under the in-\nﬂuence of uniform but time-dependent mag-\nnetization . This leads to torques with time\nderivativeof n, namely,Gilbertdampingand\nspin renormalization.\n(b) Nonequilibrium states with current\nﬂowunder static but spatially-varying mag-\nnetization . This leads to current-induced\ntorques, namely, spin-transfertorqueandthe\nβ-term.\n2.4.Small-amplitude method\nIn the presence of spin rotational symmetry\nin the electron system (except for Hsd), adi-\nabatic spin torques are expressed as\ntsd=aµ∂µn+bµ(n×∂µn),(12)\nwhereaµandbµare the coeﬃcients, and\nsumming over µ= 1,2,3(space components)\nand 0 (time) is understood. The correspond-\nings-electron spin polarization is given by\n/angb∇acketleftσ⊥/angb∇acket∇ightne=1\nM[bµ∂µn−aµ(n×∂µn)].\n(13)\nThe coeﬃcients, aµandbµ, can be deter-\nmined by considering small transverse ﬂuc-\ntuations, u= (ux,uy,0),|u| ≪1, around\na uniformly magnetized state, n= ˆz≡\n(0,0,1), such that n= ˆz+u+O(u2), and\nretain the terms ﬁrst order in uas13,14,15\n/angb∇acketleftσ⊥/angb∇acket∇ightne=1\nM[bµ∂µu−aµ(ˆz×∂µu)].(14)\nThenaµandbµare given as linear-response\ncoeﬃcients, which are evaluated in the uni-\nformly magnetized state, u=0.October 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\n4\n jv iv\nασβσ= 㧗\u0002̕̕αβK   (iω )λ  ij\n jv  ivασ\nβσ= 㧗\u0002̕̕αβK   (iω )λ  ij\nFig. 2. Diagrammatic expression for spin torque\n(upper panel) and spin motive force (lower panel).\nThe thick lines carry the external frequency iωλ.\nTo calculate current-induced torques for\nexample, we assume a static conﬁguration,\nn(r) = ˆz+u(r), andintroducead.c. electric\nﬁeldEto produce a current-carrying state.\nWe calculate σ⊥by ﬁrst applying the linear-\nresponse theory to extract Eas\n/angb∇acketleftˆσα\n⊥(q)/angb∇acket∇ightne= lim\nω→0Kα\ni(q,ω+i0)\niωEi.(15)\nThe linear-response coeﬃcient\nKα\ni(q,iωλ) =/integraldisplayβ\n0dτeiωλτ/angb∇acketleftTτˆσα\n⊥(q,τ)Ji/angb∇acket∇ight\n(16)\nis the correlation function of spin ˆ σand elec-\ntric current J, which can be non-vanishing\nin the presence of non-uniform spin texture\nu(r) =uqeiq·r. Extracting uβandqjas\nKα\ni(q,iωλ) =−eMKαβ\nij(iωλ)qjuβ\nq,(17)\nwehavecalculatedthecoeﬃcient Kαβ\nij,which\nis expressed by the upper diagram in Fig. 2.\nThe results are given, in the lowest non-\ntrivial orderin the electrondamping, by14,15\nδS=1\n2ρsa3, (18)\nvs=−a3\n2e(S+δS)js, (19)\nα=a3ν+\n4(S+δS)·/planckover2pi1\nτs+S\nS+δSα0,(20)\nβ=/planckover2pi1\n2Mτs. (21)Hereρs=n↑−n↓is thes-electron spin den-\nsity,ν±=ν↑±ν↓is the density of states,\nandjs=σsE=j↑−j↓is the spin current,\nwithσs=σ↑−σ↓being the “spin conductiv-\nity”. (σ↑(↓)is the conductivity of majority-\n(minority-) spin electrons.) We have deﬁned\nthe spin-relaxation time τsby\n/planckover2pi1\nτs=4π\n3nsu2\nsS2\nimpν+.(22)\nAs expected, only the spin scattering ( ∼\nτ−1\ns) contributes to αandβ, and the poten-\ntial scattering does not.\nThe ratio β/αcannot be unity in gen-\neral for the two-component s-dmodel, since\nit contains mutually independent quantities,\ne.g.,Sofdelectrons and δSofselec-\ntrons. For a single-band itinerant ferromag-\nnet, where δSgives the total moment, the\nresults are obtained by simply putting S= 0\nandα0= 0 in eqs.(18)-(21). We still see that\nα/negationslash=β, but it was pointed out that the ratio\nβ\nα=ρs\nMν+≃1+1\n12/parenleftbiggM\nεF/parenrightbigg2\n(23)\nis very close to unity.13Even so, if we gener-\nalize eq.(11) to the anisotropic one,\nSα\niSβ\nj=δijδαβ×/braceleftBigg\nS2\n⊥(α,β=x,y)\nS2z(α,β=z)(24)\nwe have\nβ\nα=3S2\n⊥+S2z\n2(S2\n⊥+S2z), (25)\nwhich ranges from 1 /2 (forS2\n⊥≪S2z) to 3/2\n(forS2\n⊥≫S2z). Therefore, we conclude that\nα/negationslash=βin general, and that the value β/α\nis very sensitive to the details of the spin-\nrelaxation mechanism.\nThe “β-term” due to spin relaxation\nwas ﬁrst derived by Zhang and Li based on\na phenomenological spin-diﬀusion equation.7\nTheir results can be written as\nαZL=δS\nS+δS·/planckover2pi1\n2Mτs,(26)\nandβZL=/planckover2pi1/2Mτs, thus predict “ α=β” for\na single-band itinerant ferromagnet, S= 0.October 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\n5\nSo far, all phenomenologial theories predict\nα=β, in contrast to the present microscopic\nresults14showing α/negationslash=βin general.\n2.5.Gauge-ﬁeld method\nThe treatment in the previous subsection is\nbased on the assumption of rotational sym-\nmetry in spin space of electrons; otherwise\nit is limited to small-amplitude magnetiza-\ntion dynamics around a uniformly magne-\ntized state. To treat ﬁnite-amplitude dy-\nnamics directly, we introduce in this section\na local/instantaneous spin frame (“adiabatic\nframe”)for selectrons.16,17Inthisframe, the\nspin quantization axis of selectrons is taken\nto be the local/instantaneous d-spin direc-\ntion,n. The electron spinor a(x) in the new\nframe is related to the original spinor c(x) as\nc(x) =U(x)a(x), where Uis a 2×2 unitary\nmatrix satisfying c†(n·σ)c=a†σza. The\na-electrons then obey the equation,\ni/planckover2pi1/parenleftbigg∂\n∂t+iA0/parenrightbigg\na(x) (27)\n=/bracketleftbigg\n−/planckover2pi12\n2m(∇i+iAi)2−Mσz+˜Vimp/bracketrightbigg\na(x),\nwhich is characterized by a constant magne-\ntizationMσzand an SU(2) gauge ﬁeld\nAµ=−iU†(∂µU) =Aα\nµσα≡Aµ·σ.(28)\nThis gauge ﬁeld expresses the inﬂuence of\ntemporal ( µ= 0) or spatial ( µ= 1,2,3) vari-\nation ofn.\nThe adiabatic torques in eq.(12) follow\nfrom the following expression18\n/angb∇acketleft˜σ⊥/angb∇acket∇ightne=2\nM/bracketleftBig\naµA⊥\nµ+bµ(ˆz×A⊥\nµ)/bracketrightBig\n,(29)\nobtained in the ﬁrst order in Aµ. Here\n/angb∇acketleft˜σ/angb∇acket∇ight ≡ /angb∇acketlefta†σa/angb∇acket∇ightis the electron spin density\nin the adiabatic frame, and ˜σ⊥andA⊥\nµare\nthose projected onto the xy-plane. The coef-\nﬁcientsaµandbµcan be calculated as linear-\nresponse coeﬃcients. The results for δS,vs\nandβthus obtained coincide with eqs.(18),\n(19), and (21). However, it leads to αsr= 0\nand fails to produce the Gilbert damping.Adiabatic frame : a(x)Original frame : c(x)\nMagnetization \nis in motion\nImpurity spins\nare quenched\nImpurity spins\nare in motionMagnetization\nis fixed\nFig. 3. Upper panel (lower panel) shows magneti-\nzation vector n(t) (ˆz) and impurity spins Sj(˜Sj(t))\nin the original frame (adiabatic frame).\nThis diﬃculty has been resolved18by\nnoting that the impurity spins, which are\nstatic (quenched) in the original frame, be-\ncome time-dependent in the adiabatic frame:\n˜Sj(t) =tR(t)Sj. (30)\n(See Fig.3.) Here Risa3×3orthogonalma-\ntrix representing the same rotation as Ubut\nacting on three-component vectors. From\nthe time dependence of ˜Sj(t) orR(t), the\nSU(2) gauge ﬁeld can arise as\n[R(t)t˙R(t)]αβ= 2εαβγAγ\n0(t).(31)\nIn fact, explicit evaluation of /angb∇acketleft˜σ⊥/angb∇acket∇ightnein sec-\nond order in ˜Sj(t) (nonlinear response) gives\n/angb∇acketleft˜σ⊥/angb∇acket∇ightne=−2π/planckover2pi1\n3Mnsu2\nsS2\nimpν2\n+(ˆz×A⊥\n0),(32)\nleading to the Gilbert damping which coin-\ncides with the ﬁrst term of eq.(20).\nThe above calculation provides us a new\npicture of Gilbert damping; while the spins\nofselectrons tend to follow n(t), it is at the\nsame time pinned by the quenched impurityOctober 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\n6\nspins, and this frustration gives rise to the\nGilbert damping. This picture also applies\nto the case where spin relaxation originates\nfrom spin-orbit coupling.19\n3. Spin motive force\nAs a reaction to spin torques, magneti-\nzation dynamics in turn exerts a spin-\ndependent force, called spin motive force,\non electrons.20,21,22,23,24,25,26,27,28,29Accord-\ning to Stern,22this eﬀect arises from the\ntime-dependent spin Berry phase, which we\ninterpret in our context as arising as a com-\nbined eﬀect of temporal variation and spatial\nvariation of magnetization. Here we present\na simple argument using the results obtained\nin the previous section.\nWe apply the small-amplitude method,\nand consider a small ﬂuctuation of the form,\nu(r,t) =u1e−iωt+u2eiq·r,(33)\nto calculate the current density in the ﬁrst\norder in ˙uand∇u,i.e., inωu1andqju2:\n/angb∇acketleftji(q)/angb∇acket∇ightne=−eM˜Kαβ\nij(ω)\niωωuα\n1·qjuβ\n2.(34)\nThe coeﬃcient ˜Kαβ\nij(see Fig.2) can be shown\nto be related to Kαβ\nijof the spin torque as\nKαβ\nij(iωλ) =˜Kαβ\nij(−iωλ).(35)\nTherefore, using the results of §2-4, we read-\nily obtain j=σsEs, where\nEs,i=/planckover2pi1\n2e/bracketleftBig\nn·(∂in×˙n)+β(˙n·∂in)/bracketrightBig\n.(36)\nFromj=σ↑Es+σ↓(−Es), we may identify\nEsto be a spin-dependent ‘electric’ ﬁeld, or\n−eEstobethespinmotiveforce,inthesense\nthat majority- (minority-) spin electrons feel\nan eﬀective ‘electric’ ﬁeld of Es(−Es). The\nsecond term, containing the same βparam-\neter as the spin torque, is due to spin relax-\nation, and was ﬁrst reported by Duine.24\nMore general calculation without assum-\ning the form of eq.(33),30as well as the one\nbased on the gauge-ﬁeld method31will be re-\nported elsewhere.4. Eﬀective gauge-ﬁeld action\nSpin torque and spin motive force are ac-\ntion and reaction to each other, and should\nbe derived from the same term in the eﬀec-\ntive action. This kind of study has been\ndone by Duine et al.15based on the real-\ntime, small-amplitude formalism. Here we\npresent a treatment based on the imaginary-\ntime, gauge-ﬁeld formalism. It should be\nnoted that dynamical/dissipative processes\ncan also be treated with imaginary time.\nWe introduce an electromagnetic vector\npotential Aemto drive the non-equilibrium\nOhmic current in a ferromagnet, and elimi-\nnate the a-electrons. Up to the second order\ninAemand the SU(2) gauge ﬁeld Aα, the\neﬀective action Sis obtained as32\nS=/integraldisplayβ′\n0dτ/integraldisplaydr\na3/bracketleftbigg\n2i/planckover2pi1StotAz\n0+Jeﬀ\n2(∂in)2/bracketrightbigg\n+/integraldisplayβ′\n0dτ/integraldisplayβ′\n0dτ′/integraldisplay\ndrI(τ−τ′)\n×/braceleftBigg/bracketleftbigg\nσs/planckover2pi1\neAz(τ)+σc\n2Aem(τ)/bracketrightbigg\n·Aem(τ′)\n+cαβ[R(τ)tR(τ′)]αβ/bracerightBigg\n. (37)\nHereβ′≡(kBT)−1=∞is the inverse tem-\nperature, Jeﬀ=JddS2+Jss(δS)2,17and\ncαβ=π\n6nsu2\nsS2\nimp/bracketleftbig\n2ν↑ν↓δαβ+ν2\n−δαzδβz/bracketrightbig\n.\n(38)\nThe kernel I(τ−τ′) =−[π(τ−τ′)2]−1\ndescribes dissipative processes characterized\nby Ohmic damping, as is familiar since the\nwork by Caldeira and Leggett33on macro-\nscopic quantum tunneling. The coupling\nAz·Aemdescribes the spin-transfer torque\nand spin motive force. The term containing\nR(τ)tR(τ′) describes Gilbert damping.\nIn fact, by taking the variation of Swith\nrespect to n, and performan analyticcontin-\nuation,τ→it, we obtain the LLG equation\nconsistent with eqs.(7), (18)-(20) but with\nβ= 0. Similarly, the electric current den-October 1, 2018 16:11 WSPC/Trim Size: 10in x 7in for Proceedi ngs ISQM-Tokyo08\n7\nsity is obtained from j=−δS/δAemas\nj=−/planckover2pi1\neσs˙Az−σc˙Aem,(39)\nfrom which we can read the existence of the\nspin motive force as −eEs=/planckover2pi1˙Az= (/planckover2pi1/2)n·\n(˙n×∂in). The eﬀective coupling describing\ntheβ-term remains to be derived.\n5. Summary and remarks\nWe have developed a microscopic theory of\nspin torques and spin motive force, and their\nuniﬁed description. Although the present\nmagnetic impurity model may not be quite\nrealistic as the origin of spin relaxation, we\nexpect the present calculation already cap-\ntures the essential features of the current-\nspin interaction including spin-relaxation ef-\nfects. For quantitative information such as\nthe value of β/α, calculations with realistic\nspin-relaxation mechanisms are necessary.\nAcknowledgments\nWewouldliketothankG. Bauer,A.Brataas,\nR. Duine, H. Fukuyama, A.H. MacDonald,\nS. Maekawa, Y. Nakatani, Q. Niu, H. Ohno,\nT. Ono, E. Saitoh, J. Sinova, M. Stiles, Y.\nSuzuki, A.Thiaville and Y. Tserkovnyak for\nvaluable discussions. H. K. is indebted to K.\nMiyake for his continual encouragement.\nReferences\n1.Concepts in Spin Electronics , Ed. S. Maeka-\nwa (Oxford University Press, Oxford, 2006).\n2.Spintronic Materials and Technology , Eds.\nY.B. Xu and S.M. Thompson, (Taylor &\nFrancis, 2007).\n3. L. Berger, J. Appl. Phys. 55, 1954 (1984).\n4. L. Berger, J. Appl. Phys. 71, 2721 (1992).\n5. A. Yamaguchi et al.,Phys. Rev. Lett. 92,\n077205 (2004); E. Saitoh et al.,Nature432,\n203 (2004); M. Yamanouchi et al.,Nature\n428, 539 (2004); M. Kl¨ aui et al.,Phys. Rev.\nLett.94, 106601 (2005); M. Hayashi et al.,\nNature Phys. 3, 21 (2007).\n6. G. Tatara andH.Kohno, Phys. Rev. Lett. 92,\n086601 (2004); 96189702 (2006).7. S. Zhang and Z. Li, Phys. Rev. Lett. 93,\n127204 (2004).\n8. A. Thiaville, Y. Nakatani, J. Miltat and Y.\nSuzuki,Europhys. Lett. 69, 990 (2005).\n9. S.E. Barnes andS.Maekawa, Phys. Rev. Lett.\n95, 107204 (2005); 96, 189701 (2006).\n10. G. Tatara, H. Kohno and J. Shibata, Phys.\nRep.468, 213 (2008).\n11. There are non-adiabatic torques in addition\nif the magnetization varies rapidly, which we\ndo not discuss in this report. See G. Tatara\net al.,J. Phys. Soc. Jpn. 76, 054707 (2007).\n12. Ya. B. Bazaliy, B. A. Jones, and S.-C.\nZhang,Phys. Rev. B 57, R3213 (1998).\n13. Y. Tserkovnyak, H.J. Skadsem, A. Brataas\nand G.E.W. Bauer, Phys. Rev. B74, 144405\n(2006).\n14. H. Kohno, G. Tatara and J. Shibata,\nJ. Phys. Soc. Jpn. 75, 113706 (2006).\n15. R.A.Duine, A.S. N´ u˜ nez, J. Sinovaand A.H.\nMacDonald, Phys. Rev. B75, 214420 (2007).\n16. V. Korenman et al.,Phys. Rev. B16, 4032\n(1977).\n17. G. Tatara and H. Fukuyama, J. Phys. Soc.\nJpn.63, 2538 (1994).\n18. H. Kohno and J. Shibata, J. Phys. Soc. Jpn.\n76, 063710 (2007).\n19. S. Kawabata, Master thesis (Osaka Univ.,\n2008).\n20. L. Berger, Phys. Rev. B33, 1572 (1986).\n21. G.E. Volovik, J. Phys. C 20, L83 (1987).\n22. A. Stern, Phys. Rev. Lett. 68, 1022 (1992).\n23. S. E. Barnes and S. Maekawa, Chap.7 of\nref.1;Phys. Rev. Lett. 98, 246601 (2007).\n24. R.A.Duine, Phys. Rev. B77, 014409 (2008).\n25. M. Stamenova, T.N. Todorov and S. San-\nvito, cond-mat/0708.1167.\n26. S.A. Yang, D. Xiao and Q. Niu, cond-\nmat/0709.1117.\n27. Y. Tserkovnyak and M. Mecklenburg, Phys.\nRev.B77, 134407 (2007).\n28. S.A. Yang, G. Beach, C. Knutson, D. Xiao,\nQ. Niu, M. Tsoi and J.L. Erskine, preprint.\n29. A similar phenomenon in a FN junc-\ntion system was studied in, A. Brataas,\nY. Tserkovnyak, G. Bauer and B.I. Halperin,\nPhys. Rev. B66, 060404 (2002).\n30. T. Noguchi, Master thesis (Osaka Univ.,\n2008).\n31. J. Shibata and H. Kohno, in preparation.\n32. S. Ueta, Master thesis (Osaka Univ., 2008).\n33. A.O. Caldeira and A.J. Leggett, Phys. Rev.\nLett.46, 211 (1981); Ann. Phys. 149, 374\n(1983)."    },    {        "title": "2006.08253v1.Control_of_Spin_Relaxation_Anisotropy_by_Spin_Orbit_Coupled_Diffusive_Spin_Motion.pdf",        "content": "Control of Spin Relaxation Anisotropy by Spin-Orbit-Coupled Di\u000busive Spin Motion\nDaisuke Iizasa,1Asuka Aoki,1Takahito Saito,1Junsaku Nitta,1, 2, 3Gian Salis,4and Makoto Kohda1, 2, 3, 5\n1Department of Materials Science, Tohoku University, Sendai 980{8579, Japan\n2Center for Spintronics Research Network, Tohoku University, Sendai 980{8577, Japan\n3Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai 980{8577, Japan\n4IBM Research-Zurich, S aumerstrasse 4, 8803 R uschlikon, Switzerland.\n5Division for the Establishment of Frontier Sciences, Tohoku University, Sendai 980-8577, Japan\n(Dated: June 16, 2020)\nSpatiotemporal spin dynamics under spin-orbit interaction is investigated in a (001) GaAs two-\ndimensional electron gas using magneto-optical Kerr rotation microscopy. Spin polarized electrons\nare di\u000bused away from the excited position, resulting in spin precession because of the di\u000busion-\ninduced spin-orbit \feld. Near the cancellation between spin-orbit \feld and external magnetic \feld,\nthe induced spin precession frequency depends nonlinearly on the di\u000busion velocity, which is unex-\npected from the conventional linear relation between the spin-orbit \feld and the electron velocity.\nThis behavior originates from an enhancement of the spin relaxation anisotropy by the electron ve-\nlocity perpendicular to the di\u000bused direction. We demonstrate that the spin relaxation anisotropy,\nwhich has been regarded as a material constant, can be controlled via di\u000busive electron motion.\nPrecise control of spin motion is a prerequisite from\nfundamental physics to spintronics and quantum infor-\nmation technology [1{4]. In a semiconductor quantum\nwell (QW), Rashba [5, 6] and Dresselhaus [7] spin{orbit\n(SO) interactions act as e\u000bective magnetic \felds for mov-\ning electrons, enabling coherent spin control via preces-\nsion, whereas spin relaxation occurs simultaneously be-\ncause of an interplay between the SO \feld and the ran-\ndom motion of electrons [8]. Both spin precession and\nrelaxation processes are closely tied to one another solely\nby SO interaction [9]. For stationary electrons with mean\nzero velocity, the correlation between precession and re-\nlaxation triggers a modulation of spin precessional mo-\ntion, known as spin relaxation anisotropy [10{19]. For\nspin rotation by external and/or SO \felds in a QW, spins\nalong growth and in-plane orientations do not experience\nidentical torques because of the in-plane orientation of\nthe SO \felds. This situation induces anisotropic spin\nrelaxation [10{19] and modulates the spin precession fre-\nquency [13{16, 18, 19]. Because SO \felds are well de\fned\nfor stationary electrons, the spin relaxation anisotropy\nhas been regarded as a material constant. However, for\nmoving electrons with a \fnite net velocity induced by\ndrift [20{25] and di\u000busion [24{27], the electron trajectory\nfurther modulates SO \felds and directly a\u000bects the spin\nrelaxation anisotropy through the momentum-dependent\nspin precession. Moreover, the spin relaxation anisotropy\nis not limited to particular materials such as III{V semi-\nconductors because the anisotropic SO \felds are ubiqui-\ntous in solid states, with spin-momentum locking in topo-\nlogical insulators [28, 29], Rashba interface in oxides [30],\nmetal interfaces [31], and Zeeman-type SO \feld in 2D ma-\nterials [32]. Consequently, unveiling the e\u000bects of moving\nelectrons on the modulation of spin relaxation anisotropy\nand induced precession frequency are expected to be cru-\ncially important for future spintronics, topological elec-\ntronics, and quantum information technologies. Despitethis, earlier studies of spin relaxation anisotropy have re-\nmained limited only to stationary cases [13{16, 18, 19].\nHere, we experimentally manifest control of spin pre-\ncessional motion via spin relaxation anisotropy by dif-\nfusive spin motion in a GaAs-based QW. When the SO\n\feld under di\u000busive motion is nearly compensated by a\nconstant external magnetic \feld, the spin precession fre-\nquency is no longer linear to the di\u000busion velocity. This\nbehavior cannot be anticipated from a conventional spin\ndrift/di\u000busion model. It is explained by a modulation\nof the spin relaxation anisotropy. The evaluated spin re-\nlaxation anisotropy, which exhibits six-fold enhancement\nfrom the stationary case, is explained by a tilting of the\nspin precession axis from the direction of external mag-\nnetic \feld caused by the electron di\u000busive motion. We\nin\ruence the spin relaxation anisotropy, as reported for\nthe \frst time, by precisely controlling the electron mo-\ntion.\nThe structure examined for this study was an n-doped\n20-nm-thick (001) GaAs QW. In this system, we obtain\nSO \felds characterized by the Rashba parameter \u000b(<0),\nthe Dresselhaus parameter \f=\f1\u0000\f3(>0), with linear\n\f1=\u0000\rhk2\nziand cubic term \f3=\u0000\rk2\nF=4. Here,hk2\nzi\ndenotes the expected value of the squared wavenumber\nin the QW. The bulk Dresselhaus coe\u000ecient is \r < 0.\nThe Fermi wavenumber is kF=p2\u0019ns. The carrier\ndensity and mobility measured using a Hall bar device\nwerens= 1:72\u00021011cm\u00002and 11:2\u0002104cm2V\u00001s\u00001,\nrespectively, at 4.2 K. To detect the di\u000busive spin dy-\nnamics, spatiotemporal Kerr rotation microscopy is per-\nformed using a mode-locked Ti:sapphire laser emitting\n2-ps-long pulses at a 79.2 MHz repetition rate. Fig-\nure 1(a) depicts an experimental con\fguration for pump\nand probe beams with Rashba and Dresselhaus SO \felds.\nTherein, \nRand\nDrespectively represent the spin pre-\ncession frequency vectors. A circularly polarized pump\nbeam with Gaussian sigma-width \u001bppis focused onto thearXiv:2006.08253v1  [cond-mat.mes-hall]  15 Jun 20200\nr\nx|| [110]-y|| [110]\nΩex,yΩex,x\nΩD\nΩRFIG. 1. (a) Sketch of a pump-probe scanning Kerr microscopy\nsetup with Rashba ( \nR) and Dresselhaus ( \nD) \felds as pre-\ncession vectors. An external magnetic \feld is depicted as\n\nex;xand\nex;yas a precession vector for y- andx-scans con-\n\fgurations, respectively. A circularly polarized pump beam\nexcites a spin polarization sz. A linearly polarized probe\nbeam detects szat a delay time tand a position ( x;y). (b)\nMeasuredszat di\u000berent xpositions highlighted as colored\ncircles in (a).\nsample surface to excite spin polarization szalong the\ngrowth direction. A linearly polarized probe beam (spot\nsize\u001bpr) detectsszat delay time tand arbitrary posi-\ntion by a motor-controlled scanning mirror. All optical\nmeasurements are taken at 30 K.\nThe spin precession frequency induced by a velocity\nv= (vx;vy) in an external magnetic \feld Bex= (Bx;By)\nis generally described as\n\nx;y(vy;x) =2m\n~2(\u0006\u000b+\f)vy;x+g\u0016B\n~Bx;y: (1)\nHereg <0 stands for the electron gfactor,\u0016Bdenotes\nthe Bohr magneton, ~is the reduced Plancks constant,\nandm= 0:067m0expresses e\u000bective electron mass of\nGaAs. The di\u000busion velocity vdif, which is controlled by\nthe center-to-center distance rbetween pump and probe\nspots, is de\fned as\nvdif= 2Dsr=(2Ds\u001cs+\u001b2\ne\u000b); (2)\nwhereDsis the spin di\u000busion constant, \u001csrepresents the\nD'yakonov-Perel' spin relaxation time, and the convo-\nluted spot size \u001be\u000bis de\fned by \u001b2\ne\u000b=\u001b2\npp+\u001b2\npr[26].\nAlso,\u001csis a result of the replacement of t=\u001csbe-\ncause our system satis\fes 2 Ds\u001cs\u001c\u001b2\ne\u000band small\u001cs.\nBy changing the probe position along the x-axis (y-axis)\n[26], i.e., the distance rin Eq. (2) , one can set the dif-\nfusion velocity vdif=vx(vdif=vy) and thereby modu-\nlate the spin precession frequency [\n y(vx) or \n x(vy) in\nEq. (2)]. Figure 1(b) shows the time evolution of the ex-\nperimental Kerr signal ( sz) at di\u000berent probe positions\n(x= 9:7;0:8 and\u000012:5\u0016m) in anx-scan (\u001be\u000b= 8:1\n\u0016m andBy= +0:45 T). The spin precession frequency\ndepends strongly on the probe position, re\recting the\nmomentum (velocity) dependent SO \feld induced by\nthe \fnite di\u000busion velocity. We systematically measured\nKerr signals with di\u000berent positions on the x- andy-axes\nwith several spot sizes \u001be\u000b. We extracted the precession\n1\nΩD\n0x|| [110]\nΩR-\n0y|| [110]\nΩRΩD10,167,209\nFIG. 2. Measured spin precession frequency j\nmeasjobtained\nfor di\u000berent \u001be\u000bandBexand for scans of the pump-probe\nseparation along x(a) andy(b). Di\u000busing spins experience\nstrong SO \felds for x-scan, but weak \felds for y-scan. All\nsymbols represent experimental data. All solid lines show\nlinear \fts. Dashed lines in (a) correspond to the nonlinear\n\fts based on Eq. (3) with \u0000 at=\u00000:076 GHz.\nfrequencyj\nmeasjby \ftting the normalized Kerr signal\nsz= exp (\u0000t=\u001cs) cos (2\u0019j\nmeasjt+\u001e) with phase shift \u001e.\nFigures 2(a) and 2(b) summarize extracted j\nmeasj\ninx- andy-scans. For the y-scan [Fig. 2(b)], j\nmeasj\nvaries linearly with the yposition for all conditions of\nBxand\u001be\u000b, re\recting the linear dependence of vdifon\ntheyposition, as presented in Eq. (2). In addition, when\n\u001be\u000bdecreases from 11.6 to 6.8 \u0016m, the slope d\nmeas=dy\nincreases gradually, which agrees well with Eq. (2) and\nwhich is consistent with earlier reports of the literature\n[20, 22, 23, 25{27]. For the x-scan [Fig. 2(a)], however,\na linear variation of j\nmeasjon thexposition is only ob-\nserved for\u001be\u000b= 9:8\u0016m andBy= +0:45 T (diamond\nsymbols). Reducing \u001be\u000bto 8.1 and 5.6 \u0016m exhibits a de-\nviation from a linear variation; notably most pronounced\nwhenj\nmeasjapproaches zero. This cannot be explained\nusing the conventional linear relation between electron\nvelocity and SO \feld. To understand this e\u000bect, we \frst\nevaluate the SO parameters from the linear frequency\nvariation. From linear \fts depicted as solid lines in\nFigs. 2(a) and 2(b), we obtain \u000b=\u00002:89\u000210\u000013eVm,\n\f1= 1:86\u000210\u000013eVm, and \f3= 0:22\u000210\u000013eVm.\nAlso,g=\u00000:268 is estimated at r= 0 (vdif= 0).\nWe assume g < 0 based on the QW thickness [23].\nAlso,Ds= 0:0195 m2=s is derived from the measured\n\u001cs= 75 ps at Bex=0T [33]. Using evaluated \f1;\f3,\nandns, we obtain \r=\u00008:31 eV \u0017A3which is consistent\nwith values reported in the literature [34]. To explain our\nobservation, we introduce in analogy to anisotropic spin\nrelaxation for stationary electrons modi\fed spin preces-\n2sion frequencies [13{16, 18, 19],\n\n\u0003\nx=q\n\nx(vdif)2\u0000\u00002\nat;\n\u0003\ny=q\n\ny(vdif)2\u0000\u00002\nat;(3)\nwhere the anisotropic term [15, 18] is\n\u0000at(\u0002) =\u0000(\u0000xcos2\u0002 + \u0000 ysin2\u0002)=2: (4)\nHere the relaxation rate of spins oriented along x- and\ny-axes is \u0000 x;y= (4Dsm2=~4)[(\u0007\u000b+\f)2+\f2\n3], respec-\ntively, and \u00022[0;2\u0019] is the direction of the spin pre-\ncession axis, de\fned as in-plane polar angle from + x- to-\nward +y-axis. The term \u0000 at(\u0002) describes the relaxation\nanisotropy between the two relevant orthogonal crystal\naxes and is responsible for a correction of the precession\nfrequency [Eq. (3)]. For the y-scan ( Bex= (Bx;0)) spins\nprecess in the y-zplane, and \u0000 at(\u0002 = 0) =\u0000\u0000x=2 =\n(\u0000y\u0000\u0000z)=2 denotes half of the di\u000berence of the relax-\nation rate between y- andz-axes, where \u0000 z= \u0000x+ \u0000y\nis the relaxation rate along the z-axis. For the x-scan\n(Bex= (0;By)), \u0000 at(\u0006\u0019=2) =\u0000\u0000y=2. Because \u0000 at(\u0002)\nadditionally contributes to the spin precession frequency\nshown in Eq. (3), \n\u0003\nx;yshows a nonlinear dependence on\nthe probe position r, which becomes pronounced when\nthe precession frequency induced by external and SO\n\felds becomes comparable to \u0000 x;y=2. Based on the ex-\nperimentally evaluated values for \u000b;\f1;\f3, andDs, we\ncalculate\u0000\u0000y=2 =\u00000:076 and\u0000\u0000x=2 =\u00000:99 GHz.\nFor\u001be\u000b= 5:6 and 8.1\u0016m, the calculated \n\u0003\nyare shown\nas dashed lines in Fig. 2(a). The calculated values only\nreproduce the experimental data in a linear frequency\nregion. The rapid decrease of \n\u0003\nythat occurs below 0.8\nGHz cannot be explained by \u0000\u0000y=2 =\u00000:076 GHz.\nAccording to Eq. (4), the frequency modulation caused\nby the relaxation anisotropy depends on the direction of\nthe precession axis (\u0002). For stationary electrons under\nBex, where the SO \feld does not contribute to frequency\nmodulation, \u0002 is well-de\fned by the direction of Bex.\nHowever, for moving electrons, the precession axis is de-\n\fned by the sum of Bexand the SO \feld, implying that\nthe electron trajectories under di\u000busion further modu-\nlate \u0002. Because various electron trajectories can lead\nfrom the pump spot to the probe spot, the precession\naxis is no longer well de\fned by Bexbecause of di\u000berent\ndi\u000busion velocity vectors v= (vx;vy) in time\u001cs[di\u000ber-\nent arrows in Fig. 3(a)]. Speci\fcally examining one single\ntrajectory with average velocity v, its direction of aver-\nage precession axis \u0002( vx;vy) = arctan(\n y(vx)=\nx(vy))\ncan be obtained from Eq. (1). Entering \u0002( vx;vy) into\nEq. (4) directly reveals the velocity-dependent form of\nthe anisotropic term\n\u0000at(vx;vy) =\u0000\u0000x\nx(vy)2+ \u0000y\ny(vx)2\n2 (\nx(vy)2+ \ny(vx)2): (5)\nIt is noteworthy that the precession frequencies caused\nby opposite velocities ( \u0006vy;x) do not cancel each other\n2\n(a)\nx|| [110]-y\n(x, 0)\nvyvx1\n2\n3\n PumpProbe(b)\n(x, 0)\nx xy(c)FIG. 3. (a) x-scan con\fguration where the probe center is\nseparated by a vector ( x;0) from the pump center (0 ;0). Elec-\ntron trajectories exist with an average velocity that is tilted\nfrom thex-axis, contrary to the macroscopic di\u000busion veloc-\nity [Eq. (2)]. (b) Mean velocity components of horizontal and\nvertical directions, v\u0006\nhandv\u0006\nv. (c) Calculated v\u0006\nh,v\u0006\nvbased\non Eqs. (6) and (7). v\u0006\nvare constant with respect to rhere,\nwhereasv\u0006\nhdepends on the position.\nbecause \n x;yenter as squares. This \fnding contrasts\nto the notion of a single (averaged) di\u000busion velocity for\ngiven pump and probe spots [described by Eq. (2)], corre-\nsponding to the mean value of all velocity vectors leading\nfrom the pump to the probe spots [Fig. 3(a)]. Actually,\nEq. (5) rather suggests that the anisotropic term is de-\n\fned by the microscopic behavior of the electron motion\n(velocity). Therefore, we sort all di\u000busion velocity vec-\ntors along the x- andy-axes according to their sign and\nevaluate the mean values of horizontal and vertical ve-\nlocities at probe positions v\u0006\nhandv\u0006\nv, respectively, as\nindicated by silver bold arrows in Fig. 3(b). Each veloc-\nity is described as\nv\u0006\nh=\u0006\u001be\u000bs\n\u0006\n\u0019\u001cse\u0000r2\u0006\u001cs\n\u001b2\neff+ \u0006rerfc \n\u0007rp\u0006\u001cs\n\u001be\u000b!\n;(6)\nv\u0006\nv=\u0006\u001be\u000bp\n\u0006=(\u0019\u001cs); (7)\nwhere \u0006 = Ds=(2Ds\u001cs+\u001b2\ne\u000b). The complementary error\nfunction is denoted by erfc. The \u0006signs inv\u0006\nhandv\u0006\nv\nrespectively correspond to the positive/negative velocity\ncomponents along the horizontal or vertical axes. Fig-\nure 3(c) shows calculated v\u0006\nhandv\u0006\nv. Thev+\nhincreases\nrapidly in the + xregion, whereas v\u0000\nhhas the opposite\ntendency because of radial di\u000busion of electrons from the\nexcited pump spot.\nWhen the probe spot is displaced along the \u0000x-axis\n[Fig. 3(a)], the average velocity vector points to the \u0000x-\naxis because there, jv\u0000\nhj>jv+\nhj. Remarkably, vertical\nvelocitiesv\u0006\nvare independent of probe position and are\nof similar size as v\u0006\nh. This suggests that the tilted veloc-\nity vectors should contribute to \u0000 atand the precession\naxis (\u0002) no longer points along Bex. For the total mean\nvelocityv+\nh+v\u0000\nh+v+\nv+v\u0000\nvas sketched in Fig. 3(b), the\nv\u0006\nvcancel out each other and are linear in the position\n[dashed line in Fig. 3(c)], corresponding to the conven-\ntional macroscopic di\u000busion velocity [Eq. (2)].\nBy considering both horizontal and vertical velocities\n[Eqs. (6) and (7)] depicted in Fig. 3(b), we average over\n301530\nx ( 7m)0!atx (GHz)\n01530\ny ( 7m)!aty (GHz)<eff = 5.6 7m\n By<eff = 8.1 7m\n By = 0.45 T<eff = 9.8 7m\n By = 0.45 T\n<eff = 6.8 7m\n Bx = 0.4 T\n<eff = 8.1 7m\n Bx = 0.55 T<eff = 11.6 7m\n Bx = 0.65 T(a) (b)FIG. 4. (a) Calculated anisotropy term \u0000x;y\natforx-scan (a)\nandy-scan (b) as obtained from using Eq. (8). Both \u0000x;y\nat\nare modulated by r. Parameters for the calculation are all\nexperimentally determined values.\nall contributions to the anisotropic term [Eq. (5)] for the\nx-scan with\n\u0000x\nat=\u0002\n\u0000at(v+\nh;v+\nv) + \u0000 at(v+\nh;v\u0000\nv)\n+ \u0000at(v\u0000\nh;v+\nv) + \u0000 at(v\u0000\nh;v\u0000\nv)\u0003\n=4: (8)\nFor they-scan con\fguration, \u0000y\natis obtained from Eq. (8)\nby \ripping vhandvvwith each other. We calculate \u0000x;y\nat\nin Figs. 4(a) and 4(b) with parameters evaluated from the\nexperimental conditions. The \u0000x\natexhibits a peak struc-\nture corresponding to the cancellation between external\nand SO \felds, i.e. \n y(vdif) = 0. At this position, \u0000x\natis\nenhanced by more than six times from \u0000\u0000y=2 =\u00000:076\nGHz. Such a peak structure is observed consistently for\ndi\u000berent spot sizes and Byvalues. The enhanced \u0000x\natis\nthe consequence of a tilting of the spin precession axis\naway from Bexdirection due to the v\u0006\nvcomponents that\nintroduce a precession contribution \n x(v\u0006\nv)2. As seen\nfrom Eq. (5), \u0000 x\nx(v\u0006\nv)2is introduced in the numera-\ntor of the expression for \u0000 at. Fory-scan, \u0000y\natis only\ngently modulated with ybecause, in this case, the ad-\nditional contribution proportional to \u0000 y\ny(v\u0006\nv) is weak\ncompared to the case of x-scan (because \u0000 x\u001d\u0000y). In\nother words, a small SO \feld along the y-axis does not\ntilt the spin precession axis signi\fcantly. In both x- and\ny-scans, when the magnitude of Bexbecomes su\u000eciently\nlarge compared to the SO \feld, \u0000x;y\natconverges respec-\ntively to the stationary cases of \u00000:076 GHz and\u00000:99\nGHz.\nFinally, we reproduce the nonlinear behavior of the\nx-scan quantitatively by considering the calculated \u0000x;y\nat\nin Fig. 4. Figure 5(a) shows the experimentally obtained\nresults of color-coded szin anx-scan and a frequency\nanalysis (circles) for \u001be\u000b= 5:6\u0016m withBy=\u00000:4 T.\nSolid and dashed lines respectively correspond to the cal-\nculated \n\u0003\nybased on Eq. (3) with our new \u0000x\nat[Eq. (8)]\nand the conventional \u0000\u0000y=2 =\u00000:076 GHz. The re-\nmaining parameters of Eq. (3) are all experimentally ob-\ntained. The \n\u0003\nyobtained with the new \u0000x\natshows excel-\nlent agreement with the experimental values, including\nthe nonlinear variation. To con\frm the enhancement of\n\u0000atfurther, we conducted Monte Carlo (MC) simulation.\nFigure 5(b) presents the simulated Kerr traces, showing\n01\nsz (arb. units)00.20.40.60.8\nt (ns)\n000.20.40.60.8t (ns)\n000.511.5\n| +meas| (GHz)\n00.5100.20.40.60.8\nt (ns)\n015\nx ( 7m)00.20.40.60.8t (ns)\n015\nx ( 7m)0123\n| +meas| (GHz)\n00.51\nsz (arb. units)00.20.40.60.8\nt (ns)\n02040\ny ( 7m)00.20.40.60.8t (ns)\n012\n| +meas| (GHz)\n01\nsz (arb. units)x = -8.7 7m\nx = -8.6 7m(a) (b) (c)x = 17.2 7m\nx = 17.2 7m(d) (e) (f)Experiment Monte Carlo\nExperiment I Monte Carlo Iy = -18.2 7m\n(g)(h) Monte CarloFIG. 5. Experimental and Monte Carlo simulated times-\npace records of szinx-scan for (a) and (b) with \u001be\u000b= 5:6\n\u0016m atBy=\u00000:4 T, and for (d) and (e) with \u001be\u000b= 8:1\n\u0016m atBy= 0:45 T. All solid lines are \n\u0003\nycalculated with\nnew \u0000x\nat, and dashed lines with conventional anisotropic term\n\u0000\u0000y=2 =\u00000:076 GHz. The time evolution of szis shown at\n(c) forx\u0019\u00008:6\u0016m and in (f) for x=\u000017:2\u0016m for ex-\nperimental data (diamonds) and for Monte Carlo simulation\nresults (red solid lines). (g) Monte Carlo simulated time-space\nrecords ofszin ay-scan with\u001be\u000b= 5:6\u0016m,Bx= 0:4 T and\ng=\u00000:268: the solid line shows \n\u0003\nxwith new \u0000y\nat; the dashed\nline is obtained with conventional value \u0000\u0000x=2 =\u00000:99 GHz.\nThe gray diamond is the extracted frequency from the (h)\ntime evolution of szaty=\u000018:2 m with Monte Carlo simu-\nlation (red solid) and the \ftted curve (dotted black).\ngood agreement with the experimentally obtained result.\nAtx\u0019\u00008:6\u0016m, whereas \n\u0003\nywith conventional \u0000\u0000y=2\nsuggests \fnite precession with 0.19 GHz, \n\u0003\nywith the\nnew \u0000x\natshows a halt of spin precession. As presented in\nFig. 5(c), the time evolution of szfor both the experiment\n(diamond) and MC simulation (red solid) at x\u0019\u00008:6\u0016m\ncon\frms a simple exponential decay with no oscillatory\nbehavior. Similar results also hold for \u001be\u000b= 8:1\u0016m with\nBy= 0:45 T in Figs. 5(d){5(f), supporting enhancement\nof the anisotropic term. We also compare y-scan by MC\nsimulation for parameters \u001be\u000b= 5:6\u0016m andBx= 0:4 T\n[Fig. 5(g)]. Solid and dashed lines are calculated values of\n\n\u0003\nxwith new \u0000y\natand conventional \u0000\u0000x=2 =\u00000:99 GHz,\nrespectively, where \n\u0003\nxis enhanced for negative yvalues\nfor new \u0000y\nat. This point is con\frmed further in Fig. 5(h)\nby plotting the time evolution of szaty=\u000018:2\u0016m.\nThe spin precession frequency obtained from MC simula-\ntion aty=\u000018:2\u0016m [grey diamond in Fig. 5(g)] shows\ngood agreement with our new model. The quantitative\n4agreement shown above reveals clearly that precession by\nthe relaxation anisotropy is not a material constant pa-\nrameter, but is rather controlled by di\u000busive spin motion.\nIn conclusion, we have experimentally observed an en-\nhancement of the spin relaxation anisotropy by di\u000busive\nspin motion. We measured the precession frequency in an\nexternal magnetic \feld by changing the relative distance\nbetween excited pump and detected probe positions in\na spatiotemporal Kerr rotation microscope. Because of\nvarious electron trajectories for electrons travelling from\nthe pump to the probe position, the spin precession axis\nis tilted substantially from the external magnetic \feld di-\nrection when the di\u000busion-induced SO \feld nearly com-\npensates the magnitude of the external magnetic \feld.\nSuch a SO-coupled spin-di\u000busive motion controls the re-\nlaxation anisotropy. It is detected as a nonlinear pre-\ncession frequency modulation. Whereas the relaxation\nanisotropy is regarded as a constant parameter for sta-\ntionary electrons, it becomes controllable for moving elec-\ntrons. This e\u000bect also points out a threshold to start spin\nprecession at a certain velocity. Because this e\u000bect is not\nlimited only to di\u000busive motion, but also can be con-\ntrolled by drift and ballistic transport, our \fndings link\nthe e\u000bect of the precise control of spin states to future\nspintronics and quantum information technology.\nWe acknowledge \fnancial support from the Japanese\nMinistry of Education, Culture, Sports, Science, and\nTechnology (MEXT) Grant-in-Aid for Scienti\fc Research\n(Nos. 15H02099, 15H05854, 25220604, and 15H05699),\nEPSRC-JSPS Core-to-Core program, and the Swiss Na-\ntional Science Foundation through the National Center of\nCompetence in Research (NCCR) QSIT. D.I. thanks the\nGraduate Program of Spintronics at Tohoku University,\nJapan, for \fnancial support.\n[1] S. A. Wolf, D. D. 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